Transmutations, Singular and Fractional Differential Equations with Applications to Mathematical Physics (Mathematics in Science and Engineering) [1 ed.] 0128197811, 9780128197813

Table of contents :
Contents
Acknowledgments and thanks
Introduction
1 Basic definitions and propositions
1.1 Special functions
1.1.1 Gamma function, beta function, Pochhammer symbol, and error function
1.1.2 Bessel functions
1.1.3 Hypergeometric type functions
1.1.4 Polynomials
1.2 Functional spaces
1.2.1 Orthant Rn+, Cevm, Sev, and Lpγ spaces
1.2.2 Weighted measure, space L∞γ, and definition of weak (p,q)γ type operators
1.2.3 Space of weighted generalized functions Sev', absolutely continuous functions, and unitary operators
1.2.4 Mixed case
1.3 Integral transforms and Lizorkin-Samko space
1.3.1 One-dimensional integral transforms with Bessel functions in the kernels and Mellin transform
1.3.2 Properties of composition of integral transforms with Bessel functions in the kernel
1.3.3 Multi-dimensional integral transforms
1.4 Basic facts and formulas
1.4.1 Kipriyanov's classification of second order linear partial differential equations
1.4.2 Divergence theorem and Green's second identity for B-elliptic and B-hyperbolic operators
1.4.3 Tricomi equation
1.4.4 Abstract Euler-Poisson-Darboux equation
2 Basics of fractional calculus and fractional order differential equations
2.1 Short history of fractional calculus and fractional order differential equations
2.1.1 One-dimensional fractional derivatives and integrals
2.1.2 Fractional derivatives in mechanics
2.1.3 Fractional powers of multi-dimensional operators
2.1.4 Differential equations of fractional order
2.2 Standard fractional order integro-differential operators
2.2.1 Riemann-Liouville fractional integrals and derivatives on a segment
2.2.2 Riemann-Liouville fractional integrals and derivatives on a semiaxis
2.2.3 Gerasimov-Caputo fractional derivatives
2.2.4 Dzrbashian-Nersesyan fractional operators and sequential order fractional operators
2.3 Some more fractional order integro-differential operators
2.3.1 The Erdélyi-Kober operators
2.3.2 Fractional integrals and fractional derivatives of a function with respect to another function
2.3.3 Averaged or distributed order fractional operators
2.3.4 Saigo, Love, and other fractional operators with special function kernels
2.4 Integral transforms and basic differential equations of fractional order
2.4.1 Integral transforms of fractional integrals and derivatives
2.4.1.1 Laplace transform of Riemann-Liouville fractional integrals and derivatives on semiaxes
2.4.1.2 Mellin transform of Riemann-Liouville fractional integrals and derivatives on semiaxes
2.4.1.3 Laplace transform of Gerasimov-Caputo fractional derivatives on semiaxes
2.4.2 Laplace transform method for the homogeneous equations with constant coefficients with the left-sided Riemann-Liouville fractional derivatives of the order α on a semiaxis (0,∞)
2.4.3 Laplace transform method for homogeneous equations with constant coefficients with the left-sided Gerasimov-Caputo fractional derivatives of the order α on a semiaxis [0,∞)
2.4.4 Mellin integral transform and nonhomogeneous linear differential equations of fractional order
3 Essentials of transmutations
3.1 Definition of the transmutation operator, some examples of classical transmutations
3.1.1 Introduction to transmutation theory
3.1.2 Some examples of classical transmutations
3.2 Transmutations for Sturm-Liouville operator
3.2.1 Description of the problem and terminology
3.2.2 Transmutations in the form of the second kind Fredholm operators
3.2.3 Transmutations in the form of the second kind Volterra operators
3.2.4 Transmutations in the form of the first kind Volterra operators
3.3 Transmutations for different potentials
3.3.1 Kernel of transmutation intertwining operators of the Sturm-Liouville type
3.3.2 Cases when potential q(x) is an exponential function
3.3.3 Cases when potential q(x) is constant
3.3.4 Estimates of kernels and point formulas for estimating the error for calculating transmutation operators
3.4 Transmutations for singular Bessel operator
3.4.1 One-dimensional Poisson operator
3.4.2 Multi-dimensional Poisson operator
3.4.3 Generalized translation
3.4.4 Weighted spherical mean
4 Weighted generalized functions generated by quadratic forms
4.1 The weighted generalized function associated with a positive quadratic form and concentrated on a part of a cone
4.1.1 B-ultrahyperbolic operator
4.1.2 Weighted generalized function associated with a positive quadratic form
4.1.3 Weighted generalized function δγ(P)
4.2 Weighted generalized functions realized by the degrees of quadratic forms
4.2.1 Weighted generalized functions Pγ,±λ
4.2.2 The weighted generalized function Pλγ and (P±i 0)γλ associated with a quadratic form with complex coefficients
4.3 Other weighted generalized functions associated with a quadratic form
4.3.1 Functions (w2-|x|2)+,γλ and (c2+P±i0)λγ
4.3.2 General weighted generalized functions connected with quadratic form
4.4 Hankel transform of weighted generalized functions generated by the quadratic form
4.4.1 Hankel transform of rλγ
4.4.2 Hankel transforms of functions Pλγ, (P±i0)λγ, and Pλγ,±
4.4.3 Hankel transforms of functions (w2-|x|2)+,γλ and (c2+P±i0)λγ
5 Buschman-Erdélyi integral and transmutation operators
5.1 Buschman-Erdélyi transmutations of the first kind
5.1.1 Sonine-Poisson-Delsarte transmutations
5.1.2 Definition and main properties of Buschman-Erdélyi transmutations of the first kind
5.1.3 Factorizations of the first kind Buschman-Erdélyi operators and the Mellin transform
5.2 Buschman-Erdélyi transmutations of the second and third kind
5.2.1 Second kind Buschman-Erdélyi transmutation operators
5.2.2 Sonine-Katrakhov and Poisson-Katrakhov transmutations
5.2.3 Buschman-Erdélyi transmutations of the third kind with arbitrary weight function
5.2.4 Some applications of Buschman-Erdélyi transmutations
5.3 Multi-dimensional integral transforms of Buschman-Erdélyi type with Legendre functions in kernels
5.3.1 Basic definitions
5.3.2 The n-dimensional Mellin transform and its properties
5.3.3 Lν,2-theory and the inversion formulas for the modified H-transform
5.3.4 Inversion of H1σ,κ
5.4 Representations in the form of modified H-transform
5.4.1 Mellin transform of auxiliary functions K1( x) and K2( x)
5.4.2 Mellin transform of Pγδ,1( x) and Pγδ,2( x)
5.4.3 Lν,2-theory of the transforms Pγδ,kf (k=1,2)
5.4.4 Inversion formulas for transforms Pγδ,kf (k=1,2)
6 Integral transforms composition method for transmutations
6.1 Basic ideas and definitions of the integral transforms composition method for the study of transmutations
6.1.1 Background of ITCM
6.1.2 What is ITCM and how to use it?
6.2 Application of the ITCM to derive transmutations connected with the Bessel operator
6.2.1 Index shift for the Bessel operator
6.2.2 Poisson and "descent" operators, negative fractional power of the Bessel operator
6.2.3 ITCM for generalized translation and the weighted spherical mean
6.2.4 Integral representations of transmutations for perturbed differential Bessel operators
6.3 Connection formulas for solutions to singular differential equations via the ITCM
6.3.1 Application of transmutations for finding general solutions to Euler-Poisson-Darboux type equations
6.3.2 Application of transmutations for finding solutions to general Euler-Poisson-Darboux type equations
6.3.3 Application of transmutations for finding general solutions to singular Cauchy problems
7 Differential equations with Bessel operator
7.1 General Euler-Poisson-Darboux equation
7.1.1 The first Cauchy problem for the general Euler-Poisson-Darboux equation
7.1.2 The second Cauchy problem for the general Euler-Poisson-Darboux equation
7.1.3 The singular Cauchy problem for the generalized homogeneous Euler-Poisson-Darboux equation
7.1.4 Examples
7.2 Hyperbolic and ultrahyperbolic equations with Bessel operator in spaces of weighted distributions
7.2.1 The generalized Euler-Poisson-Darboux equation and the singular Klein-Gordon equation
7.2.2 Iterated ultrahyperbolic equation with Bessel operator
7.2.3 Generalization of the Asgeirsson theorem
7.2.4 Descent method for the general Euler-Poisson-Darboux equation
7.3 Elliptic equations with Bessel operator
7.3.1 Weighted homogeneous distributions
7.3.2 Extension of the weighted homogeneous distributions
7.3.3 Weighted fundamental solution of the Laplace-Bessel operator
7.3.4 The Dirichlet problem for an elliptic singular equation
8 Applications of transmutations to different problems
8.1 Inverse problems and applications of Buschman-Erdélyi transmutations
8.1.1 Inverse problems
8.1.2 Copson lemma
8.1.3 Norm estimates and embedding theorems in Kipriyanov spaces
8.1.4 Other applications of Buschman-Erdélyi operators
8.2 Applications of the transmutation method to estimates of the solutions for differential equations with variable coefficients and the problem of E. M. Landis
8.2.1 Applications of the transmutations method to the perturbed Bessel equation with a potential
8.2.2 The solution of the basic integral equation for the kernel of the transmutation operator
8.2.3 Application of the method of transmutation operators to the problem of E. M. Landis
8.2.4 The solution to the E. M. Landis problem belongs to T (λ+ε)
8.3 Applications of transmutations to perturbed Bessel and one-dimensional Schrödinger equations
8.3.1 Formulation of the problem
8.3.2 Solution of the basic integral equation for the kernel of a transmutation operator
8.3.3 Estimates for the case of a power singular at zero potential
8.3.4 Asymptotically exact inequalities for Legendre functions
8.4 Iterated spherical mean in the computed tomography problem
8.4.1 Iterated weighted spherical mean and its properties
8.4.2 Application of identity for an iterated spherical mean to the task of computed tomography
9 Fractional powers of Bessel operators
9.1 Fractional Bessel integrals and derivatives on a segment
9.1.1 Definitions
9.1.2 Basic properties of fractional Bessel integrals on a segment
9.1.3 Fractional Bessel integrals and derivatives on a segment of elementary and special functions
9.1.4 Fractional Bessel derivatives on a segment as inverse to integrals
9.2 Fractional Bessel integral and derivatives on a semiaxis
9.2.1 Definitions
9.2.2 Basic properties of fractional Bessel integrals on a semiaxis
9.2.3 Factorization
9.2.4 Fractional Bessel integrals on semiaxes of elementary and special functions
9.3 Integral transforms of fractional powers of Bessel operators
9.3.1 The Mellin transform
9.3.2 The Hankel transform
9.3.3 The Meijer transform
9.3.4 Generalized Whittaker transform
9.4 Further properties of fractional powers of Bessel operators
9.4.1 Resolvent for the right-sided fractional Bessel integral on a semiaxis
9.4.2 The generalized Taylor formula with powers of Bessel operators
10 B-potentials theory
10.1 Definitions of hyperbolic B-potentials, absolute convergence, and boundedness
10.1.1 Negative fractional powers of the hyperbolic expression with Bessel operators
10.1.2 Absolute convergence and boundedness
10.1.3 Semigroup properties
10.1.4 Examples
10.2 Method of approximative inverse operators applied to inversion of the hyperbolic B-potentials
10.2.1 Method of approximative inverse operators
10.2.2 General Poisson kernel
10.2.3 Representation of the kernel gαε,δ
10.2.4 Inversion of the hyperbolic B-potentials
10.3 Mixed hyperbolic Riesz B-potentials
10.3.1 Definition and basic properties of the mixed hyperbolic Riesz B-potential
10.3.2 Homogenizing kernel
10.4 Inversion of the mixed hyperbolic Riesz B-potentials
10.4.1 Auxiliary lemma
10.4.2 Property of Lrγ-boundedness of the function gα,γ,ε
10.4.3 Inversion theorems
11 Fractional differential equations with singular coefficients
11.1 Meijer transform method for the solution to homogeneous fractional equations with left-sided fractional Bessel derivatives on semiaxes of Gerasimov-Caputo type
11.1.1 General case
11.1.2 Particular cases and examples
11.2 Mellin transform method
11.2.1 Ordinary linear nonhomogeneous differential equations of fractional order on semiaxes
11.2.2 Example
11.3 Hyperbolic Riesz B-potential and its connection with the solution of an iterated B-hyperbolic equation
11.3.1 General algorithm
11.3.2 Definition
11.3.3 Variables in Lorentz space
11.3.4 Identity operator
11.4 The Riesz potential method for solving nonhomogeneous equations of Euler-Poisson-Darboux type
11.4.1 General nonhomogeneous iterated Euler-Poisson-Darboux equation
11.4.2 Mixed truncated hyperbolic Riesz B-potential
11.4.3 Nonhomogeneous general Euler-Poisson-Darboux equation with homogeneous conditions
11.4.4 Examples
12 Conclusion
References
Index

Citation preview

Transmutations, Singular and Fractional Differential Equations With Applications to Mathematical Physics

Mathematics in Science and Engineering

Transmutations, Singular and Fractional Differential Equations With Applications to Mathematical Physics Elina Shishkina Sergei Sitnik Series Editor

Goong Chen

Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1650, San Diego, CA 92101, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom Copyright © 2020 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-819781-3 For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Candice Janco Editorial Project Manager: Aleksandra Packowska Production Project Manager: Joy Christel Neumarin Honest Thangia Designer: Mark Rogers Typeset by VTeX

Contents

Acknowledgments and thanks Introduction 1

2

Basic definitions and propositions 1.1 Special functions 1.1.1 Gamma function, beta function, Pochhammer symbol, and error function 1.1.2 Bessel functions 1.1.3 Hypergeometric type functions 1.1.4 Polynomials 1.2 Functional spaces m , S , and Lγ spaces 1.2.1 Orthant Rn+ , Cev ev p γ 1.2.2 Weighted measure, space L∞ , and definition of weak (p, q)γ type operators  , absolutely 1.2.3 Space of weighted generalized functions Sev continuous functions, and unitary operators 1.2.4 Mixed case 1.3 Integral transforms and Lizorkin–Samko space 1.3.1 One-dimensional integral transforms with Bessel functions in the kernels and Mellin transform 1.3.2 Properties of composition of integral transforms with Bessel functions in the kernel 1.3.3 Multi-dimensional integral transforms 1.4 Basic facts and formulas 1.4.1 Kipriyanov’s classification of second order linear partial differential equations 1.4.2 Divergence theorem and Green’s second identity for B-elliptic and B-hyperbolic operators 1.4.3 Tricomi equation 1.4.4 Abstract Euler–Poisson–Darboux equation Basics of fractional calculus and fractional order differential equations 2.1 Short history of fractional calculus and fractional order differential equations 2.1.1 One-dimensional fractional derivatives and integrals 2.1.2 Fractional derivatives in mechanics 2.1.3 Fractional powers of multi-dimensional operators

xiii xv 1 1 1 3 6 12 13 13 14 18 20 22 22 26 37 39 39 42 49 50

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2.1.4 Differential equations of fractional order 2.2 Standard fractional order integro-differential operators 2.2.1 Riemann–Liouville fractional integrals and derivatives on a segment 2.2.2 Riemann–Liouville fractional integrals and derivatives on a semiaxis 2.2.3 Gerasimov–Caputo fractional derivatives 2.2.4 Dzrbashian–Nersesyan fractional operators and sequential order fractional operators 2.3 Some more fractional order integro-differential operators 2.3.1 The Erdélyi–Kober operators 2.3.2 Fractional integrals and fractional derivatives of a function with respect to another function 2.3.3 Averaged or distributed order fractional operators 2.3.4 Saigo, Love, and other fractional operators with special function kernels 2.4 Integral transforms and basic differential equations of fractional order 2.4.1 Integral transforms of fractional integrals and derivatives 2.4.2 Laplace transform method for the homogeneous equations with constant coefficients with the left-sided Riemann–Liouville fractional derivatives of the order α on a semiaxis (0, ∞) 2.4.3 Laplace transform method for homogeneous equations with constant coefficients with the left-sided Gerasimov–Caputo fractional derivatives of the order α on a semiaxis [0, ∞) 2.4.4 Mellin integral transform and nonhomogeneous linear differential equations of fractional order 3

Essentials of transmutations 3.1 Definition of the transmutation operator, some examples of classical transmutations 3.1.1 Introduction to transmutation theory 3.1.2 Some examples of classical transmutations 3.2 Transmutations for Sturm–Liouville operator 3.2.1 Description of the problem and terminology 3.2.2 Transmutations in the form of the second kind Fredholm operators 3.2.3 Transmutations in the form of the second kind Volterra operators 3.2.4 Transmutations in the form of the first kind Volterra operators 3.3 Transmutations for different potentials 3.3.1 Kernel of transmutation intertwining operators of the Sturm–Liouville type

63 65 65 69 71 73 74 74 74 75 76 76 76

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3.3.2 Cases when potential q(x) is an exponential function 3.3.3 Cases when potential q(x) is constant 3.3.4 Estimates of kernels and point formulas for estimating the error for calculating transmutation operators 3.4 Transmutations for singular Bessel operator 3.4.1 One-dimensional Poisson operator 3.4.2 Multi-dimensional Poisson operator 3.4.3 Generalized translation 3.4.4 Weighted spherical mean 4

5

Weighted generalized functions generated by quadratic forms 4.1 The weighted generalized function associated with a positive quadratic form and concentrated on a part of a cone 4.1.1 B-ultrahyperbolic operator 4.1.2 Weighted generalized function associated with a positive quadratic form 4.1.3 Weighted generalized function δγ (P ) 4.2 Weighted generalized functions realized by the degrees of quadratic forms 4.2.1 Weighted generalized functions Pγλ,± 4.2.2 The weighted generalized function Pγλ and (P ± i0)λγ associated with a quadratic form with complex coefficients 4.3 Other weighted generalized functions associated with a quadratic form 4.3.1 Functions (w 2 − |x|2 )λ+,γ and (c2 + P ± i0)λγ 4.3.2 General weighted generalized functions connected with quadratic form 4.4 Hankel transform of weighted generalized functions generated by the quadratic form 4.4.1 Hankel transform of rγλ 4.4.2 Hankel transforms of functions Pγλ , (P ± i0)λγ , and Pγλ,± 4.4.3 Hankel transforms of functions (w 2 − |x|2 )λ+,γ and (c2 + P ± i0)λγ Buschman–Erdélyi integral and transmutation operators 5.1 Buschman–Erdélyi transmutations of the first kind 5.1.1 Sonine–Poisson–Delsarte transmutations 5.1.2 Definition and main properties of Buschman–Erdélyi transmutations of the first kind 5.1.3 Factorizations of the first kind Buschman–Erdélyi operators and the Mellin transform 5.2 Buschman–Erdélyi transmutations of the second and third kind 5.2.1 Second kind Buschman–Erdélyi transmutation operators 5.2.2 Sonine–Katrakhov and Poisson–Katrakhov transmutations

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5.2.3 Buschman–Erdélyi transmutations of the third kind with arbitrary weight function 5.2.4 Some applications of Buschman–Erdélyi transmutations 5.3 Multi-dimensional integral transforms of Buschman–Erdélyi type with Legendre functions in kernels 5.3.1 Basic definitions 5.3.2 The n-dimensional Mellin transform and its properties 5.3.3 Lν,2 -theory and the inversion formulas for the modified H -transform 1 5.3.4 Inversion of Hσ,κ 5.4 Representations in the form of modified H -transform 5.4.1 Mellin transform of auxiliary functions K1 (x) and K2 (x) γ γ 5.4.2 Mellin transform of Pδ,1 (x) and Pδ,2 (x) γ 5.4.3 Lν, 2 -theory of the transforms Pδ,k f (k = 1, 2) γ 5.4.4 Inversion formulas for transforms Pδ,k f (k = 1, 2) 6

7

Integral transforms composition method for transmutations 6.1 Basic ideas and definitions of the integral transforms composition method for the study of transmutations 6.1.1 Background of ITCM 6.1.2 What is ITCM and how to use it? 6.2 Application of the ITCM to derive transmutations connected with the Bessel operator 6.2.1 Index shift for the Bessel operator 6.2.2 Poisson and “descent” operators, negative fractional power of the Bessel operator 6.2.3 ITCM for generalized translation and the weighted spherical mean 6.2.4 Integral representations of transmutations for perturbed differential Bessel operators 6.3 Connection formulas for solutions to singular differential equations via the ITCM 6.3.1 Application of transmutations for finding general solutions to Euler–Poisson–Darboux type equations 6.3.2 Application of transmutations for finding solutions to general Euler–Poisson–Darboux type equations 6.3.3 Application of transmutations for finding general solutions to singular Cauchy problems Differential equations with Bessel operator 7.1 General Euler–Poisson–Darboux equation 7.1.1 The first Cauchy problem for the general Euler–Poisson–Darboux equation 7.1.2 The second Cauchy problem for the general Euler–Poisson–Darboux equation

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7.1.3 The singular Cauchy problem for the generalized homogeneous Euler–Poisson–Darboux equation 7.1.4 Examples 7.2 Hyperbolic and ultrahyperbolic equations with Bessel operator in spaces of weighted distributions 7.2.1 The generalized Euler–Poisson–Darboux equation and the singular Klein–Gordon equation 7.2.2 Iterated ultrahyperbolic equation with Bessel operator 7.2.3 Generalization of the Asgeirsson theorem 7.2.4 Descent method for the general Euler–Poisson–Darboux equation 7.3 Elliptic equations with Bessel operator 7.3.1 Weighted homogeneous distributions 7.3.2 Extension of the weighted homogeneous distributions 7.3.3 Weighted fundamental solution of the Laplace–Bessel operator 7.3.4 The Dirichlet problem for an elliptic singular equation 8

Applications of transmutations to different problems 8.1 Inverse problems and applications of Buschman–Erdélyi transmutations 8.1.1 Inverse problems 8.1.2 Copson lemma 8.1.3 Norm estimates and embedding theorems in Kipriyanov spaces 8.1.4 Other applications of Buschman–Erdélyi operators 8.2 Applications of the transmutation method to estimates of the solutions for differential equations with variable coefficients and the problem of E. M. Landis 8.2.1 Applications of the transmutations method to the perturbed Bessel equation with a potential 8.2.2 The solution of the basic integral equation for the kernel of the transmutation operator 8.2.3 Application of the method of transmutation operators to the problem of E. M. Landis 8.2.4 The solution to the E. M. Landis problem belongs to T (λ + ε) 8.3 Applications of transmutations to perturbed Bessel and one-dimensional Schrödinger equations 8.3.1 Formulation of the problem 8.3.2 Solution of the basic integral equation for the kernel of a transmutation operator 8.3.3 Estimates for the case of a power singular at zero potential 8.3.4 Asymptotically exact inequalities for Legendre functions 8.4 Iterated spherical mean in the computed tomography problem

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8.4.1 Iterated weighted spherical mean and its properties 8.4.2 Application of identity for an iterated spherical mean to the task of computed tomography 9

10

Fractional powers of Bessel operators 9.1 Fractional Bessel integrals and derivatives on a segment 9.1.1 Definitions 9.1.2 Basic properties of fractional Bessel integrals on a segment 9.1.3 Fractional Bessel integrals and derivatives on a segment of elementary and special functions 9.1.4 Fractional Bessel derivatives on a segment as inverse to integrals 9.2 Fractional Bessel integral and derivatives on a semiaxis 9.2.1 Definitions 9.2.2 Basic properties of fractional Bessel integrals on a semiaxis 9.2.3 Factorization 9.2.4 Fractional Bessel integrals on semiaxes of elementary and special functions 9.3 Integral transforms of fractional powers of Bessel operators 9.3.1 The Mellin transform 9.3.2 The Hankel transform 9.3.3 The Meijer transform 9.3.4 Generalized Whittaker transform 9.4 Further properties of fractional powers of Bessel operators 9.4.1 Resolvent for the right-sided fractional Bessel integral on a semiaxis 9.4.2 The generalized Taylor formula with powers of Bessel operators B-potentials theory 10.1 Definitions of hyperbolic B-potentials, absolute convergence, and boundedness 10.1.1 Negative fractional powers of the hyperbolic expression with Bessel operators 10.1.2 Absolute convergence and boundedness 10.1.3 Semigroup properties 10.1.4 Examples 10.2 Method of approximative inverse operators applied to inversion of the hyperbolic B-potentials 10.2.1 Method of approximative inverse operators 10.2.2 General Poisson kernel α 10.2.3 Representation of the kernel ∓ gε,δ 10.2.4 Inversion of the hyperbolic B-potentials 10.3 Mixed hyperbolic Riesz B-potentials

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10.3.1 Definition and basic properties of the mixed hyperbolic Riesz B-potential 10.3.2 Homogenizing kernel 10.4 Inversion of the mixed hyperbolic Riesz B-potentials 10.4.1 Auxiliary lemma γ 10.4.2 Property of Lr -boundedness of the function gα,γ ,ε 10.4.3 Inversion theorems 11

12

Fractional differential equations with singular coefficients 11.1 Meijer transform method for the solution to homogeneous fractional equations with left-sided fractional Bessel derivatives on semiaxes of Gerasimov–Caputo type 11.1.1 General case 11.1.2 Particular cases and examples 11.2 Mellin transform method 11.2.1 Ordinary linear nonhomogeneous differential equations of fractional order on semiaxes 11.2.2 Example 11.3 Hyperbolic Riesz B-potential and its connection with the solution of an iterated B-hyperbolic equation 11.3.1 General algorithm 11.3.2 Definition 11.3.3 Variables in Lorentz space 11.3.4 Identity operator 11.4 The Riesz potential method for solving nonhomogeneous equations of Euler–Poisson–Darboux type 11.4.1 General nonhomogeneous iterated Euler–Poisson–Darboux equation 11.4.2 Mixed truncated hyperbolic Riesz B-potential 11.4.3 Nonhomogeneous general Euler–Poisson–Darboux equation with homogeneous conditions 11.4.4 Examples Conclusion

References Index

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483 483 488 491 491 492 498 498 499 500 502 512 512 514 519 520 525 527 553

Acknowledgments and thanks

There are many colleagues to whom we are indebted for their support and many kinds of help. Unfortunately some of our thanks are late now. First of all we are grateful to our teachers – I. A. Kipriyanov, V. V. Katrakhov, and L. N. Lyakhov. We received invaluable help, friendly support and useful discussions from A. V. Borovskikh (Moscow, Russia), K. Brahim (Tunis, Tunisia), V. I. Burenkov (Moscow, Russia), R. W. Carroll (Delaware, USA), A. Dzhrbashian (Colombia and Armenia), I. H. Dimovski (Sofia, Bulgaria), V. E. Fedorov (Chelyabinsk, Russia), A. Fitouhi (Tunis, Tunisia), V. Gaidey (Kiev, Ukraine), A. V. Glushak (Belgorod, Russia), V. P. Glushko (Voronezh, Russia), M. L. Goldman (Moscow, Russia), S. Gorlov (Voronezh, Russia), V. Guliev (Baku, Azerbaijan), I. Jebabli (Tunis, Tunisia), S. T. Karimov (Fergana, Uzbekistan), D. B. Karp (Holon, Israel and Vladivostok, Russia), A. A. Kilbas (Minsk, Belarus), V. Kiryakova (Sofia, Bulgaria), V. V. Kravchenko (Querétaro, Mexico), A. Kufner (Praha, Czech Republic), A. G. Kusraev (Vladikavkaz, Russia), N. V. Kuznetsov (Vladivostok, Russia), V. P. Leksin (Suzdal, Russia), Yu. Luchko (Berlin, Germany), G. V. Lyahovetskii (Vladivostok, Russia), K. Mehrez (Tunis, Tunisia), V. Z. Meshkov (Voronezh, Russia), A. B. Muravnik (Voronezh, Russia), A. M. Nakhushev (Nalchik, Russia), A. I. Nazarov (St. Petersburg, Russia), T. Poganj (Rijeka, Croatia), I. P. Polovinkin (Voronezh, Russia), A. V. Pskhu (Nalchik, Russia), L. S. Pulkina (Samara, Russia), E. V. Radkevich (Moscow, Russia), V. Radulescu (Craiova, Romania), O. A. Repin (Samara, Russia), B. Rubin (Louisiana, USA), S. Rutkauskas (Vilnius, Lithuania), M. V. Shitikova (Voronezh, Russia), O. V. Skoromnik (Polotsk, Belarus), A. P. Soldatov (Moscow and Belgorod, Russia), A. L. Skubachevskii (Moscow, Russia), N. A. Virchenko (Kiev, Ukraine), V. Volovich (Voronezh, Russia), A. K. Urinov (Fergana, Uzbekistan), S. N. Ushakov (Voronezh, Russia), N. I. Yurchuk (Minsk, Belarus), V. A. Yurko (Saratov, Russia), Ya. I. Zhitomirskii (Haifa, Israel). We are also thankful to participants of mathematical seminars at Voronezh State University (ongoing since 1990) and at Belgorod State University (ongoing since 2017). Many discussions, disputes, and considerations during these seminars were devoted to transmutation theory and its applications, and they have led to many ideas and results that have been incorporated into this book.

Introduction

Transmutation operators theory is one of the attempts to create a general approach to different mathematical problems. Let us start with the main definition. Definition 1. For a given pair of operators (A, B), an operator T is called a transmutation (or intertwining) operator if on elements of some functional spaces the following property is valid: T A = B T.

(1)

It is obvious that the notion of transmutation is a direct and far reaching generalization of the matrix similarity from linear algebra. But transmutations do not reduce to similar operators because intertwining operators often are not bounded in classical spaces and the inverse operator may not exist or not be bounded in the same space. As a consequence, spectra of intertwining operators are not the same as a rule. Moreover, transmutations may be unbounded. It is the case for the Darboux transformations which are defined for a pair of differential operators and are differential operators themselves; in this case all three operators are unbounded in classical spaces. But the theory of the Darboux transformations is included in transmutation theory too. Also, a pair of intertwining operators may not be differential ones. In transmutation theory there are problems for the following various types of operators: integral, integro-differential, difference-differential (e.g., the Dunkl operator), differential or integro-differential of infinite order (e.g., in connection with Schur’s lemma), general linear operators in functional spaces, and pseudodifferential and abstract differential operators. All classical integral transforms due to Definition 1 are also special cases of transmutations; they include the Fourier, Petzval (Laplace), Mellin, Hankel, Weierstrass, Kontorovich–Lebedev, Meijer, Stankovic, Obrechkoff, finite Grinberg, and other transforms. In quantum physics, in the study of the Schrödinger equation and inverse scattering theory, the underlying transmutations are called wave operators. The commuting operators are also a special class of transmutations. The most important class consists of operators commuting with derivatives. In this case transmutations as commutants are usually in the form of formal series or pseudodifferential or infinite order differential operators. Finding commutants is directly connected with finding all transmutations in the given functional space. For these problems works a theory of operator convolutions, including the Berg–Dimovski convolutions [89]. Also, more and more applications are developed that are connected with the transmutation theory for commuting differential operators; such problems are based on

xvi

Introduction

classical results of J. L. Burchnall and T. W. Chaundy. The transmutations are also connected with factorization problems for integral and differential operators. A special class of transmutations are the so-called Dirichlet-to-Neumann and Neumannto-Dirichlet operators, which link together solutions of the same equation but with different kinds of boundary conditions. How do transmutations usually work? Suppose we study properties for a rather complicated operator A. But suppose also that we know the corresponding properties for a more simple model operator B and transmutation (1) readily exists. Then we usually may copy results for the model operator B to corresponding ones for the more complicated operator A. This is the main idea of transmutations. Let us consider for example an equation Au = f . Then applying to it a transmutation with property (1), we consider a new equation Bv = g, with v = T u, g = Tf . So if we can solve the simpler equation Bv = g, then the initial one is also solved and has solution u = T −1 v. Of course, it is supposed that the inverse operator exists and its explicit form is known. This is a simple application of the transmutation technique for finding and proving formulas for solutions of ordinary and partial differential equations. The monographs [51,571] are completely devoted to transmutation theory and its applications (note also the author’s survey [532]). Moreover, essential parts of monographs [56,89,252,259], among others, include material on transmutations; the complete list of books which investigate some transmutational problems is now near of 100 items. We use the term “transmutation” due to [53]: “Such operators are often called transformation operators by the Russian school (Levitan, Naimark, Marchenko, etc.), but transformation seems a too broad term, and since some of the machinery seems ‘magical’ at times, we have followed Lions and Delsarte in using the word ‘transmutation’.” Now transmutation theory is a completely formed part of the mathematical world in which methods and ideas from different areas are used, i.e., differential and integral equations, functional analysis, function theory, complex analysis, special functions, and fractional integro-differentiation. In the history of transmutation theory, three main periods can be distinguished. In the first initial period, basic ideas and definitions were formulated. Their source was the theory of similarity of finite matrices (see [175,212,573]), some ideas on similarity of operators, and some results for the simplest differential equations. It is believed that the idea of transmutations in the operator formulation was expressed by Friedrichs [153]. But in fact, the method of transmutation operators for obtaining representations of solutions to differential equations was developed and first applied much earlier in the 19th century in the works of A. V. Letnikov. In addition, it was essentially the first real application of fractional integro-differentiation as transmutations to problems of differential equations [273,498]. The second period conventionally continued during 1940–1980. This can be called the classic period. Numerous results in the theory of transmutation operators and their applications were obtained. We list the main directions and results of this period. The methods of transmutations were successfully applied in the theory of inverse problems, defining the generalized Fourier transform, the spectral function, and

Introduction

xvii

solutions of the famous Gelfand–Levitan equation (see the works by Z. S. Agranovich and V. A. Marchenko [4,368–374] and B. M. Levitan [316–318,321,322, 325–327]). In scattering theory, the no less famous Marchenko equation was presented (see B. M. Levitan [316–318,321,322,325–327], V. A. Marchenko [4,373,374], and L. D. Fadeev [131,132]). For both classes of inverse problems, transmutation operators are the main tool, since the listed classical equations are written out for the kernels of the transmutation operators, and the values of the kernels on the diagonal reconstruct unknown potentials in the inverse problem from the spectral function (see [31,68,308,309,417,420,421,497]). For the Sturm–Liouville operators, classical transmutations on the interval were constructed by A. Ya. Povzner [454] and on the half-axis by B. Ya. Levin [313]. In spectral theory, well-known trace formulas and the asymptotics of the spectral function were obtained by V. A. Marchenko [373,374] and B. M. Levitan [316–318,321, 322,325–327]. Estimates of the kernels of transmutation operators responsible for the stability of inverse problems and scattering problems were given by V. A. Marchenko [4,373,374]. Estimates of Jost’s solutions in quantum scattering theory were obtained by Z. S. Agranovich and V. A. Marchenko [4,373,374], B. M. Levitan [316–318,321, 322,325–327], V. V. Stashevskaya [557,558], and A. S. Sokhin [546–549]. As a result of applying the transmutations, we can say that the theory of Sturm–Liouville operators with a variable coefficient was trivialized to the level of the simplest equation with trigonometric or exponential solutions. The Dirac system and other matrix systems of differential equations were studied by B. M. Levitan and I. S. Sargsyan [326]. The theory of generalized analytic functions was developed. Such theory can be interpreted as a part of the transmutation operators theory that intertwines the unperturbed and perturbed Cauchy–Riemann equations (see L. Bers [26,27], S. Bergman [25], I. N. Vekua [579,582], B. Boyarsky [36], and G. N. Polozhy [450–452]). It has applications to mechanics problems and the theory of elasticity and gas dynamics. Based on the methods of transmutation operators, a new part of harmonic analysis was created. This part contains various modifications of generalized translation operators and generalized convolutions (see J. Delsarte [83,84], I. I. Zhitomirsky [609,610], and B. M. Levitan [321,327]). A deep connection between transmutation operators and Paley–Wiener type theorems was established (see V. V. Stashevskaya [557,558], A. I. Akhiezer [5], H. Chablis [62–65], and H. Trimesh [569,570]). The theory of transmutation operators allowed us to give a new classification of special functions and integral operators with special functions in kernels (see R. Carroll [51–53] and T. Corvinder [273]). Moreover, to find the transmutation operator kernels, the existence and explicit form of the Green or Riemann functions for various classes of differential equations are used [545,588, 589], stimulating the finding of these functions for various problems. In the theory of nonlinear differential equations, the Lax method was developed. This method uses transmutation operators to prove the existence of and construct solutions to nonlinear differential equations (see [1,54,605,617]). Darboux transformation has also been widely used as transformation operator in the case when both the intertwining and intertwined operators are differential [366]. For a connection between Darboux transformation theories and transmutation operators, see [11]. In quantum

xviii

Introduction

physics, when considering the Schrödinger equation and problems of theory scattering, a special class of transmutation operators, so-called wave operators, were studied. General scattering problems and inverse problems were considered from the point of view of transmutations in [131,132,375]. In [213] the wave operators were constructed for problems of scattering theory with the Stark potential. Unfortunately this paper by V. P. Kachalov and Ya. V. Kurylyova (1989) is practically forgotten. For example, in the article [324] (1995), B. M. Levitan formulates the problem of constructing the corresponding transmutation operator as unsolved. In the theory of transmutation operators, restrictions related to the order of the differential operator were discovered. It was shown that for differential operators of orders higher than the third, classical Volterra operators exist only in the case of analytic coefficients (see V. I. Matsaev [365], L. A. Sakhnovich [488–490], and M. M. Malamud [356–360]). In the general case, transmutations have a more complicated structure that requires access to the complex plane even for constructing real solutions (see A. F. Leontiev [328], Yu. N. Valitsky [578], I. G. Khachatryan [254, 255], M. M. Malamud [356–360], and A. P. Khromov [256]). At the same time, in the spaces of analytic functions, the equivalence of differential operators of the same order was proved and a number of problems were studied (see D. K. Fage [133–139], B. A. Marchenko [370–372], Yu. F. Korobeinik [269,270], and M. K. Fishman [144]). Operator theory was applied to the theory of solubility for the well-known Bianchi equation (see D. K. Fage [139]). Transmutation theory is strongly connected with many applications in different fields of mathematics. Transmutation operators are applied in inverse problems via the generalized Fourier transform, the spectral function, and the famous Levitan equation; in scattering theory, the Marchenko equation is formulated in terms of transmutations; in spectral theory, transmutations help to prove trace formulas and asymptotics for the spectral function; estimates for transmutational kernels control stability in inverse and scattering problems; for nonlinear equations via the Lax method, transmutations for Sturm–Lioville problems lead to proving existence and explicit formulas for solutions. Special kinds of transmutations are the generalized analytic functions, generalized translations and convolutions, and Darboux transformations. In the theory of partial differential equations, the transmutations work for proving explicit correspondence formulas among solutions of perturbed and nonperturbed equations, singular and degenerate equations, pseudodifferential operators, problems with essential singularities at inner or corner points, and estimates of solution decay for elliptic and ultraelliptic equations. In function theory, transmutations are applied to embedding theorems and generalizations of Hardy operators, Paley–Wiener theory, and generalizations of harmonic analysis based on generalized translations. Methods of transmutations are used in many applied problems: investigation of Jost solutions in scattering theory, inverse problems, Dirac and other matrix systems of differential equations, integral equations with special function kernels, probability theory and random processes, stochastic random equations, linear stochastic estimation, inverse problems of geophysics, and transsound gas dynamics. Also a number of applications of the transmutations to nonlinear equations is permanently increased.

Introduction

xix

In fact, the modern transmutation theory originated from two basic examples (see [532]). The first is the transmutation T for Sturm–Liouville problems with some potential q(x) and natural boundary conditions T (D 2 y(x) + q(x)y(x)) = D 2 (T y(x)), D 2 y(x) = y  (x). In this book we pay a lot of attention to equations with the singular Bessel differential operator Bγ , (Bγ )t =

∂ ∂2 γ ∂ 1 ∂ + = γ tγ , 2 t ∂t t ∂t ∂t ∂t

t > 0,

γ ∈ R,

and the second example of transmutation is a problem of intertwining the Bessel operator Bγ and the second derivative: T (Bγ )f = (D 2 )Tf. This class of transmutations includes the Sonine–Poisson–Delsarte and Buschman– Erdélyi operators and their generalizations. Such transmutations found many applications for a special class of partial differential equations with singular coefficients. It should be noted here that the first fundamental paper that began the study of degenerate and singular partial differential equations with variable coefficients is the article by M. V. Keldysh [235] (Problem E). Let u = u(x1 , ..., un ), f = f (x1 , ..., xn ). In accordance with I. A. Kipriyanov’s terminology, the equation n 

(Bγi )xi u = f

(2)

i=1

is classified as B-elliptic, the equation  ∂ u− (Bγi )xi u = f ∂x1 n

i=2

is classified as B-parabolic, and the equation (Bγ1 )x1 u −

n  (Bγi )xi u = f k=2

is classified as B-hyperbolic. Singular elliptic equations containing the Bessel operator are mathematical models of axial and multi-axial symmetry of the most diverse processes and phenomena of the world. Difficulties in the study of such equations are associated, inter alia, with the presence of singularities in the coefficients. The foundation of a systematic study of equations of B-elliptic type was laid in the works [592,594,596,598,599], where Weinstein’s theory of generalized axially symmetric potential (GASPT) was created.

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Introduction

In his papers Weinstein made a link between the B-elliptic equation and Tricomi equations and their fundamental solutions. I. A. Kipriyanov, together with V. V. Katrakhov (see [225,247,364]), studied boundary value problems for elliptic equations, with singularities of the type of essential singularities of analytic functions at isolated boundary points. L. N. Lyakhov studied the questions regarding fractional powers of the B-elliptic operator (elliptic operator with the Bessel operator instead of all or some second derivatives) and realized a solution to the B-elliptic equation and other questions (see [343–347,351,352]). In the paper of M. B. Kapilevic [217], the theory of degenerate elliptic differential equations of Bessel class were considered. The first who applied the Fourier–Bessel (Hankel) transformation to equations with the Bessel operator Bγ was Yakov Isaakovich Zhitomirsky. At the beginning of the 1950s, the rapid development of the theory of generalized functions by Gelfand and Shilov made it possible to establish the uniqueness class for the solution of the Cauchy problem for a system of linear partial differential evolution equations with constant coefficients that depend only on the order of the system. Further attempts to extend these results to equations with variable coefficients depending on spatial coordinates were made. In search of such equations, Ya. I. Zhitomirsky came to parabolic equations with the Bessel operator (B-parabolic equations). For such equations, he developed and used the theory of Fourier–Bessel (Hankel) integral transform in the corresponding function spaces to obtain results on uniqueness classes. These results were obtained in the thesis by Ya. I. Zhitomirsky (1954) and published in articles [609,610]. He found classes of the correct solvability of problems for parabolic systems with increasing coefficients [611]. Exact uniqueness classes were established for solving the Cauchy problem for linear evolutionary systems with variable coefficients, and a new boundary for the growth of coefficients was found that guarantees the stability of uniqueness classes in [612–615]. Subsequently, Ya. I. Zhitomirsky turned to questions of the existence and uniqueness of a solution to the Cauchy problem in terms of the general theory of differential equations. A. B. Muravnik studied parabolic differential equations and their generalizations. In [390,392,394,396,397,399–402], the Cauchy problem for parabolic differentialdifference equations are investigated. In [391,403], these investigations are extended to more general cases where second derivatives and translation operators act with respect to an arbitrary amount of nonspecial spatial variables, while Bessel operators and the corresponding generalized translation operators act with respect to an arbitrary amount of special spatial variables; thus, the considered functional-differential equations become differential-difference and integro-differential at the same time. In the monograph [9], the above investigations are summarized and developed. In [407–409,412], elliptic differential-difference equations in the half-plane are investigated. In [85,86,393,395,398,404–406,410,411], qualitative properties of solutions (including blow-up phenomena) are investigated for various quasilinear partial differential equations and inequalities (including singular and degenerate ones) with Kardar–Parisi–Zhang nonlinearities arising in numerous applications. In [386–389], specific properties of Fourier–Bessel transforms of measures are studied and applied to singular differential equations.

Introduction

xxi

The class of B-hyperbolic equations was first studied by Euler, Poisson, and Darboux and this study was continued by Weinstein [593,595,597,599]. In [593,595] the Cauchy problem for (7.1) is considered with k ∈ R, the first initial condition being nonzero and the second initial condition equaling zero. A solution of the Cauchy problem (7.1)–(7.2) in the classical sense was obtained in [595,596,599,602] and in the distributional sense in [38,56]. S. A. Tersenov in [564] solved the Cauchy problem for (7.1) in the general form where the first and the second conditions are nonzeros. Different problems for Eq. (7.1) with many applications to gas dynamics, hydrodynamics, mechanics, elasticity and plasticity, and so on, were also studied in [7,32,38,39,56,61,74,88,96,97,127,140,148–150,159,203,306,383,461,462, 539,550,552,553,559,564,581,602,616], and of course the above list of references is incomplete. Problems for operator-differential (abstract) equations including hyperbolics with Bessel operator appeared in the well-known monograph [56] A. V. Glushak studied abstract differential equations with a Bessel operator such as B-hyperbolic equations (see [182,185,188–190,192,193]). In particular, he investigated the stability of the property of uniform well-posedness of the Cauchy problem for the indicated equations and studied the solvability conditions for such problems with the Fredholm operator with derivatives. In the most detailed and complete way, equations with Bessel operators were studied by the Voronezh mathematician Kipriyanov and his disciples Ivanov, Ryzhkov, Katrakhov, Arhipov, Baidakov, Bogachov, Brodskii, Vinogradova, Zaitsev, Zasorin, Kagan, Katrakhova, Kipriyanova, Kononenko, Kluchantsev, Kulikov, Larin, Leizin, Lyakhov, Muravnik, Polovinkin, Sazonov, Sitnik, Shatskii, and Yaroslavtseva. The essence of Kipriyanov’s school results was published in [242]. For classes of equations with Bessel operators, Kipriyanov introduced special functional spaces which were named after him [243]. In this field, interesting results were investigated by Katrakhov and his disciples; now these problems are considered by Gadjiev, Guliev, Glushak, Lyakhov, and Shishkina with their coauthors and students. Abstract equations of the form (2) originating from the monograph [56] were considered by Egorov, Repnikov, Kononenko, Glushak, Shmulevich, and others. To describe the classes of solutions to the corresponding equations, I. A. Kipriyanov introduced and studied the functional spaces [243], later named after him (see the monographs by H. Tribel [568] and L. D. Kudryavtsev and S. M. Nikolsky [304] in which separate sections are devoted to Kipriyanov’s spaces). Transmutations are still one of the basic tools for equations with Bessel operators; they are applied in the construction of solutions and fundamental solutions, the study of singularities, and new boundary value and other problems. Transmutation operators for numerous generalizations of the Bessel operator were also considered. An important generalization of the Sonin–Poisson–Delsart operator is the transmutation operator for hyper-Bessel functions. The theory of such functions was originally laid down in the works of Kummer and Deleru. A complete study of hyper-Bessel functions, differential equations for them, and the corresponding transformation operators was exhaustively carried out by I. Dimovsky and his students [89,92,93]. The corresponding operators deservedly received in the literature the names of Sonin–Dimovsky and Poisson–Dimovsky operators; they were also stud-

xxii

Introduction

ied by V. Kiryakova [92,93,252,253]. The central role in the theory of hyper-Bessel functions, differential equations, and transmutation operators for them is played by the famous Obreshkov integral transform, introduced by the Bulgarian mathematician N. Obreshkov. This transformation, whose core is expressed in the general case in terms of the Meijer G-function, is a simultaneous generalization of the Laplace, Mellin, sine- and cosine-Fourier, Hankel, Meijer, and other classical integral transforms. Various forms of hyper-Bessel functions, differential equations, and transformation operators for them, as well as special cases of the Obreshkov transform, were subsequently rediscovered many times. Obreshkov’s transform was historically the first integral transform whose kernel is expressed in terms of the Meijer G-function but cannot be expressed in terms of one generalized hypergeometric function. Another important integral transformation, the Stankovic transform, was introduced by the Serbian mathematician B. Stankovic. The core of the Stankovic transform is expressed in terms of the Wright–Fox H-function, but is not expressed in terms of the simpler Meijer G-function. This transformation finds important applications in the study of fractional differential equations of the type of fractional diffusion [118,264, 265,459,460]. At the same time, similar theories were also constructed for some other model operators, such as [51–53,601] A=

1 d d v(x) , v(x) dx dx

(3)

v(x) = sin2ν+1 x, sh2ν+1 x, (ex − e−x )2ν+1 (ex + e−x )2μ+1 . The importance of A operators of the form (3) for the theory lies in the fact that, according to the famous Gelfand formula, they represent the radial part of the Laplace operator on symmetric spaces [162]. Here the Bessel operator is obtained by choosing v(x) = x γ in (3). Another model operator for which the transmutations are constructed is the Airy operator D 2 + x. In [213] its perturbed version related to the Stark effect from quantum mechanics was also considered. We studied the shift operators with respect to the spectral parameter Erdélyi–Vekua–Lowndes [337–339]. Papers from the 1990s to the present can be attributed to the third period of development of transmutation theory, which can be called the modern period. In this period, many important studies have been received and continue to appear (see, for example, reviews [55,234,375,528,532,533]). We list some of them. The development of the theory of generalized analytic functions was continued (see A. P. Soldatov [551], S. B. Klimentov [260–263], and V. V. Kravchenko [277]). Applications of transmutation operators to embedding function spaces and a generalization of Hardy operators were found [522,524,525]. Various constructions of a generalized translation and the generalized versions of harmonic analysis based on them were studied by A. D. Gadzhiev, V. Guliev, and A. Serbetci [174,206,207], S. S. Platonov [442–444], and L. N. Lyakhov and E. L. Shishkina [351–353], as well as in [510]. The use of transmutation operators and related methods in the theory of inverse problems and scattering theory continued [59,439,467,604]. For differential equations, the development of the Darboux method and its modifications continues (see V. B. Matveev

Introduction

xxiii

[366]), new classes of problems for solutions with significant features on the part of the boundary at internal or corner points are considered (see V. V. Katrakhov [225,227] and I. A. Kipriyanov [248–250]), and exact estimates of the rate of decrease of solutions of some elliptic and ultraelliptic equations have been obtained (see V. Z. Meshkov and S. M. Sitnik [379,380,520]). A separate topic is the use of operators in the study of various fractional integro-differentiation operators (see I. Dimovski and V. Kiryakova [89,93,252] and N. A. Virchenko [583,584]). Using the methods of transmutation operators, singular and degenerate boundary value problems, pseudodifferential operators (see V. V. Katrakhov [225,227], I. A. Kipriyanov [249,250], L. N. Lyakhov [343,344], and O. A. Repin [468]), and operator equations (see A. V. Glushak [184–186] and V. E. Fedorov [141,142]) were studied. The equations with Bessel operator and related questions were studied by A. V. Glushak [185,188,189], V. S. Guliev [208], L. N. Lyakhov, I. P. Polovinkin, and E. L. Shishkina [349,350,354], L. S. Pulkina [463], K. B. Sabitov [486], and V. V. Volchkov [587]. A separate class of problems is comprised of problems of the Dirichlet-to-Neumann and Neumann-to-Dirichlet types, under which the transformation operator acts on the boundary or initial conditions, preserving the differential expression; such tasks have found important applications in mechanics (see O. E. Yaremko [600]). Enough completed modifications of harmonic analysis for Bessel operators were constructed in the works of S. S. Platonov [442–444]. For a perturbed Bessel type operator with variable coefficients, see Kh. Triméche [571,572]. Recently, harmonic analysis has been actively created for differential-difference operators of the Dunkl type [115–117,479,480] based on appropriate generalizations of Sonin–Poisson– Darboux operators. The existence of transmutation operators corresponding to generalized translation also allows us to determine the generalized convolution and new algebraic and group structures, and consider various problems of approximating functions [69]. The ideas of M. K. Fage developed for the Bianchi equation in connection with the construction of transmutations for higher order differential equations found their continuation in the study of more general equations in the works of V. I. Zhegalov, A. N. Mironov, and E. A. Utkina [606,607]. In the theory of fractional order equations, papers that can be interpreted as considering the transmutations method for representing solutions of fractional order equations through solutions of integer order equations have appeared (see A. Pskhu [459,460], Ya. Pruss [458], and A. N. Kochubey [264,265]). Transmutation operators find applications in the theory of the Radon transform and mathematical tomography [162,416,485], as well as in the expansion of functions in various series in special functions [214]. In the works of V. A. Marchenko, the application of transmutations to quantum theory continued [375,376]. An important section of transmutation theory is a special class of Bushman–Erdélyi operators. This is a class of transmutation which, with a certain choice of parameters, is a generalization of the Sonin–Poisson–Dardoux operators and their conjugates, the fractional integro-differentiation operators of Riemann–Liouville and Erdélyi–Kober, and the Mehler–Fock integral transforms.

xxiv

Introduction

Integral operators of the indicated form with Legendre functions in kernels were first encountered in the works of E. T. Copson according to the Euler–Poisson– Darboux equation in the late 1950s [70–72]. The first detailed study of the solvability and reversibility of these operators was started in the 1960s in the works of R. Bushman [41,42] and A. Erdélyi [123–127]. Bushman–Erdélyi operators and their analogues were also studied by T. P. Higgins [167], Ta Li [561,562], E. R. Love [335,336], G. M. Habibullah, K. N. Srivastava, V. I. Smirniv, B. Rubin, N. A. Virchenko, and I. Fedotova [584], A. A. Kilbas and O. B. Skoromnik [240], and others. Moreover, the problems of solving integral equations with these operators and their factorization and inversion were studied. The results are partially mentioned in the monograph [494], although the case of the integration limits chosen by us is considered special there and is not considered, with the exception of one set of composition formulas (see also [234,522,537]). The term “Bushman–Erdélyi operators” is the most historically justified. It was introduced by S. M. Sitnik in [522,523], and later it was used by other authors. Earlier, in [494], the term “Bushman operators” was proposed by O. I. Marichev. The term “Chebyshev–Gegenbauer operators” [485] is also used in the theory of Radon transforms and mathematical tomography. The most complete study of the Bushman– Erdélyi operators, in our opinion, was carried out in the 1980s and 1990s [521–525] and then continued in [234,528,533–535]. It should be noted that the role of the Bushman–Erdélyi operators as transmutations before these works has never been noted or considered before. Recently, V. Kravchenko and S. Torba together with their colleagues have taken up the problem of efficient construction of the integral kernels of the transmutation operators. They tried to use the fact that the result of application of a transmutation operator to any nonnegative integer power of the independent variable can be obtained without knowledge of the operator itself. The powers of the independent variable are transmuted to so-called formal powers arising in the spectral parameter power series (SPPS) method (see [257,276,277,283]). This mapping property of the transmutation operator [45,286] leads to the possibility to transmute any polynomial into a corresponding generalized polynomial in terms of formal powers. V. Kravchenko and S. Torba proved a completeness property of so-called wave polynomials in a class of solutions of the wave equation and, as a corollary, the completeness of the system of the transmuted wave polynomials in a class of solutions of the hyperbolic equation satisfied by the transmutation kernel [287,288]. This result led to a method of approximation of the transmutation kernels and consequently of solutions of the Sturm–Liouville equation. It was observed in [287,288] that an approximate representation of the solution based on an approximation of the transmutation kernel admits estimates independent of the real part of the square root of the spectral parameter. This feature makes such representations especially valuable for solving spectral problems and allows one to compute large sets of eigendata. The next step was made in [281], where the authors managed to pass from approximation of the transmutation kernels to their exact representation in the form of functional series involving Legendre polynomials with easily computable expansion coefficients. As a corollary, new representations for solutions of the one-dimensional

Introduction

xxv

Schrödinger equation and later on for the Sturm–Liouville equation [290] in the form of the Neumann series of Bessel functions were obtained revealing the same attractive feature: They admit truncation estimates independent of the real part of the square root of the spectral parameter and allow one to compute in practice thousands of eigendata applying minimal computational efforts. It is worth mentioning that the mapping property of the transmutation operators allowing one to obtain the images of the powers of the independent variable without knowledge of the transmutation operator itself was used in several publications [45–48,258,274,275,282] for obtaining complete systems of solutions of partial differential equations and for using them when solving different boundary value problems. Besides regular Sturm–Liouville equations, singular perturbed Bessel equations were studied in [284,291,293]. In [278] and [292] the authors explored the possibility to expand the transmutation kernels into series in terms of other systems of orthogonal polynomials obtaining different series representations for solutions of the one-dimensional Schrödinger equation. In [280], V. Kravchenko found a way to obtain a functional series representation for the transmutation operator with a condition at infinity, the Levin transmutation operator arising in the Gelfand–Levitan–Marchenko scattering theory. In [82] this result was developed and led to an attractive representation of the Jost solutions and as a consequence to an efficient method of practical solution of spectral problems on infinite intervals, allowing one to compute spectral (or scattering) data corresponding not only to the discrete part of the spectrum but also to its continuous part, a computationally challenging problem. In [279], V. Kravchenko discovered an application of the Fourier–Legendre series representation for the transmutation kernel from [281] to the solution of the classical inverse Sturm–Liouville problem on a finite interval. Their idea is based on the observation that the potential can be recovered from the very first coefficient of the Fourier–Legendre series, and to find this coefficient a system of linear algebraic equations can be obtained directly from the Gelfand–Levitan equation. In contrast to existing methods for solving inverse Sturm–Liouville problems, the method derived by V. Kravchenko is not iterative. The inverse spectral problem is reduced directly to a linear system of algebraic equations. The same approach was developed in [280] for the inverse scattering problem on the line and in [81] for the inverse Sturm–Liouville problem on the half-line. Thus, as was shown by V. Kravchenko and his group, the transmutation operator method is an important tool for practical solution of forward and inverse spectral problems. Thus, the methods of transmutation theory and related problems were applied to one degree or another in the works of many mathematicians. We list some of them: A. I. Aliev, H. Begehr, J. Betancor, A. Boumenir, B. Braaksma, L. Bragg, R. Carroll, H. Chebli, I. Dimovski, C. Dunkl, J. Delsarte, A. Fitouhi, R. Gilbert, V. Hristov, V. Hutson, G. K. Kalish, S. L. Kalla, T. H. Koornwinder, V. Kiryakova, J. Löffström, J. Lions, M. M. Moro, J. S. Pym, B. Rubin, F. Santosa, J. Siersma, H. S. V. de Snoo, K. Stempak, V. Thyssen, K. Triméche, M. Voit, Vu Kim Tuan, Z. S. Agranovich, A. A. Androshchuk, A. G. Baskakov, L. E. Britvina, Yu. N. Val-

xxvi

Introduction

itsky, V. Ya. Volk, V. V. Volchkov, A. D. Gadzhiev, A. V. Glushak, M. L. Gorbachuk, I. Ts. Gokhberg, V. S. Guliev, I. M. Huseynov, Ya. I. Zhytomyrskii, L. A. Ivanov, M. S. Eremin, D. B. Karp, V. V. Katrakhov, A. P. Kachalov, A. A. Kilbas, I. A. Kipriyanov, M. I. Klyuchantsev, V. I. Kononenko, Yu. F. Korobeinik, V. V. Kravchenko, M. G. Krein, P. P. Kulish, I. F. Kushnirchuk, G. I. Laptev, B. Ya. Levin, B. M. Levitan, A. F. Leontiev, N. E. Lynchuk, S. S. Lynchuk, L. N. Lyakhov, G. V. Lyakhovetsky, M. M. Malamud, V. A. Marchenko, V. I. Matsaev, A. B. Muravnik, N. I. Nagnibida, L. P. Nizhnik, M. N. Olevsky, S. S. Platonov, A. Ya. Povzner, B. Rubin, F. S. Rofe-Beketov, K. B. Sabitov, L. A. Sakhnovich, A. S. Sokhin, V. V. Stashevskaya, S. M. Torba, L. D. Faddeev, D. K. Fage, K. M. Fishman, I. G. Khachatryan, A. P. Khromov, E. L. Shishkina, S. D. Shmulevich, and V. Ya. Yaroslavtseva. Of course, this list is not complete and could be significantly expanded. We must note that the term “operator” is used in this book for brevity in the broad and sometimes not exact meaning, so appropriate domains and function classes are not always specified. It is easy to complete and make strict for every special result. Now let us list the content of the book briefly by chapter. In Chapter 1, basic definitions and propositions are presented. First we give definitions of some special functions such as the gamma function, beta function, Pochhammer symbol, error function, Bessel functions, hypergeometric type functions, and some orthogonal polynomials. Next, some functional spaces and integral transforms are considered. Also Kipriyanov’s classification of second order linear partial differential equations, the divergence theorem and Green’s second identity for B-elliptic and B-hyperbolic operators, the Tricomi equation, and the abstract Euler–Poisson– Darboux equation are discussed. In Chapter 2, we collect the basic facts about fractional calculus and fractional order differential equations. First we give a brief history of fractional calculus and fractional order differential equations, which include one-dimensional fractional derivatives and integrals, fractional derivatives in mechanics, fractional powers of multi-dimensional operators such as Riesz potentials, and differential equations of fractional order. We list some standard fractional order integro-differential operators, such as Riemann– Liouville fractional integrals and derivatives on a segment and a semiaxis, Gerasimov– Caputo fractional derivatives, Dzrbashian–Nersesyan fractional operators, sequential order fractional operators, and others. Also integral transforms and basic differential equations of fractional order are considered. Chapter 3 contains information about transmutations. We give a definition of the transmutation operator, some examples of classical transmutations, transmutations for the Sturm–Liouville operator, and transmutations for the singular Bessel operator such as the Poisson operator, the generalized translation, and the weighted spherical mean. Chapter 4 contains detailed studies of weighted generalized functions generated by quadratic forms. First we define the weighted generalized function associated with a positive quadratic form concentrated on a part of a cone and obtain its properties; next, we obtain the Hankel transform of weighted generalized functions generated by quadratic forms.

Introduction

xxvii

Chapter 5 covers one- and multi-dimensional Buschman–Erdélyi transmutation operators theory. It includes Buschman–Erdélyi transmutations of the first, second, and third kinds with properties, Sonine–Katrakhov and Poisson–Katrakhov transmutations, and generalizations to the multi-dimensional case. In Chapter 6, we present the integral transform compositions method (ITCM) for constructing different transmutations. We give basic ideas, a background, and a definition of the ITCM and apply the ITCM to derive transmutations connected with the Bessel operator. Also some examples of the use of the ITCM to the solution of differential equations are given. In Chapter 7, differential equations with Bessel operator without fractional power operators are considered. Firstly, hyperbolic and ultrahyperbolic equations with Bessel operator such as the general and generalized Euler–Poisson–Darboux equation and the singular Klein–Gordon equation are solved. The rest of Chapter 7 contains the solution to the problem for elliptic equations with Bessel operator. In this chapter we also give a short historical introduction on differential equations with Bessel operators and a rather detailed reference list of monographs and papers on mathematical theory and applications of this class of differential equations. Chapter 8 introduces the applications of transmutations to different problems. It includes applications of Buschman–Erdélyi transmutation to the Copson lemma, norm estimates and embedding theorems in Kipriyanov spaces, and the Radon transform. Next, applications of the transmutation method to estimations of the solutions for differential equations with variable coefficients and the E. M. Landis problem and to the perturbed Bessel and the one-dimensional Schrödinger equation are given. Finally, we present identities for iterated weighted spherical means, which are necessary in various applied problems of tomography and integral geometry. In Chapter 9, fractional powers of Bessel operators are studied. We consider fractional Bessel integrals and derivatives on a segment and on a semiaxis, and some of their integral transforms, such as the Mellin transform, the Hankel transform, and the generalized Whittaker transform. Moreover, resolvents for the right-sided fractional Bessel integral on a semiaxis and the generalized Taylor formula with powers of Bessel operators are given. In Chapter 10, the theory of fractional powers of hyperbolic operators with Bessel operators instead of all or some second derivatives is developed. Such operators are called hyperbolic B-potentials. First, we give definitions of the hyperbolic Bpotentials and prove their absolute convergence and boundedness. Next, using the idea of approximative inverse operators, we construct an inverse-to-hyperbolic B-potential operator. Also mixed hyperbolic Riesz B-potentials and their inversions are considered. In Chapter 11, we solve fractional differential equations with singular coefficients. We apply the Meijer transform method for solution of homogeneous fractional equations with left-sided fractional Bessel derivatives on semiaxes of Gerasimov–Caputo type and the Mellin transform method for the solution of ordinary linear nonhomogeneous differential equations of fractional order on semiaxes. Next, we use the Riesz B-potential method for solution of nonhomogeneous hyperbolic equations with Bessel operators.

Basic definitions and propositions

1.1

1

Special functions

1.1.1 Gamma function, beta function, Pochhammer symbol, and error function The gamma function, also called the Euler integral of the second kind, is one of the extensions of the factorial function (see [2], p. 255). The gamma function (z) is defined via a convergent improper integral ∞ (z) =

y z−1 e−y dy,

(1.1)

0

which converges for all z ∈ C such that Re z > 0. Function (1.1) is extended by analytic continuation to all complex numbers except the nonpositive integers (where the function has simple poles). Integration by parts of expression (1.1) yields the recurrent formula (z + 1) = z(z).

(1.2)

Rewriting formula (1.2) in the form (z − 1) =

(z) , z−1

(1.3)

we get an expression that allows us to determine the gamma function of z ∈ C such that Re z ≤ 0, for which the definition (1.1) is unacceptable. Formula (1.3) shows that (z) has simple poles at z = 0, −1, −2, −3, .... From (1.3) we get (z + m + 1) = z(z + 1) · · · (z + m)(z),

m ∈ N.

(1.4)

For the gamma function Euler’s reflection formula (z)(1 − z) =

π , sin zπ

the formula     1 1 π  +z  −z = , 2 2 cos(πz) Transmutations, Singular and Fractional Differential Equations With Applications to Mathematical Physics https://doi.org/10.1016/B978-0-12-819781-3.00007-0 Copyright © 2020 Elsevier Inc. All rights reserved.

(1.5)

(1.6)

2

Transmutations, Singular and Fractional Differential Equations

and the Legendre duplication formula   22z−1 1 (2z) = √ (z) z + 2 π

(1.7)

are valid. The beta function, also called the Euler integral of the first kind, is closely related to the gamma function (see [2], p. 258). The beta function B(z, w) for z, w ∈ C, Re z>0, Re w>0 is a special function defined by 1 t z−1 (1 − t)w−1 dt.

B(z, w) =

(1.8)

0

The beta function is related to the gamma function by the formula B(z, w) =

(z)(w) . (z + w)

(1.9)

The Pochhammer symbol (z)n for integer n is defined by (z)n = z(z + 1)...(z + n − 1),

n = 1, 2, ...,

(z)0 ≡ 1

(1.10)

(see [2], p. 256). The following equalities are true: (z)n = (−1)n (1 − n − z)n ,

(1)n = n!,

and (z)n =

(z + n) . (z)

(1.11)

Equality (1.11) can be used to extend (z)n to real or complex values of n. The error function (also called the probability integral) is defined as 2 erf (x) = √ π

x

e−t dt, 2

x ∈ R,

(1.12)

0

(see [2], p. 297). The error function’s Maclaurin series holds for every complex number z and has the form   ∞ z3 2  (−1)n z2n+1 2 z5 z7 z9 z− erf (z) = √ =√ + − + − ··· . n!(2n + 1) 3 10 42 216 π π n=0

The error function is an entire function.

Basic definitions and propositions

3

1.1.2 Bessel functions Bessel functions, named after the German astronomer Friedrich Bessel, are defined as solutions of the Bessel differential equation x2

d 2y dy + (x 2 − α 2 )y = 0, +x dx dx 2

where α is a complex number. The Bessel functions of the first kind, denoted by Jα (x), are solutions of Bessel’s differential equation that are finite at the origin x = 0. The Bessel function Jα (x) can be defined by the series Jα (x) =

∞  m=0

 x 2m+α (−1)m . m! (m + α + 1) 2

(1.13)

For noninteger α the functions Jα (x) and J−α (x) are linearly independent. If α is integer the following relationship is valid: J−α (x) = (−1)α Jα (x). The Bessel functions of the second kind, denoted by Yα (x), for noninteger α are related to Jα (x) by the formula Yα (x) =

Jα (x) cos(απ) − J−α (x) . sin(απ)

In the case of integer order n, the function Yα (x) is defined by taking the limit as a noninteger α tends to n: Yn (x) = lim Yα (x). α→n

Functions Yα (x) are also called Neumann functions and are denoted by Nα (x). The linear combination of the Bessel functions of the first and second kinds represents a complete solution of the Bessel equation: y(x) = C1 Jα (x) + C2 Yα (x). (1)

(2)

Hankel functions of the first and second kind, denoted by Hα (x) and Hα (x), respectively, are defined by the equalities Hα(1) (x) = Jα (x) + iYα (x)

(1.14)

Hα(2) (x) = Jα (x) − iYα (x).

(1.15)

and

4

Transmutations, Singular and Fractional Differential Equations

Modified Bessel functions (or occasionally the hyperbolic Bessel functions) of the first and second kind Iα (x) and Kα (x) are defined as Iα (x) = i −α Jα (ix) =

∞  m=0

Kα (x) =

 x 2m+α 1 , m! (m + α + 1) 2

π I−α (x) − Iα (x) , 2 sin(απ)

(1.16) (1.17)

where α is noninteger. In the case of integer order α, the functions Iα (x) and Kα (x) are defined by taking the limit as a noninteger α tends to n ∈ Z: In (x) = lim Iα (x),

Kn (x) = lim Kα (x).

α→n

α→n

It is obvious that Kα (x) = K−α (x). Function Iν (r) is exponentially growing when r → ∞ and Kν (r) is exponentially decaying when r → ∞ for real r and ν:    1 ez π |Arg(z)| < , |z| → ∞, 1+O , Iν (z) ∝ √ z 2 2πz     −z πe 1 |z| → ∞. , Kν (z) ∝ √ 1+O 2 z z √ For small arguments 0 < |r|  ν + 1, we have  r ν 1 Iν (r) ∼ , (ν + 1) 2   r − ln − ϑ if ν = 0, 2 (1.18) Kν (r) ∼ (ν) −ν r if ν > 0, 21−ν where

ϑ = lim

n→∞

− ln n +

n  1 k=1

k



 ∞ 1 1 = dx − + x x 1

is the Euler–Mascheroni constant [121]. Here are some of the important particular cases of Bessel functions:   2 2 sin(z), J− 1 (z) = cos(z), J 1 (z) = 2 2 πz πz   2 2 I 1 (z) = sinh(z), I− 1 (z) = cosh(z), 2 2 πz πz  π −z K 1 (z) = K− 1 (z) = e . 2 2 2z

Basic definitions and propositions

5

The normalized Bessel function of the first kind jν is defined by the formula (see [242], p. 10, [317]) 2ν (ν + 1) Jν (x), xν

jν (x) =

(1.19)

where Jν is a Bessel function of the first kind. Operator function of the type (1.19) was considered in [183,187]. The normalized modified Bessel function of the first kind iν is defined by the formula 2ν (ν + 1) Iν (x), xν

iν (x) =

(1.20)

where Iν is a modified Bessel function of the first kind. The normalized modified Bessel function of the second kind kν is defined by the formula kν (x) =

1 2ν (1 + ν)x ν

(1.21)

Kν (x),

where Kν is a modified Bessel function of the second kind. We have dkν (x) 1 Kν+1 (x). =− ν dx 2 (1 + ν)x ν

(1.22)

Here are some of the important particular cases of normalized Bessel functions: sin(z) , j− 1 (z) = cos(z), 2 z sinh(z) i 1 (z) = , i− 1 (z) = cosh(z), 2 2 z −z e k 1 (z) = , k− 1 (z) = e−z . 2 2 z j 1 (z) = 2

Using formulas (9.1.27) from [2] we obtain that jν (t) is an eigenfunction of operd2 ν d ator (Bν )t = dt 2 + t dt : (Bν )t j ν−1 (τ t) = −τ 2 j ν−1 (τ t),

(1.23)

(Bν )t i ν−1 (τ t) = τ 2 i ν−1 (τ t),

(1.24)

(Bν )t k ν−1 (τ t) = τ 2 k ν−1 (τ t).

(1.25)

2

2

2

2

2

2

Normalized Bessel functions have the following properties: jν (0) = 0, iν (0) = 1, 1 lim x 2ν kν (x) = , ν > 0, x→0 2ν

jν (0) = 1,

iν (0) = 0, (1.26)

6

Transmutations, Singular and Fractional Differential Equations

lim kν (x) =

x→0

(−ν) 22ν+1 (1 + ν)

lim x α k0 (x) = 0,

x→0

lim x 2ν+1

x→0

ν < 0,

,

α > 0,

dkν (x) = −1, dx

lim

x→0

−ν ∈ / N,

1 k0 (x) = −1, ln x

ν > −1.

(1.27) (1.28) (1.29)

We will use notations jγ (x, ξ ) =

n i=1

j γi −1 (xi ξi )

(1.30)

i γi −1 (xi ξi ),

(1.31)

2

and iγ (x, ξ ) =

n i=1

2

where γ = (γ1 , ..., γn ), γ1 > 0, ..., γn > 0. Information about the Bessel functions is taken from [591]. The Struve function is a solution y(x) of the nonhomogeneous Bessel differential equation:

α+1  4 x2 dy  2 2  . x +x + x −α y = √ dx dx 2 π α + 12 2d

2y

Struve functions, denoted as Hα (x), have the power series form Hα (x) =

 x 2m+α+1 (−1)m    . 3 3 2 m=0  m + 2  m + α + 2 ∞ 



(1.32)

Another definition of the Struve function, for values of α satisfying Re α > − 12 , is possible using an integral representation:

α  π 2 2 x2   sin(x cos τ ) sin2α (τ ) dτ. Hα (x) = √ 1 π α + 2 0

1.1.3 Hypergeometric type functions The hypergeometric Gauss function inside the circle |z| 0, ξ ≥ 0, t > 0, t 

t 2 − u2

α−1

u1−γ Jγ (uξ )dt =

0

ξ γ t 2α 2γ +1 α(γ + 1)   t 2ξ 2 × 1 F2 1; α + 1, γ + 1; − 4

and for γ < 2, α > 0, ξ ≥ 0, t > 0, t 

t 2 − u2

α−1

u1−γ Iγ (uξ )dt =

0

ξ γ t 2α 2γ +1 α(γ

+ 1)

  t 2ξ 2 . × 1 F2 1; α + 1, γ + 1; 4

The Appell hypergeometric function F4 (a, b, c1 , c2 ; x, y) (see [456], p. 658) for |x|1/2 + |y|1/2 −1

l=1

is satisfied, the series in (1.38) is convergent for any z ∈ C. Let δ=

p

μ=

|αl |−αl

l=1 q 

q

|βj |βj ,

j =1

bj −

j =1

p 

al +

l=1

p−q . 2

If q  j =1

βj −

p 

αl = −1,

l=1

then the series in (1.38) is absolutely convergent for |z| < δ and for |z| = δ and Re μ > 12 . The Mittag-Leffler function Eα,β (z) is the entire function of order 1/α defined by the following series when the real part of α is strictly positive: Eα,β (z) =

∞  n=0

zn , z ∈ C, α, β ∈ C, Re α > 0, Re β > 0. (αn + β)

(1.39)

Function (1.39) was introduced by Gesta Mittag-Leffler in 1903 for α = 1 and by A. Wiman in 1905 in the general case. The first applications of these functions by

Basic definitions and propositions

11

Mittag-Leffler and Wiman were applications in complex analysis (nontrivial examples of entire functions with noninteger orders of growth and generalized summation methods). In the USSR, these functions became mainly known after the publication of the famous monograph by M. M. Dzhrbashyan [98] (see also his later monograph [106]). The most famous application of the Mittag-Leffler functions in the theory of integro-differential equations and fractional calculus is the fact that through them the resolvent of the Riemann–Liouville fractional integral is explicitly expressed in accordance with the famous Hille–Tamarkin–Dzhrbashyan formula [494]. In view of the numerous applications to the solution of fractional differential equations, this function was deservedly named in [202] “Royal function of fractional calculus.” The derivative of the Mittag-Leffler function is calculated by the formula Eα,β (z) =

∞

Eα,β (z)  (1 + k)zk = . dz (β + α(1 + k)) k=0

Note that Eα,β (0) = 1, E0,1 (z) =

∞ 

zk =

k=0

E1,1 (z) = ez ,

1 , 1−z

E1,2 (z) =

ez − 1 , z

E2,2 (z) =

√ sinh( z) . √ z

Using the Fox–Wright function (1.38) we can write  Eα,β (z) = 1 1

  (1, 1)  z . (β, α)

(1.40)

A general definition of the Meijer G-function is given by the following line integral in the complex plane (see [20], p. 206):   a1 , . . . , ap  z b 1 , . . . , bq  n m   (bj − s) (1 − aj + s)  1 j =1 j =1 zs ds. = q p   2πi (1 − bj + s) (aj − s) L 

m,n Gp,q

j =m+1

j =n+1

The general Legendre equation reads  (1 − x 2 ) y − 2xy + λ(λ + 1) −

 μ2 y = 0, 1 − x2

(1.41)

12

Transmutations, Singular and Fractional Differential Equations

where the numbers λ and μ may be complex. This differential equation has two linearly independent solutions, which can both be expressed in terms of the hypergeometric function 2 F1 :     1 1 + z μ/2 1−z F , −λ, λ + 1; 1 − μ; 2 1 (1 − μ) 1 − z 2 for |1 − z| < 2, μ

Pλ (z) =

(1.42)

μ

Qλ (z) = √   π (λ + μ + 1) eiμπ (z2 − 1)μ/2 3 1 λ+μ+1 λ+μ+2 , ;λ + ; 2 , 2 F1 2 2 2 z 2λ+1 (λ + 3/2) zλ+μ+1 (1.43) for |z| > 1. μ

μ

Functions Pλ and Qλ are generally known as Legendre functions of the first and second kind of noninteger degree, with the additional qualifier “associated” if μ is nonzero. We will use also Legendre functions with z = x, where −1 < x < 1 (see [19]). μ The interval (−1, 1) is “the cut.” If μ is even integer, then the values of Pλ (z) on both sides of the cut are equal and in this case it is sufficient to take the branch cut along the μ μ real axis from −1 to −∞. In all other cases Pλ (x − i0) and Pλ (x + i0) are different (here f (x ± i0) means lim f (x ± iε), ε > 0). In order to avoid ambiguity it is usual to ε→0

μ

introduce slightly modified Legendre functions. These will be denoted by Pν (z) and μ Qν (z):  μπ 1  i μπ μ e 2 Pν (x + i0) + e−i 2 Pνμ (x − i0) , 2   μπ 1 μ i μπ 2 Qμ (x − i0) . (x + i0) + e Qν (x) = e−iμπ e−i 2 Qμ ν ν 2

Pμ ν (x) =

(1.44) (1.45)

1.1.4 Polynomials Gegenbauer polynomials or ultraspherical polynomials Cn(α) (x) are orthogonal poly1 nomials on the interval [−1, 1] with respect to the weight function (1 − x 2 )α− 2 that can be defined by the recurrence relation C1α (x) = 2αx, C0α (x) = 1, 1 α α Cnα (x) = [2x(n + α − 1)Cn−1 (x) − (n + 2α − 2)Cn−2 (x)]. n The next decomposition is valid: ∞

 1 = Cn(α) (x)t n . 2 α (1 − 2xt + t ) n=0

Basic definitions and propositions

13

Gegenbauer polynomials are particular solutions of the Gegenbauer differential equation (1 − x 2 )y − (2α + 1)xy + n(n + 2α)y = 0. When α = 12 , this equation reduces to the Legendre equation, and the Gegenbauer polynomials reduce to the Legendre polynomials. When α = 1, the equation reduces to the Chebyshev differential equation, and the Gegenbauer polynomials reduce to the Chebyshev polynomials of the second kind. They are given as Gauss hypergeometric functions in certain cases where the series is in fact finite (see [2], p. 561):   (2α)n 1 1−z Cn(α) (z) = F ; −n, 2α + n; α + 2 1 n! 2 2 =

n/2  k=0

(−1)k

(n − k + α) (2z)n−2k . (α)k!(n − 2k)!

The Gegenbauer polynomial can also be represented by the Rodrigues formula Cn(α) (x) =

1.2

 n  (−1)n (α + 12 )(n + 2α) 2 −α+1/2 d 2 n+α−1/2 ) ) (1 − x . (1 − x 2n n! (2α)(α + n + 12 ) dx n

Functional spaces γ

m , S , and L spaces 1.2.1 Orthant Rn+ , Cev ev p

Suppose that Rn is the n-dimensional Euclidean space, Rn+ ={x=(x1 , . . . , xn ) ∈ Rn , x1 >0, . . . , xn >0}, R n+ ={x=(x1 , . . . , xn ) ∈ Rn , x1 ≥0, . . . , xn ≥0}, γ =(γ1 , ..., γn ) is a multi-index consisting of positive fixed real numbers γi , i=1, ..., n, and |γ |=γ1 +. . .+γn . The part of the sphere of radius r with center at the origin belonging to Rn+ we will denote Sr+ (n): Sr+ (n)={x ∈ R n+ : |x|=r} ∪ {x ∈ R n+ : xi = 0, |x|≤r, i = 1, ..., n}. Let  be a finite or infinite open set in Rn symmetric with respect to each hyperplane xi =0, i = 1, ..., n, + =  ∩ Rn+ , and + =  ∩ Rn+ , where R n+ = {x = (x1 , . . . , xn ) ∈ Rn , x1 ≥0, . . . , xn ≥ 0}. We deal with the class C m (+ ) consisting of m times differentiable on + functions and denote by C m (+ ) the subset of functions from C m (+ ) such that all derivatives of these functions with respect to xi

14

Transmutations, Singular and Fractional Differential Equations

m for any i = 1, ..., n are continuous  up to xi =0. Class Cev (+ ) consists of all functions  2k+1 from C m (+ ) such that ∂ 2k+1f  = 0 for all nonnegative integers k ≤ m−1 2 (see ∂xi

xi =0

m (Rn ) by C m . We set [610] and [242], p. 21). In the following we will denote Cev + ev ∞ (+ ) = Cev



m Cev (+ )

∞ (R ) = C ∞ . with intersection taken for all finite m and Cev + ev As the space of basic functions we will use the subspace of the space of rapidly decreasing functions:   α β  ∞ n   Sev = f ∈ C : sup x D f (x) < ∞ ∀α, β ∈ Z , ev

+

x∈Rn+

where α = (α1 , ..., αn ), β = (β1 , ..., βn ), α1 , ..., αn , β1 , ..., βn are integer nonnegative β β numbers, x α = x1α1 x2α2 . . . xnαn , D β = Dx11 ...Dxnn , Dxj = ∂x∂ j . γ

γ

Let Lp (Rn+ ) = Lp , 1≤p t} = {x: |f (x)|>t}+

where {x:|f (x)|>t}+ ={x∈Rn+ :|f (x)|>t}. We will call the function μγ = μγ (f, t) a weighted distribution function of |f (x)|. Statement 1. For any function f ∈ Lp (R+ n ) the following equality is correct: γ

⎛ ||f ||Lγp = ⎝p

∞

⎞1/p t p−1 μγ (f, t)dt ⎠

(1.46)

.

0

Proof. Let us first suppose that the function f is continuous in R+ n and has a limited support + . This area is divided into parts as follows. Let m = t0 < t1 < t2 < . . . < t = M, where m and M are the largest and the smallest value of the function |f | on the + . We introduce the following partition of this area: + =

 #

+ i =

i=1

 #

{x : ti−1 < |f (x)| < ti }+ .

i=1

It is easy to see that the weight measure of a subset + i is represented as a difference of the weight distribution functions at the points ti and ti+1 : mesγ {+ i } = μγ (f ; ti ) − μγ (f ; ti+1 ). Then



 |f (x)|p x γ dx =

R+ N

|f (x)|p x γ dx

+

= lim

λ→0

 

|f (ξi )|p mesγ i = lim

λ→0

i=1

= − lim

 

λ→0

p

ti

$

i=1

i=1

∞ t dt μγ (f, t)dt = − p

m

$ % |f (ξi )|p μγ (f ; ti+1 ) − μγ (f ; ti )

% μγ (f ; ti )−μγ (f ; ti+1 )

M =−

 

t p dt μγ (f, t)dt, 0

16

Transmutations, Singular and Fractional Differential Equations

where λ is the maximum partition size and ξi ∈ + i is the midpoint of the i-th partition and we took into account that μγ (f ; t) for the continuous function f is a differentiable function. Now equality (1.46) is obtained by integration by parts. γ Now let f ∈ Lp (R+ n ). In this case an infinitely differentiable function fε exists such that lim f − fε Lγp = 0.

ε→0

It remains to write equality (1.46) for the function fε and passing to the limit at ε → 0 taking into account that both functions μγ (fε ; t) and μγ (f ; t) are monotone and that lim μγ (fε ; t) = μγ (f ; t).

ε→0

The proof is complete. γ

γ

The space L∞ (Rn+ )=L∞ is the space of all measurable in Rn+ functions even with respect to each variable xi , i = 1, ..., n, for which the norm ||f ||Lγ∞ (Rn ) = ||f ||∞,γ = ess supγ |f (x)| = inf {μγ (f, a) = 0} +

a∈R

x∈Rn+

is finite. γ

γ

Statement 2. Norms of spaces Lp and L∞ related by equality γ

||f ||∞,γ = lim ||f ||p,γ ,

f ∈ L∞ .

p→∞

(1.47)

Proof. If ||f ||∞,γ = 0, then equality (1.47) is obvious. γ Let 0 0. If q = ∞, then a linear operator A is an operator of weak type (p, q)γ when it has strong type (p, q)γ .

18

Transmutations, Singular and Fractional Differential Equations

, absolutely 1.2.3 Space of weighted generalized functions Sev continuous functions, and unitary operators γ

γ

For 1 ≤ p ≤ ∞, Lp,loc (Rn+ ) = Lp,loc is the set of functions u(x) defined almost evγ erywhere in Rn+ such that uf ∈ Lp for any f ∈ Sev . ◦

k (X) with compact Definition 2. Let C kev (X) be the space of all functions u ∈ Cev support.

Definition 3. Let L1loc,γ (X), X ⊂ Rn+ , be the space of all functions integrable with the weight x γ on compact subsets in X:  1 |f (x)|x γ dx < ∞. f ∈ Lloc,γ (X) ⇔ X (Rn ) = S is a class Definition 4. The space of weighted generalized functions Sev + ev of continuous linear functionals that map a set of test functions f ∈ Sev into the set γ of real numbers. Each function u(x) ∈ L1,loc will be identified with the functional (Rn ) = S acting according to the formula u ∈ Sev + ev  f ∈ Sev . (1.52) (u, f )γ = u(x) f (x) x γ dx, Rn+ acting by formula (1.52) will be called regular Generalized functions u ∈ Sev will be called weighted generalized functions. All other generalized functions u ∈ Sev singular weighted generalized functions.

As we have seen, a singular weighted generalized function cannot be identified with any locally integrable function. The simplest example of a singular weighted generalized function is the weighted delta function. is defined by the equality (by analogy with The weighted delta function δγ ∈ Sev [177], p. 247) (δγ , ϕ)γ = ϕ(0),

ϕ(x) ∈ Sev .

The fact that this generalized function is weighted is explained as follows. Let ⎧ 2 ⎨ − 2ε 2 ε −|x| lrC e |x| ≤ ε, ε ωε (x) = ⎩0 |x| > ε, where Cε is selected such that  ωε (x)x γ dx = 1. Rn+

Basic definitions and propositions

Since

19

 lim

ε→+0 Rn+

ωε (x)ϕ(x)x γ dx = ϕ(0),

ϕ ∈ Sev ,

we have (ωε (x), ϕ(x))γ → (δγ (x), ϕ(x))γ ,

ε → +0,

ϕ ∈ Sev .

Considering that for convenience we will write  (δγ , ϕ)γ = δγ (x)ϕ(x)x γ dx = ϕ(0) Rn+

and understand it in the sense of the limit of delta-shaped sequences. Following [494] we give the definition of the space AC() of absolutely continuous functions. Let  = [a, b]. Definition 5. A function f (x) is called absolutely continuous on an interval  if for any ε > 0 there exists a δ > 0 such that for any finite set of pairwise n nonintersecting n ) intervals [ak , bk ] ⊂ , k = 1, 2, ..., n, such that for (bk − ak ) < δ, the inequality n )

k=1

|f (bk ) − f (ak )| < ε holds. The space of these functions is denoted by AC().

k=1

The space AC() coincides with the space of primitives of Lebesgue summable functions (see [266], p. 338): x

b ϕ(t)dt + c,

f (x) ∈ AC() ⇔ f (x) = a

|ϕ(t)|dt < ∞. a

Definition 6. Let us denote by AC n (), where n = 1, 2, ..., the space of functions f (x) which have continuous derivatives up to order n − 1 on  with f (n−1) (x) ∈ AC(). It is clear that AC 1 () = AC(). The space AC n () consists of those and only those functions f (x) which are represented in the form 1 f (x) = (n − 1)!

x (x − t)n−1 ϕ(t)dt + a

n−1 

ck (x − a)k ,

k=0

where ϕ(t) ∈ L1 (a, b), ck are arbitrary constants. A unitary operator is a surjective bounded operator on a Hilbert space preserving the inner product. Unitary operators are usually taken as operating on a Hilbert space, but the same notion serves to define the concept of isomorphism between Hilbert spaces.

20

Transmutations, Singular and Fractional Differential Equations

1.2.4 Mixed case In this subsection we give a summary of the basic notations, terminology, and results connected with the case when the Bessel operator acts on all variables except the first one. Suppose that Rn+1 is the (n + 1)-dimensional Euclidean space, n+1 , x1 >0, . . . , xn >0}, Rn+1 + ={(t, x)=(t, x1 , . . . , xn ) ∈ R

γ =(γ1 , ..., γn ) is a multi-index consisting of positive fixed real numbers γi , i=1, ..., n, and |γ |=γ1 +. . .+γn . Let  be a finite or infinite open set in Rn+1 symmetric with ren+1 spect to each hyperplane xi =0, i = 1, ..., n, and + =  ∩ Rn+1 + and + =  ∩ R+ , n+1 n+1 where R + ={(t, x)=(t, x1 , . . . , xn )∈R , x1 ≥0, . . . , xn ≥0}. We deal with the class C m (+ ) consisting of m times differentiable on + functions and denote by C m (+ ) the subset of functions from C m (+ ) such that all existing derivatives of these functions with respect to xi for any i = 1, ..., n are continuous up to xi =0 and all existing m derivatives with respect to t are continuous for  t ∈ R. Class Cev (+ ) consists of 2k+1  all functions from C m (+ ) such that ∂ 2k+1f  = 0 for all nonnegative integers ∂xi x=0

k ≤ m−1 2 and for i = 1, ..., n (see [610] and [242], p. 21). In the following we will n+1 m denote Cm ev (R+ ) by Cev . We set C∞ ev (+ ) =



Cm ev (+ ) ◦

n+1 ∞ ∞ with intersection taken for all finite m. Let C∞ ev (R + ) = Cev . Assume that C ev (+ ) ∞ is the space of all functions f ∈Cev (+ ) with a compact support. We will use the ◦

notation C ∞ ev (+ )=D+ (+ ). γ Let Lp (+ ), 1≤p σ }  = x γ dtdx, {(t,x): |f (t,x)|>σ }+

where {(t, x):|f (t, x)|>σ }+ ={(t, x)∈Rn+1 + :|f (t, x)|>σ }. γ Let a space L∞ (+ ) be defined as a set of measurable on + functions f (t, x) such that ||f ||Lγ∞ (+ ) = ess supγ |f (t, x)| = inf {Mγ (f, σ ) = 0} < ∞. σ ∈+

(t,x)∈+

γ

For 1 ≤ p ≤ ∞, Lp,loc (+ ) is the set of functions u defined almost everywhere in γ + such that uf ∈ Lp (+ ) for any f ∈ D+ (+ ). Let us define D + (+ ) as a set of continuous linear functionals on + . Each γ function u ∈ L1,loc (+ ) will be identified with the functional u ∈ D + (+ ) acting according to the formula  (u, f )γ =

u(t, x) f (t, x) x γ dtdx,

f ∈ D+ (+ ).

(1.53)

+

Functionals u ∈ D + (+ ) acting by formula (1.53) will be called mixed regular weighted functionals. All other continuous linear functionals u ∈ D + (+ ) will be called mixed singular weighted functionals. The generalized function δγ is defined by the equality (by analogy with [242], p. 12) (δγ , ϕ)γ = ϕ(0),

ϕ ∈ D+ (+ ).

As the space of basic functions we will use the subspace of rapidly decreasing functions: ⎧ ⎫ ⎨ ⎬   ∞ t α0 x α D β f (t, x) < ∞ , ) = f ∈ C : sup Sev (Rn+1 + ev ⎩ ⎭ (t,x)∈Rn+1 +

where α = (α1 , ..., αn ), β = (β0 , β1 , ..., βn ), α0 , α1 , ..., αn , β0 , β1 , ..., βn are arbitrary β β β integer nonnegative numbers, and x α = x1α1 x2α2 . . . xnαn , D β = Dt 0 Dx11 ...Dxnn , Dt = ∂ ∂ ∂t , Dxj = ∂xj , j = 1, ..., n. In the same way as D+ we introduce the space Sev . In with a subspace of D since D is dense in S . fact we identify Sev + ev +

22

Transmutations, Singular and Fractional Differential Equations

1.3 Integral transforms and Lizorkin–Samko space 1.3.1 One-dimensional integral transforms with Bessel functions in the kernels and Mellin transform In this subsection, following [180], we consider some one-dimensional integral transforms which we will use later. Definition 7. The one-dimensional Fourier transform of an integrable function f : R → C is ∞ F [f ](ξ ) =

f (x) e−ixξ dx,

−∞

for any real number ξ . Under suitable conditions, f is determined by F [f ] via the inverse transform: 1 f (x) = 2π

∞ F [f ](ξ ) eixξ dξ, −∞

for any real number x. In F [f ], instead of the kernel e−ixξ , sometimes eixξ , e−2πixξ , or (2π)−1/2 e±ixξ is chosen, as in certain instances these kernels are more convenient. Theorem 1. [180] Let f ∈ L1 be piecewise smooth in each interval [a, b] ⊂ R. Then we have for every x0 ∈ R 1 2π

∞ F [f ](ξ ) eix0 ξ dξ = f (x0 ) −∞

if f is continuous at x0 , and we have 1 2π

∞ F [f ](ξ ) eix0 ξ dξ = −∞

f (x0 + 0) − f (x0 − 0) 2

if f is discontinuous at x0 , and the integral in this case has to be understood in the sense of Cauchy’s principal value. Let f, g ∈ L1 ∩ L2 . Then the Parseval formula follows ∞

∞ f (x)g(x) dx =

−∞

−∞

F [f ](ξ )F [g](ξ ) dξ,

Basic definitions and propositions

23

where the bar denotes complex conjugation. The Plancherel theorem, which follows from the above, states that ∞

∞ |f (x)| dx =

|F (f )(ξ )|2 dξ.

2

−∞

−∞

Definition 8. The Laplace transform of a function f (t), defined for all real numbers t > 0, is the function F (s), which is a unilateral transform defined by ∞ L[f ](s) = F (s) =

f (t)e−st dt,

(1.54)

0

where s is a complex number frequency parameter s = σ + iω, with real numbers σ and ω. -∞

Let Ea , a ∈ R, be the space of functions f : R → C, f ∈ Lloc 1 (R), such that |f (t)|e−at dt < ∞ and f (t) vanishes if t < 0.

0

Let f ∈ Ea . Then the Laplace integral (1.54) is absolutely and uniformly convergent on H¯ a = {p : p ∈ C, Re p ≥ a}. The Laplace transform of function f ∈ Ea is bounded on H¯ a and it is an analytic function on Ha = {p : p ∈ C, Re p > a}. Let f ∈ Ea be smooth on every interval (a, b) ∈ R+ . Then in points t of continuity the complex inversion formula L

−1

1 [F ](t) = f (t) = 2πi

c+i∞ 

F (s)ets ds,

c > a,

c−i∞

holds. The Laplace transform of the Mittag-Leffler function multiplied by the power function is (see [241], p. 47, formula (1.9.13), where ρ = 1) L[x β−1 Eα,β (λx α )](s) =

s α−β . sα − λ

(1.55) γ

Definition 9. The one-dimensional Hankel transform of a function f ∈L1 (R1+ ) is expressed as ∞ Fγ [f ](ξ ) = Fγ [f (x)](ξ ) = f (ξ ) =

f (x) j γ −1 (xξ )x γ dx,

(1.56)

2

0

where γ > 0 and the symbol jν is used for the normalized Bessel function of the first kind (1.19).

24

Transmutations, Singular and Fractional Differential Equations γ

Let f ∈L1 (R+ ) be of bounded variation in a neighborhood of a point x of continuity of f . Then for γ > 0 the inversion formula Fγ−1 [f.(ξ )](x) = f (x) =

21−γ  

2

γ +1 2

∞

j γ −1 (xξ )f.(ξ )ξ γ dξ

(1.57)

2

0

holds. Definition 10. For functions f the integral transform involving the normalized modified Bessel function of the second kind k γ −1 , γ ≥ 1, as kernel is the Meijer transform 2 defined by ∞ Kγ [f ](ξ ) = F (ξ ) =

k γ −1 (xξ ) f (x)x γ dx.

(1.58)

2

0

  β− γ2 as t → +0, where β > Let f ∈ Lloc 1 (R+ ) and f (t) = o t

γ 2

− 2 if γ > 1 and

β > −1 if γ = 1. Furthermore, let f (t) = 0(eat ) as t → +∞. Then its Meijer exists a.e. for Re ξ > a (see [180], p. 94). Since k− 1 (z) = e−z , 2

∞ K0 [f ](ξ ) = F (ξ ) =

e−xξ f (x) dx = L[f ](ξ )

0

is a Laplace transform (1.54). Let S be the space of rapidly decreasing functions on (0, ∞),    S = f ∈ C ∞ (0, ∞) : sup x α D β f (x) < ∞ ∀α, β ∈ Z+ , x∈(0,∞)

and f ∈ S. We also will use the following transforms with Bessel functions in the kernel: Hν [f ](x) =

∞ 1 (xt) 2 Jν (xt)f (t)dt,

1 ν≥− , 2

(1.59)

0

and ∞ 1 Yν [f ](x) = (xt) 2 Yν (xt)f (t)dt, 0

(1.60)

Basic definitions and propositions

25

where Jν is the Bessel function of the first kind of order ν and Yν is the Bessel function of the second kind of order ν. Transforms Hν and Yν are well studied in [477,478,166]. In these papers, their boundedness in Lp,μ -spaces have been completely examined. The Lp,μ -space is the space of functions (or, more precisely, equivalence classes) such that their Lebesgue -∞ integral |x μ f (x)|p dx x is finite. The Lp,μ -norm is defined by 0

 1 dx 2 . ||f ||p,μ = |x μ f (x)|p x

(1.61)

Here 1 ≤ p < ∞ and μ is any number. Definition 11. The Mellin transform of a function f : R+ → C is the function f ∗ defined by ∞

∗

f (s) = M[f ](s) =

x s−1 f (x)dx, 0

where s = σ + iτ ∈ C, provided that the integral exists. As space of originals we choose the space Pab , −∞ < a < b < ∞, which is the linear space of R+ → C functions such that x s−1 f (x) ∈ L1 (R+ ) for every s ∈ {p ∈ C : a ≤ Re p ≤ b}. If additionally f ∗ (c + iτ ) ∈ L1 (R) with respect to τ , then the complex inversion formula holds: M

−1

c+i∞ 

1 [ϕ](x) = f (x) = 2πi

x −s ϕ(s) ds.

c−i∞

In fact, the gamma function (z) corresponds to the Mellin transform of the negative exponential function: (z) = M[e−x ](z). If an operator A is acting in the images of the Mellin transform as a multiplication to a function M[Af ](s) = mA M[f ](s),

(1.62)

then we name mA a multiplier of operator A. The Mellin convolution (f ∗ g)M (y) of two functions f and g is given by ∞ f (x)g 0

 y  dx x

x

.

(1.63)

26

Transmutations, Singular and Fractional Differential Equations

We have ⎤ ⎡∞  x  dt f (t) ⎦ (s) = M[K](s) M[f ](s), M⎣ K t t

(1.64)

0

so M[K](s) is the multiplier for the Mellin convolution (see [361]).

1.3.2 Properties of composition of integral transforms with Bessel functions in the kernel The next lemma, which allows to reduce the question of the boundedness of some operator in L2 to study its multiplier, is well known (see [114,98,565]). We give this lemma here. Lemma 1. (1) Let for operator A equality (1.62) be true. Then, in order for expansion of A to the bounded operator in L2 to be allowed, it is necessary and sufficient that     1   (1.65) = M1 < ∞. sup mA iξ + 2  ξ ∈R Thus we have ||A||L2 = M1 . (2) In order for expansion of the inverse operator A−1 to the bounded operator in L2 to be allowed, it is necessary and sufficient that     1   (1.66) = m1 < ∞. inf mA iξ + ξ ∈R  2  Thus we have ||A−1 ||L2 = m11 . (3) Let operators A and A−1 be defined and bounded in L2 . Then in order for A and A−1 to be unitary it is necessary and sufficient that an equality      mA iξ + 1  = 1 (1.67)  2  is true for almost all ξ ∈ R. Here, we give some more known results from [477,478,166] which we will use later. Lemma 2. (1) The integral transform Hν bijectively maps the space L2,μ into itself when 12 ≤ μ < ν + 32 . (2) Let f ∈ L2,μ , 12 ≤ μ < 32 − ν. Then for Re s = μ we have M[Hν f ](s) = mν (s) M[f ](1 − s),

(1.68)

Basic definitions and propositions

where

  1

mν (s) = 2s− 2





27

2s+2ν+1 4 2s−2ν+3 4

 .

(1.69)

Lemma 3. (1) The integral transform Yν bijectively maps the space L2,μ into itself when 12 ≤ μ < 32 − |ν|, except for the case μ = 12 − ν. For μ = 12 − ν we have Yν [L2, 1 −ν ] = Hν [L2, 1 −ν ], and Yν = C1ν Hν + C2ν H−ν when − 12 < ν < 0 and 2 2 Hν [L2, 1 −ν ] = H−ν [L2, 1 −ν ] = L2, 1 −ν . It is obvious that Yν [L2, 1 −ν ] ⊂ L2, 1 −ν . For 2 2 2 2 2 Y0 it is proved in [478]. (2) Let f ∈ L2,μ , 12 ≤ μ < ν + 32 . Then for Re s = μ we have   1 π M[Yν f ](s) = −mν (s) ctg s + − |ν| M[f ](1 − s), (1.70) 2 2 

where mν (s) = 2

s− 12







2s+2ν+1 4 2s−2ν+3 4

 .

Now let us study the composition of operators Hν and Yν . Theorem 2. Let f ∈ L2,μ , 12 ≤ μ < 32 − ν. Then: 1) Operator Hν Yν acts in Mellin images according to (1.62) with the multiplier   2s + 2ν − 1 m(s) = − tg π, Re s = μ. (1.71) 4 2) The formula 2 Hν [Yν f ](x) = π

∞ 0

tf (t)  x ν+ 12 dt t 2 − x2 t

(1.72)

is valid. Proof. 1) According to [478,166], if f ∈ L2,μ , then Yν f ∈ L2,μ for and we can apply formula (1.70) to Yν f :

1 2

≤μ
−1 operator Hν is unitary in L2 . Proof. The space L2 is obtained from L2,μ when μ = 12 ; therefore the multiplier of operator Hν for ν > 12 is defined on the line Re s = 12 . Let us write its values on this line:         1+ν−iξ      m iξ + 1  =   2   .  2    1−ν−iξ  2 ¯ Considering that |z| = |¯z| and (z) = (¯z) we obtain      m iξ + 1  = 1  2  for any ξ ∈ R. That means unitarity of Hˆ ν and, consequently, Hν by Lemma 1. Remark 1. It is easy to see that for complex ν the operator Hν is not unitary in L2 . Indeed, let ν = λ + iμ, μ = 0. Then         λ+1+i(μ+ξ )      2 m iξ + 1  =     .    2   λ+1+i(μ−ξ )  2

It is obvious that the modules of imaginary parts of the arguments of gamma functions are equal here if and only if ξ = 0 and, consequently, equality (1.67) is not true for almost all ξ ∈ R. Theorem 5. For ν ∈ (−1, 1) the operator Yν is bounded in L2 and 1 − 12  ≤ ν ≤ 12 ,     

||Yν ||L2 =  nν  ν ∈ −1, − 12 ∪ 12 , 1 . tg n Proof. Let us write the multiplier of operator Yν on the line Re s = 12 :           1+ν+iξ  1−ν+iξ            2 m iξ + 1  =   2  = tg iξ − ν n .          2 2   1−ν+iξ  ν−iξ  2

2

Here we use formulas (1.5), (1.6), and |(z)| = |(¯z)|. Since for k ∈ Z lim

z→ π2 +πk

tg(z) = ∞

we require that ν = 2k + 1. For −1 β. In virtue of the asymptotic formula (see [20], p. 62, formula (4)) (iξ + α) ∼ (iξ )α−β , (iξ + β)

|ξ | → ∞,

we obtain

   (iξ + α)  = ∞  lim  |ξ |→∞ (iξ + β)  0

α > β, α 0, ..., γn > 0,

2

and the symbol jν is used for the normalized Bessel function of the first kind (1.19).

38

Transmutations, Singular and Fractional Differential Equations γ

Let f ∈L1 (R+ ) be of bounded variation in a neighborhood of a point x of continuity of f . Then for γ > 0 the inversion formula  2n−|γ | . F−1 [ f (ξ )](x) = f (x) = jγ (x, ξ )f.(ξ )ξ γ dξ   γ n  γj +1 2  Rn+ 2 j =1

holds. Definition 13. The multi-dimensional Fourier–Bessel transform of a function γ f ∈L1 (Rn+1 + ) is  . Fγ [f ](τ, ξ ) = f (τ, ξ ) = f (t, x) e−itτ jγ (x; ξ )x γ dtdx, Rn+1 +

where jγ (x; ξ ) =

n

j γi −1 (xi ξi ),

γ1 > 0, ..., γn > 0.

2

i=1

, then the equality If g ∈ Sev

(Fγ g, ϕ)γ = (g, Fγ ϕ)γ ,

ϕ ∈ Sev ,

(1.83)

. defines the Hankel transform of functional g ∈ Sev In [491] the space V consisting of functions vanishing on a given closed set V of measure zero was considered. The Lizorkin–Samko space V is dual to V in the γ sense of Fourier transforms. We introduce the space V of functions Sev vanishing with all their derivatives on a given closed set V : γ

V = {ψ ∈ Sev (Rn+ ) : (D k ψ)(x) = 0, x ∈ V , |k| = 0, 1, 2, ...}. γ

γ

Space V is dual to V in the sense of Hankel transforms, γ

γ

V = {ϕ : Fγ ϕ ∈ V }.

(1.84)

1γ ,V denote the following class of functions: For the mixed case let  k 1γ ,V = {ψ ∈ Sev (Rn+1  + ) : (D ψ)(x) = 0, x ∈ V , |k| = 0, 1, 2, ...}

and 1γ ,V }. 1γ ,V = {ϕ : Fγ ϕ ∈   We will consider weighted generalized functions over  and . The Hankel transform of f ∈  is (Fγ f, ψ)γ = (f, Fγ ψ)γ , ψ ∈ .

(1.85)

Basic definitions and propositions

39

If g ∈  , then (Fγ g, ϕ)γ = (g, Fγ ϕ)γ , ϕ ∈ .

(1.86)

Definition 14. [163] Let f be a function on Rn , integrable on each hyperplane in Rn . Let Pn denote the space of all hyperplanes in Rn . The Radon transform of f (x) : Rn → R is defined as the function Rf : S n−1 × R → R on Pn given by  Rf (ω, s) =

f (x)dm(x),

x·ω=s

where dm is the Euclidean measure on the hyperplane x · ω = s. Along with the transformation f → Rf we consider also the dual transform R∗ ϕ which associates with a continuous function ϕ on Rn the function R∗ ϕ on Rn given by  ϕ(ξ )dμ(ξ ), R∗ ϕ(x) = x∈ξ

where ξ = {x · ω = s} is the hyperplane incident with the point x ∈ Rn and dμ is the measure on the compact set {ξ ∈ Pn : x ∈ ξ } which is invariant under the group of rotations around x and for which the measure of the whole set is 1.

1.4

Basic facts and formulas

1.4.1 Kipriyanov’s classification of second order linear partial differential equations We will deal with the singular Bessel differential operator Bγ (see, for example, [242], p. 5): (Bγ )t =

∂ ∂2 γ ∂ 1 ∂ + = γ tγ , t ∂t t ∂t ∂t ∂t 2

γ ∈ R.

t > 0,

(1.87)

The Russian mathematician I. A. Kipriyanov introduced convenient classifications for linear differential operators and linear partial differential equations with operator (1.87). Let ak > 0, k = 1, ..., n. In accordance with I. A. Kipriyanov’s terminology the operator n  k=1

ak (Bγk )xk =

n  k=1

ak

γk ∂ ∂2 + ∂xk2 xk ∂xk

 =

n  k=1

ak

1 ∂ γk ∂ x γ xk k ∂xk k ∂xk

40

Transmutations, Singular and Fractional Differential Equations

is classified as B-elliptic operator, the operator  ∂ − ak (Bγi )xi a1 ∂x1 n

k=2

is classified as B-parabolic operator, the operator a1 (Bγ1 )x1 −

n 

ak (Bγk )xk

k=2

is classified as B-hyperbolic operator, and p 

ak (Bγk )xk −

k=1

n 

ak (Bγk )xk ,

1 < p < n,

k=p+1

is classified as B-ultrahyperbolic operator. We will use notations γ = (γ )x =

n 

(Bγk )xk

(1.88)

k=1

and γ = (γ )x = (Bγ1 )x1 −

n  (Bγi )xi .

(1.89)

i=2

For γ the term Laplace–Bessel operator is used. Let u = u(x) = u(x1 , . . . , xn ), f = f (x) = f (x1 , . . . , xn ). B-elliptic linear differential equations have the form n 

ak (Bνk )xk u = f.

(1.90)

k=1

Generalized axisymmetric potential theory (GASPT) for particular cases of (1.90) was studied by A. Weinstein [592,594,599,598]. Also Eq. (1.90) and related ones were considered by L. D. Kudryavtsev [303], P. I. Lizorkin and S. M. Nikol’skii [334], I. A. Kipriyanov [242], V. V. Katrakhov [226], and others. B-parabolic linear differential equations ∂u  a1 − ak (Bνk )xk u = f ∂x1 n

(1.91)

k=2

were studied by Ya. I. Zhitomirskii [610,609], M. I. Matiichuk [364,363], and A. B. Muravnik [394–397].

Basic definitions and propositions

41

B-hyperbolic linear differential equations have the form ∂ 2u  − ak (Bνk )xk u = f. ∂x12 k=2 n

a1

(1.92)

The study of this class of equations was begun in the works of L. Euler [128], S. D. Poisson [447], and J. G. Darboux [77] and continued in the works of R. Carroll and R. Showatler [56], A. Weinstein [599,595,593,596], D. Fox [147], I. A. Kipriyanov and L. A. Ivanov [246], S. A. Tersenov [564], etc. In [349,350,354] the B-ultrahyperbolic linear differential equation p 

ak (Bγi )xi u −

i=1

n 

ak (Bγi )xi u = 0

(1.93)

i=p+1

was considered. For equations of fractional order, a similar classification is adopted. Let α ∈ R, α > 0. B-elliptic linear differential equations of fractional order have the form

n α  ak (Bνk )xk u = f, k=1

B-parabolic linear differential equations of fractional order have the form α

n  ∂ − ak (Bνk )xk u = f, a1 ∂x1 k=2

and B-hyperbolic linear differential equations of fractional order have the forms

α α

n n   ∂2 ak (Bνk )xk u = f and (Bγ )t − ak (Bνk )xk u = f. a1 2 − ∂x1 k=2 k=1 Kipriyanov (see [242]) also introduced the B-polyharmonic of order p function. It is the function u = u(x) = u(x1 , ..., xn ) such that pγ u = 0,

(1.94)

where γ is operator (1.88). The B-polyharmonic of order 1 function will be called B-harmonic. The multi-dimensional Hankel transform Fγ given in Definition 12 acts to γ as the Fourier transform acts to the Laplace operator. This is proved in the next Lemma. Lemma 6. Let u ∈ Sev . Then Fγ [γ f ](ξ ) = −|ξ |2 Fγ [f ](ξ ).

(1.95)

42

Transmutations, Singular and Fractional Differential Equations

Proof. We have  Fγ [γ f ](ξ ) =

[γ f (x)] jγ (x; ξ )x γ dx

Rn+

=

2

n   i=1Rn +

3 1 ∂ γi ∂ x f (x) jγ (x; ξ )x γ dx. γ xi i ∂xi i ∂xi

Integrating by parts by variable xi and using formula (1.23), we obtain 2 3 n   1 ∂ γi ∂ f (x) x jγ (x; ξ ) x γ dx Fγ [γ f ](ξ ) = γ xi i ∂xi i ∂xi i=1Rn +

 n  2 (−ξi ) f (x) jγ (x; ξ )x γ dx = i=1

 = −|ξ |2

Rn+

f (x) jγ (x; ξ )x γ dx = −|ξ |2 Fγ [f ](ξ ).

Rn+

We will also need the formula n  (μ + n + 1) 2μ d x 2μ+2n = x . 2xdx (μ + 1)

(1.96)

1.4.2 Divergence theorem and Green’s second identity for B-elliptic and B-hyperbolic operators Here we give the generalization of the divergence theorem to the case of weighted divergence and derive Green’s second identity for γ and γ . Suppose that e = (e1 , ..., en ) is an orthonormal basis in Rn ,

 1 ∂ 1 ∂ , ..., γn ∇γ = γ x1 1 ∂x1 xn ∂xn is the first weighted nabla operator, F = F (x) = (F1 (x), ..., Fn (x)) is a vector field, and (∇γ · F ) =

1 ∂F1 1 ∂Fn + ... + γn γ x1 1 ∂x1 xi ∂xn

is the weighted divergence.

Basic definitions and propositions

43

In R n+ let us consider a domain G+ bounded by a piecewise smooth surface q 4 S + ∈R n+ . Thus, a surface can be represented as a union S + = Sk+ of a finite number k=1

of its parts Sk+ without common internal points. Let there be for each interior point a neighborhood within which the surface Sk+ is represented by parametric equations of the form xi = χi (y1 , ..., yn−1 ),

i = 1, ..., n,

where χi (y), y/ = (y1 , ..., y/n−1 ), has continuous first derivatives and the rank of the / ∂(χ1 ,...,χn ) / Jacobi matrix / ∂(y / is equal to n − 1. The vector 1 ,...,yn−1 ) / / / / / N = / / / /

e1 ∂χ1 (y) ∂y1

... ∂χ1 (y) ∂yn−1

... ... ... ...

en ∂χn (y) ∂y1

... ∂χn (y) ∂yn−1

/ / / / / / / / /

is normal to the surface S + in each point y ∈ S + with the exception of the junction points of surfaces Sk+ , k = 1, .., q, where it is not defined unambiguously and will not be considered. The vector ν =

N |N |

is determined to within sign. Of the two possible directions ν , we choose the external with respect to the domain G+ . Such a vector will be called the unit normal vector to the surface S + at the point y. We denote by ηi the angle which forms a vector ν with an axis xj . Then ν = e1 cos η1 + ... + en cos ηn . Theorem 10. Let G+ be the domain in R n+ such that each line perpendicular to the plane xi = 0, i = 1, ..., n, either does not cross G+ or has one common segment with G+ (maybe degenerating to a point) of the form αi (x ) ≤ xi ≤ βi (x ),

x =(x1 , ..., xi−1 , xi+1 , ..., xn ),

i = 1, ..., n.

If g=(g1 (x), ..., gn (x)) is a continuously differentiable in G+ vector field and γ γ F =(F1 (x), ..., Fn (x)), F1 (x)=x1 1 g1 (x), ..., Fn (x)=xnn gn (x), then the following formula is valid:   (∇γ · F ) x γ dx = ( g · ν ) x γ dS, (1.97) G+

S+

where ν is the external unit normal vector S + .

44

Transmutations, Singular and Fractional Differential Equations

Proof. Let i = 1, ..., n be fixed. If the part of the surface S + defined by the equation xi = βi (x ) is denoted by Su+ and the part of the surface S + defined by the equation xi = αi (x ) is denoted by Sd+ , then

(ν , ei ) =

⎧ −   2 ⎪ ⎪ ∂αi ⎨ 1+ +...+( ⎪ ⎪ ⎩

1

x ∈ Sd+ ,

1     2  2  ∂βi 2 ∂βi ∂β ∂β 2 1+ ∂x +...+ ∂x + ∂x i +...+ ∂xni

x ∈ Su+ .

∂x1



1

We have 

(∇γ · F ) x γ dx =

i−1

n   i=1G+

G+

2    ∂αi 2 ∂αi ∂α 2 +...+ ∂xni ∂xi−1 ) + ∂xi+1

i+1

1 ∂Fi γ x dx. γ xi i ∂xi

Let us consider  1 ∂Fi γ x dx γ xi i ∂xi G+

 =

γi−1 γi+1 γ γ x1 1 ...xi−1 xi+1 ...xnn dx1 ...dxi−1 dxi+1 ...dxn

βi (x )

αi (x )

Q

∂Fi dxi , ∂xi

where Q is a projection of G+ to xi = 0. Integrating by xi we obtain  1 ∂Fi γ x dx γ xi i ∂xi G+  γi−1 γi+1 γ x =β (x ) γ xi+1 ...xnn dx1 ...dxi−1 dxi+1 ...dxn . = Fi (x)|xii =αii (x ) x1 1 ...xi−1 Q

i−1 i+1 Let (x )γ = x1 1 ...xi−1 xi+1 ...xnn , dx = dx1 ...dxi−1 dxi+1 ...dxn . Then

γ

γ

γ

γ



1 ∂Fi γ x dx γ xi i ∂xi + G  = Fi (x1 , ..., xi−1 , βi (x ), xi+1 , ..., xn )(x )γ dx Q



−  = Q



Fi (x1 , ..., xi−1 , αi (x ), xi+1 , ..., xn )(x )γ dx

Q

Fi (x1 , ..., xi−1 , βi (x ), xi+1 , ..., xn )(ν , ei )

Basic definitions and propositions

5 ×  +



∂βi 1+ ∂x1

2

45



∂βi + ... + ∂xi−1

2



∂βi + ∂xi+1

2



∂βi + ... + ∂xn

2



(x )γ dx

Fi (x1 , ..., xi−1 , αi (x ), xi+1 , ..., xn )(ν , ei )

Q

5



     ∂αi 2 ∂αi 2 ∂αi 2 γ + ... + + + ... + (x ) dx × ∂xi−1 ∂xi+1 ∂xn   = Fi (x)(ν , ei )(x )γ dSu + Fi (x)(ν , ei )(x )γ dSd ∂αi 1+ ∂x1



2

Su+



=

 gi (x)(ν , ei )x γ dSu +

Su+

gi (x)(ν , ei )x γ dSd

Sd+



=

Sd+

gi (x) cos ηi x γ dS.

S+

Then 

(∇γ · F ) x γ dx =

n  

 gi (x) cos ηi x γ dS =

i=1S +

G+

( g · ν ) x γ dS.

S+

+ Remark 3. Suppose that a domain G+ ∈ R n+ is a union of domains G+ 1 , ..., Gm with+ out common internal points. Let each Gj be the domain in R n+ such that each line perpendicular to the plane xi = 0, i = 1, ..., n, either does not cross G+ j or has one + common segment with Gj (maybe degenerating to a point) of the form

αi (x ) ≤ xi ≤ βi (x ), j

j

x =(x1 , ..., xi−1 , xi+1 , ..., xn ),

i = 1, ..., n,

and F = (F1 (x), ..., Fn (x)), F1 (x) = x1 1 g1 (x), ..., Fn (x) = xnn gn (x), g = (g1 (x), ..., gn (x)), is a continuously differentiable in G+ vector field. Then the following formula is valid:   (∇γ · F ) x γ dx = ( g · ν ) x γ dS, (1.98) γ

G+

γ

S+

where S + ∈ R n+ is a piecewise smooth surface boundary and ν is the external unit normal vector S + . Theorem 11. Let G+ satisfy the conditions in Remark 3. If ϕ, ψ are twice continuously differentiable functions defined on G+ , such that   ∂ϕ  ∂ψ  = 0, = 0, i = 1, ..., n, ∂xi xi =0 ∂xi xi =0

46

Transmutations, Singular and Fractional Differential Equations

then Green’s second identity for the Laplace–Bessel operator has the form     ∂ψ ∂ϕ (ψγ ϕ − ϕγ ψ) x γ dx = −ϕ x γ dS. ψ ∂ ν ∂ ν G+

(1.99)

S+

Proof. If ϕ, ψ are twice continuously differentiable functions defined on a neighborhood of BR+ (n), such that   ∂ψ  ∂ϕ  = 0, = 0, i = 1, ..., n, ∂xi xi =0 ∂xi xi =0 one may choose F = ψ∇γ ϕ − ϕ∇γ ψ   γ ∂ϕ γ ∂ψ γ ∂ϕ γ ∂ψ − ϕ · x1 1 , ..., ψ · xnn − ϕ · xn n = ψ · x1 1 ∂x1 ∂x1 ∂xn ∂xn      ∂ψ ∂ψ ∂ϕ ∂ϕ γ γ = x1 1 ψ −ϕ −ϕ , ..., xnn ψ ∂x1 ∂x1 ∂xn ∂xn to obtain Green’s second identity for the Laplace–Bessel operator. In this case   ∂ϕ ∂ψ ∂ϕ ∂ψ g = ψ −ϕ , ..., ψ −ϕ ∂x1 ∂x1 ∂xn ∂xn is a continuously differentiable vector field defined on a neighborhood of BR+ (n), (∇γ · F ) = (∇γ · (ψ∇γ ϕ − ϕ∇γ ψ))

    n  1 ∂ 1 ∂ γi ∂ϕ γi ∂ψ = ψ · xi − γi ϕ · xi γ ∂xi ∂xi xi i ∂xi xi ∂xi i=1

n  1 ∂ γi ∂ϕ 1 ∂ψ γi ∂ϕ · xi + ψ · γi x = γi ∂x ∂x ∂xi i ∂xi x x i i i i i=1  1 ∂ϕ 1 ∂ γi ∂ψ γi ∂ψ − γi ·x − ϕ · γi x xi ∂xi i ∂xi xi ∂xi i ∂xi =

n 

 ψBγi ϕ − ϕBγi ψ = ψγ ϕ − ϕγ ψ,

i=1

  ∂ϕ ∂ϕ ( g · ν ) = ψ cos η1 + ... + ψ cos ηn ∂x1 ∂xn   ∂ψ ∂ψ ∂ψ ∂ϕ − ϕ cos η1 + ... + ϕ cos ηn = ψ −ϕ , ∂x1 ∂xn ∂ ν ∂ ν and applying (1.98) we obtain (1.99).

Basic definitions and propositions

47

Let

  ∂ γ γ ∂ γ ∂ ♦γ = x1 1 , −x2 p+1 , ..., −xnn . ∂x1 ∂x2 ∂xn

Then (∇γ · ♦γ ) = γ . Theorem 12. Let G+ satisfy the conditions in Remark 3. If ϕ, ψ are twice continuously differentiable functions defined on G+ , such that   ∂ϕ  ∂ψ  = 0, = 0, i = 1, ..., n, ∂xi xi =0 ∂xi xi =0 then Green’s second identity for the B-ultrahyperbolic operator has the form     ∂ψ ∂ϕ (ψγ ϕ − ϕγ ψ) x γ dx = (1.100) −ϕ x γ dS, ψ ∂ τ ∂ τ G+

S+

where τ = (cos η1 , − cos η2 , ..., − cos ηn ). Proof. Let F = ψ♦γ ϕ − ϕ♦γ ψ      ∂ψ ∂ψ ∂ϕ ∂ϕ γ1 γ2 −ϕ −ϕ , −x2 ψ , ..., = x1 ψ ∂x1 ∂x1 ∂x2 ∂x2   ∂ψ ∂ϕ γ −ϕ , −xnn ψ ∂xn ∂xn so that

     ∂ϕ ∂ψ ∂ϕ ∂ψ ∂ψ ∂ϕ g = ψ −ϕ ,− ψ −ϕ −ϕ , ..., − ψ , ∂x1 ∂x1 ∂x2 ∂x2 ∂xn ∂xn (∇γ · F ) = ψγ ϕ − ϕγ ψ,     n  ∂ψ ∂ψ ∂ϕ ∂ϕ cos η1 − ϕ cos η1 − cos ηi − ϕ cos ηi ( g · ν ) = ψ ψ ∂x1 ∂x1 ∂xi ∂xi i=2

∂ϕ ∂ψ =ψ −ϕ , ∂ τ ∂ τ where τ = (cos η1 , − cos η2 , ..., − cos ηn ). Then applying (1.98) we obtain (1.100). As a corollary of Theorem 12 we obtain the formula of integral with weight x γ = γ γ x1 1 ...xnn , γi > 0, i=1, ..., n, when the region of integration is a part of a ball belonging to the orthant Rn+ .

48

Transmutations, Singular and Fractional Differential Equations

6 We will denote the part of a ball |x| ≤ r, |x| = x12 + ... + xn2 belonging to Rn+ by Br+ (n). The boundary of Br+ (n) denoted by Sr+ (n) consists of a part of a sphere {x ∈ Rn+ : |x|=r} and of parts of coordinate hyperplanes xi =0, i=1, . . ., n, such that |x i | ≤ r. 2 (B + (n)) the following formula is valid: Corollary 2. For w ∈ Cev R





 (γ w(x)) x dx = γ

BR+ (n)

SR+ (n)

 ∂w(x) γ x dS, ∂ ν

(1.101)

where ν is external normal to the SR+ (n). Now let us consider integration by Sr+ (n) with a weight of the form  xγ =

u(x)x γ dSr ,

n

γ

xi i ,

i=1

Sr+ (n)

where dSr is a surface element of Sr+ (n). It is easy to see that the formulas  u(x)x dx = r γ

n+|γ |

Br+ (n)

 u(rx)x γ dx

(1.102)

B1+ (n)

and  u(x)x dSr = r γ

n+|γ |−1

Sr+ (n)

 u(rx)x γ dS,

(1.103)

S1+ (n)

where dS is a surface element of S1+ (n), are valid. Let a function f (x) be integrable by Br+ (n) and a function g(t) be continuous of variable t, t ∈ [0, ∞). For integration by Br+ (n) with weight x γ formulas 

r g(|x|)f (x) x dx = γ

Br+ (n)



g(λ)λn+|γ |−1 dλ

S1+ (n)

are valid (see [349]).

f (λx)x γ dω,

(1.104)

S1+ (n)

0

f (rx)x γ dS = r 1−n−|γ |



d dr

 f (z)zγ dz Br+ (n)

(1.105)

Basic definitions and propositions

-

Let us find

49

x γ dS taking into account the formula (see [73]). We have

S1+ (n)





y1α1 ...ynαn dy1 ...dyn =

... y1 +...+yn ≤1,y1 ≥0,...,yn ≥0

(α1 + 1)...(αn + 1) , (α1 + ... + αn + n + 1) (1.106)

where α1 , ..., αn ∈ R. Using new coordinates y1 = x12 , ..., yn = xn2 we rewrite (1.106) in the form  (α1 + 1)...(αn + 1) x12α1 +1 ...xn2αn +1 dx1 ...dxn = n , 2 (α1 + ... + αn + n + 1) B1+ (n)

or putting 2αi + 1 = γi , i = 1, ..., n,       γ12+1 ... γn2+1   . x γ dx = n  n+|γ | + 1 2 + 2 B1 (n)

Then from (1.102) and (1.105), for u = 1, applying the formula (z + 1) = z(z), putting r = 1, and denoting the resulting integral by |S1+ (n)|γ , we obtain

|S1+ (n)|γ =

n 

 x γ dS =

S1+ (n)

 

i=1

2n−1 



γi +1 2



n+|γ | 2

.

(1.107)

1.4.3 Tricomi equation The Tricomi equation is a second order partial differential equation of mixed elliptichyperbolic type for u(x; y) with the following form: uxx + xuyy = 0. It was first analyzed in the work by Francesco Giacomo Tricomi (1923) on the wellposedness of a boundary value problem. The equation is hyperbolic in the half-plane x < 0, is elliptic in the half-plane x > 0, and degenerates on the line x = 0. Its characteristic equation is dy 2 + xdx 2 = 0, whose solutions are 3 2 y ± (−x) 2 = C 3

50

Transmutations, Singular and Fractional Differential Equations

for any constant C, which are real for x < 0. The characteristics comprise two families of semicubical parabolas lying in the half-plane x < 0, with cusps on the line x = 0. This is of hyperbolic degeneracy, for which the two characteristic families coincide, perpendicularly to the line x = 0. 3 For ±x > 0, set τ = 23 (±x) 2 . Then the Tricomi equation becomes the classical elliptic or hyperbolic Euler–Poisson–Darboux equation: uτ τ ± uyy +

β uτ = 0. τ

The index β = 13 determines the singularity of solutions near τ = 0, equivalently, x = 0. Many important problems in fluid mechanics and differential geometry can be reduced to corresponding problems for the Tricomi equation, particularly transonic flow problems and isometric embedding problems. The Tricomi equation is a prototype of the generalized Tricomi equation: uxx + K(x)uyy = 0. For a steady-state transonic flow in R2 , u(x; y) is the stream function of the flow, K(x) and x are functions of the velocity, which are positive at subsonic and negative at supersonic speeds, and y is the angle of inclination of the velocity. The solutions u(x; y) also serve as entropy generators for entropy pairs of the potential flow system for the velocity. For the isometric embedding problem of two-dimensional Riemannian manifolds into R3 , the function K(x) has the same sign as the Gaussian curvature. A closely related partial differential equation is the Keldysh equation xuxx + uyy = 0. It is hyperbolic when x < 0, is elliptic when x > 0, and degenerates on the line x = 0. Its characteristics are 1 1 y ± (−x) 2 = C 2 for any constant C, which are real for x < 0. The two characteristic families are (quadratic) parabolas lying in the half-plane x < 0 and coincide tangentially to the degenerate line x = 0, which is of parabolic degeneracy. For ±x > 0, the Keldysh equation becomes the elliptic or hyperbolic Euler–Poisson–Darboux equation with index β = − 14 by setting τ = 12 (±x) 12 . Many important problems in continuum mechanics can also be reduced to corresponding problems for the Keldysh equation, particularly shock reflection–diffraction problems in gas dynamics.

1.4.4 Abstract Euler–Poisson–Darboux equation The abstract Euler–Poisson–Darboux equation has the form Au = (Bγ )t u,

u = u(x, t; γ ),

(1.108)

Basic definitions and propositions

51

where A is a linear operator acting only by variable x = (x1 , ..., xn ). Lemma 7. Let uk = u(x, t; k) be a solution to Eq. (1.108). The two recurrent formulas uk = t 1−k u2−k ,

(1.109)

ukt = tuk+2

(1.110)

are valid. Proof. Let us show (1.109). Putting w = t k−1 v, v = uk , we obtain k−1 w + t k−1 vt , t wtt = (k − 1)(k − 2)t k−3 v + (k − 1)t k−2 vt + (k − 1)t k−2 vt + t k−1 vtt (k − 1)(k − 2) = w + 2(k − 1)t k−2 vt + t k−1 vtt , t2 (k − 1)(k − 2) 2−k w + (2 − k)t k−2 vt , wt = − t t2 2−k wt = 2(k − 1)t k−2 vt + t k−1 vtt + (2 − k)t k−2 vt wtt + t   k = t k−1 vtt + vt t wt = (k − 1)t k−2 v + t k−1 vt =

or

  2−k k k−1 wt = t vtt + vt . wtt + t t

If w = t k−1 v satisfies the equation Aw = wtt +

2−k wt , t

then using (1.111) we get   k k−1 k−1 t Av = t vtt + vt , t which means that v satisfies the equation Av = vtt +

k vt . t

Denoting w = u2−k we obtain (1.109). Let us prove now (1.110). Putting tw = vt , v = uk , we can write wt = −

1 1 vt + vtt , 2 t t

(1.111)

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Transmutations, Singular and Fractional Differential Equations

wtt =

2 2 1 vt − 2 vtt + vttt , 3 t t t

and k+2 k+2 k+2 wt = − 3 vt + 2 vtt . t t t We have wtt +

k+2 2 2 1 k+2 k+2 wt = 3 vt − 2 vtt + vttt − 3 vt + 2 vtt t t t t t t  1 k k 1 k k = vttt − 3 vt + 2 vtt = vttt − 2 vt + vtt t t t t t t   1 ∂ k = vtt + vt t ∂t t

or k+2 1 ∂ wtt + wt = t t ∂t



 k vtt + vt . t

(1.112)

If w = 1t vt satisfies the equation Aw = wtt +

k+2 wt , t

then using (1.112) we obtain   1 ∂ k 1 ∂ Av = vtt + vt , t ∂t t ∂t t which means that v satisfies the equation k Av = vtt + vt . t Denoting w = uk+2 , v = uk , we get (1.110). For A = γ formula (1.109) is proved in [147] and formula (1.110) is proved in [16]. Both formulas are present in Weinsteins’ article [595], but in the case when in Eq. (1.108) for each of variables xi , i = 1, ..., n, the second derivative acts. These recurring formulas will allow using the solution uk to (1.108) to get a solution to the same equation, but with the parameters k + 2 and 2 − k, respectively. So, if the function u(x, t; k) is a solution of the abstract Euler–Poisson–Darboux equation Au = (Bk )t u, where A is a linear operator acting only by x = (x1 , ..., xn ), then the function t 1−k u(x, t; 2 − k) also is a solution to this equation.

Basics of fractional calculus and fractional order differential equations

2.1

2

Short history of fractional calculus and fractional order differential equations

Operators of fractional integro-differentiation play an important role in many modern fields of mathematics. For special function theory its importance is reflected in the title of the well-known paper [253] “All special functions are fractional integrals of elementary functions!” (But there is a remark of Professor A. A. Kilbas – all but Fox functions.) In this section we list essential one- and multi-dimensional fractional operators and include some historical and priority information.

2.1.1 One-dimensional fractional derivatives and integrals Euler’s introduction in 1729 of the gamma function (1.1) allowed to expand the concept of factorial to the case of a fractional value of the argument. This, in turn, allowed Euler to note that the concept of the n-th order derivative of the power function x p acquired meaning for a nonintegral n. Namely, let n, p ∈ N and p ≥ n. It is well known that (x p ) = px p−1 ,

(x p ) = p(p − 1)x p−2 , ...,

(x p )(n) = p(p − 1)...(p − n + 1)x p−n or (x p )(n) =

p! x p−n . (p − n)!

(2.1)

Expression (2.1) can also be meaningful for noninteger n and p. Namely, by virtue of the well-known equality for the gamma function (p + 1) = p(p − 1)...(p − n + 1)(p − n + 1), Transmutations, Singular and Fractional Differential Equations With Applications to Mathematical Physics https://doi.org/10.1016/B978-0-12-819781-3.00008-2 Copyright © 2020 Elsevier Inc. All rights reserved.

54

Transmutations, Singular and Fractional Differential Equations

formula (2.1) can be written as (p + 1) dn p x = x p−n dx n (p − n + 1) and can be used for all real n. Further, we note that in 1823, Liouville formally expanded the formula for the d n bx derivative of the integer order of the exponential dx n e , b ∈ R, to the derivative of α d bx the exponential of arbitrary order dx α e . Specifically, d α ebx = bα ebx . dx α

(2.2)

Based on formula (2.2) we can formally write the derivative of order α ∈ R of an arbitrary function f represented by the series f (x) =

∞ 

ck ebk x .

k=0

Therefore, ∞

d α f (x)  = ck bkα ebk x . dx α k=0

The limitations of this definition are related not only to the convergence or divergence of the series but mainly to the fact that not every function can be represented as a series in exponentials (in modern terms as Dirichlet series). In 1847 B. Riemann proposed the definition of a fractional integral which is now used as standard. His definition was based on a generalization of the formula for an n-fold integral of the form x

x dx...

a



x dx

a



a



1 f (x)dx = (n − 1)!

x (x − t)n−1 f (t)dt,

(2.3)

a

n

x ∈ [a, b],

a, b ∈ R.

Generalizing formula (2.3) to the case of real n = α > 0 one can obtain α (Ia+ f )(x) =

1 (α)

x (x − t)α−1 f (t)dt,

x > a.

a α f. Liouville in 1832 introduced a fractional integral operator in the form close to Ia+ α is called the left-sided fractional Riemann–Liouville integral of orCurrently, Ia+ der α > 0.

Basics of fractional calculus and fractional order differential equations

55

It is known that the derivative of the integral with a variable upper limit of the continuous function equals the integrand in which the integration variable is replaced by the upper limit: d dx

x f (t)dt = f (x). a

d Thus, the differentiation operator dx can be interpreted as the left inverse to the intex  gration operator . If we can interchange the integration and differentiation operators, a

then we obtain x

d f (t) dt = f (x) − f (a) dt

a

and

x

will no longer be inverse for

a

d dx

for all functions.

So we can see that in order to obtain the Riemann–Liouville left-sided fractional derivative we should solve an integral equation 1 (α)

x (x − t)α−1 f (t)dt = g(x). a

This equation was already solved by Abel in 1823 in connection with the tautochrone problem, and the solution has the form α f )(x) = (Da+

1 (n − α)



d dx

n x a

f (t)dt , (x − t)α−n+1

α f )(x) is now called the leftwhere n = [α] + 1, α > 0. The expression (Da+ sided fractional Riemann–Liouville derivative. Similarly the right-sided fractional Riemann–Liouville integral and derivative have been introduced, in which an integration is taken from x to b. Historically it were really Liouville and Riemann who considered the above forms of fractional derivatives and integrals. Riemann considered them on a finite segment and Liouville on the half-axes. But these considerations were not strict; Liouville suggested that every function is represented by the series in exponentials, and Riemann used for the derivation consciously divergent integrals. The first strict approach was developed in fact by A. V. Letnikov [498], and he was also the first researcher who applied fractional integrals for transmutation [273,498]. Important generalizations of Abel’s equation were studied among others by N. Ya. Sonine [554].

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Transmutations, Singular and Fractional Differential Equations

The resulting definition of the fractional derivative is not very similar to the definition of the ordinary derivative as a limit of the form f  (x) =

f (x) − f (x − h) df = lim , dx h→0 h

h = x.

(2.4)

However, A. Grünwald in 1867 and A. V. Letnikov in 1868 proposed the construction of fractional differentiation of the form N−1 h−α  (k − α) f (x − kh) , N ∈ N, α < N −1, N→∞ (−α) (k + 1)

α (Da+ f )(x)= lim

k=0

x −a , h= N

which is a natural generalization of formula (2.4) that is also convenient for computer approximations. The fractional Grünwald–Letnikov integral for α > 0 has the form α (Ia+ f )(x) =

N−1 hα  (k + α) x −a lim f (x − kh) , h = . N→∞ (α) (k + 1) N

(2.5)

k=0

It is known [494] that for an integrable on [a, b] function f (x) the limit (2.5) exists almost for all x and (2.5) coincides with the left-sided fractional Riemann–Liouville integral  N−1 hα  (k + α) f (t)dt 1 lim , f (x − kh) = N→∞ (α) (k + 1) (α) (x − t)1−α x

k=0

a

x −a . h= N

Riemann–Liouville and Grünwald–Letnikov integro-differential operators were adapted for x from the finite segment. It is important to note that an idea on which a definition of the Grünwald–Letnikov derivative is based, namely, the usage of finite differences, was used again after many years for a generalization of Sobolev spaces of fractional orders – Besov spaces [28]. Next many new forms of fractional integrals and derivatives appeared in particular adapted for x from the axes or semiaxes. In 1917 Weil defined fractional integration suitable for periodic functions:

I±(α) f

∼

∞  k=−∞

(±ik)

−α

fk e

ikx

,

1 fk = 2π

2π 0

e−ikt f (t)dt,

f0 = 0,

Basics of fractional calculus and fractional order differential equations

57

and showed that I±(α) for 0 < α < 1 may be written in the form (α) (I+ f )(x) =

(α)

(I− f )(x) =

1 (α) 1 (α)

x (x − t)α−1 f (t)dt, −∞ ∞

(t − x)α−1 f (t)dt. x

We should also mention the form of fractional differentiation which was presented by Marchaud in 1927 and has the form ∞ (D f )(x) = c α

(lt f )(x) dt, t α+1

α > 0,

0

where lt is the forward finite difference: (lt f )(x) =



l   l (−1)k f x + (l − k)t . k k=0

Also backward finite difference may be used, (∇tl f )(x) =



l  k l (−1) f (x − kt), k k=0

and central finite difference, (δtl f )(x) =



l  l k l (−1) −k t . f x+ 2 k k=0

To use finite differences is one of the ways of regularizing the integral in Dα . As a conclusion let us mention that important results concerning fractional operators were obtained by different mathematical schools. Let us mention just some of them. In Minsk, Belarus, the founder of a well-known school on fractional calculus and its applications to fractional differential equations was Anatoly Kilbas; his ideas are now developed by his disciples S. Ragosin, A. Koroleva, A. Grin’ko, O. Skoromnik, A. Shlapakov, and others. In Sofia, Bulgaria, important results were obtained by I. Dimovski, V. Kiryakova, J. Paneva-Konovska, and others. The town Nal’chik, Russia, is sometimes called the “Mekka” of fractional calculus due to a well-known school of fractional calculus founded and developed by A. M. Nahushev. From his books [413–415] most mathematicians in the Soviet Union

58

Transmutations, Singular and Fractional Differential Equations

and Russia first learned about and studied fractional calculus. He brought up many talented disciples now working on fractional calculus and its applications. Now this school is headed by Arsen Pskhu – also his disciple. An important impact was made by the famous Voronezh mathematical school, originally headed by M. Krasnosel’skii and S. Krein. In their books on differential equations in Banach spaces fractional powers of operators were used for the general theory (cf. [294,301]). Also let us mention works of mechanics of the Voronezh school which developed applications of fractional calculus. An essential impact was made by S. Meshkov, Yu. Rossikhin, and M. Shitikova. Now this work is continued at Voronezh Polytechnical (former Construction and Architecture) University. There, the International Scientific Center named after professor Yu. A. Rossikhin is headed by M. Shitikova. Recently results on fractional calculus and applications were summed up in eight volumes of the “Handbook of Fractional Calculus with Applications” [161]. But we need also mention that nowadays many unprofessional and inappropriate generalizations of fractional operators have appeared which cannot be considered as “true” generalizations of fractional operators. Some of them do not obey the semigroup property, some are simply reduced to multiplication by a function, and some are similar to fractional operator’s resolvents and are also not “true” generalizations. The project of Yu. Luchko [340] is devoted to critics of inappropriate generalizations of fractional operators (cf. also papers [211,432,433,556,563]). This concludes our brief review of the history of one-dimensional fractional integrals and derivatives. A detailed history of this issue is set forth, for example, in [494].

2.1.2 Fractional derivatives in mechanics Methods of fractional calculus and fractional integro-differential equations are widely used in different applied sciences. Among them these methods turned out to be very effective in theory and numerically in mechanics, especially in viscoelasticity and construction mechanics. We want to stress that a remarkable impact in these branches was made by Soviet and Russian scientists. Starting from fundamental works of A. N. Gerasimov [179] and Yu. N. Rabotnov in 1947–1948, it was continued in works and books of Yu. N. Rabotnov [464,484] and many of his disciples. Based on works and researches of Rabotnov in the USSR period, there was a standard used for foundations of buildings, so we may say that every building during the period of the 1950s–1980s was constructed with the use of fractional calculus! An essential impact on the field was also made by D. Shermegor, M. Rosovskii, A. Rzhanitsyn, and others. And also a serious impact was made by the Voronezh school of mechanics, namely, by S. Meshkov [378], Yu. Rossikhin, and M. Shitikova. For the real history of applications of fractional calculus in mechanics, cf. the survey papers of Yu. Rossikhin and M. Shitikova [482,483] and also [574].

Basics of fractional calculus and fractional order differential equations

59

2.1.3 Fractional powers of multi-dimensional operators The most developed type of fractional multi-dimensional integrals are Riesz potentials, which generalize both the Newton potential to the fractional case and the Riemann–Liouville fractional integral to the multi-dimensional case. Let us start from the classical Newton potential. If f is an integrable function with compact support, then the Newton potential of f is the convolution product (see [585])  VN f (x) = v(x − y)f (y)dy, Rn

where

v(x) =

log |x| 1 2−n n(2−n)ωn |x| 1 2π

n = 2, n = 2,

ωn is a volume of unit ball Rn .

The Newton potential VN of f is the solution to the Poisson equation VN = f,

=

n  ∂2 , ∂xi2 i=1

x = (x1 , ..., xn ) ∈ Rn ,

and therefore, it can be considered as a negative degree of the Laplace operator: VN f = −1 f. The term potential is due to Green [204] (1828) and Gauss [176] (1840) (see [267]). Along with the Newtonian potential, the wave potential of the function f has found wide applications (see [585]):  VW f (x) = ε(x − y)f (y)dy, Rn

where ε is a fundamental solution of the wave operator. For the wave potential VW the following equality is true: VW = f,

=

∂2 ∂2 ∂2 − 2 − ... − 2 , 2 ∂xn ∂x1 ∂x2

and therefore, it can be considered as a negative degree of the D’Alembert operator: VW f = −1 f . Marcel Riesz was a Hungarian mathematician who first introduced fractional powers of Laplace and D’Alembert operators. On November 1, 1933 M. Riesz presented the report “Integral of Riemann–Liouville, Potential, Waves” at the Physiographic Society of Lund, where he generalized the fractional Riemann–Liouville integral to the multi-dimensional case and also generalized Newtonian and wave potentials to the case of fractional power of a kernel. However, the first publication on this subject was

60

Transmutations, Singular and Fractional Differential Equations

made by O. Frostman in 1935 [154], who was a PhD student of M. Riesz at Lund University. Frostman wrote that in addition to the Riemann–Liouville integrals of elliptical character studied in [154], in the speech in 1933 M. Riesz also considered integrals of hyperbolic and parabolic character. On the International Mathematical Congress in Oslo in 1936 M. Riesz published four abstracts in different sections, two of which [472,473] were about the solution to the Cauchy problem for the wave equation. He found a solution to the equation u = f,

u = u(x1 , x2 , ..., xn ),

f = f (x1 , x2 , ..., xn ).

In order to solve (2.6) Riesz presented a potential  1 f (Q)rPα−n I α f (P ) = Q dQ, Hn (α)

(2.6)

(2.7)

DSP

 where r=rP Q = (x1 − ξ1 )2 − (x2 − ξ2 )2 − ... − (xn − ξn )2 is a Lorentz distance, P =(x1 , x2 , ..., xn ), Q=(ξ1 , ξ2 , ..., ξn ), DSP ={Q ∈ Rn :rP2 Q < 0, x1 − ξ1 > 0} is a retro   n−2 grade light cone, α is a positive real number, and Hn (α) = π 2 2α−1  α2  α+2−n . 2 The integral (2.7) converges for α > n − 2 and satisfies the relations I α I β = I α+β and I α+2 = I α . But in order to solve (2.6) using (2.7) is it necessary to extend analytically I α with respect to α to the values α ≤ n − 2 and show that I 0 = I under the suitable regularity conditions. Riesz emphasized that the starting point of the analytical continuation of potentials is the concept of finite-part integrals presented by J. Hadamard in [157,158]. Hadamard’s regularization reduces to dropping some divergent terms of a divergent integral and keeping the finite part. Riesz showed that this can be interpreted as taking the meromorphic continuation of a convergent integral. In [472] the Green formula for the potential (2.7) was presented in the following form: 1 α+2 α f (P ) = I f (P ) − I Hm (α + 2)    ∂rPα+2−n ∂f (Q) α+2−n Q − f (Q) r dS, × ∂ν P Q ∂ν

(2.8)

SP

where S P denotes the portion of the surface cut by S from the cone, ν is the outer conormal to S P , and S is a surface such that the generators of the retrograde cone belonging to the points P are considered cut only at a single point. M. Riesz presented the results of the abstracts [472,473] in a more detailed form in 1939 in the article [474]. In [151] Riesz potential was considered as a generalization of the Riemann– Liouville integral. In 1949 Riesz published an extended paper [475] about two forms of potentials: with Euclidean and with Lorentz distances. Such potentials are now called the elliptic and hyperbolic Riesz potentials and have the following forms, respectively:  1 α I f (P ) = f (Q)r α−n dQ γn (α) Rn

Basics of fractional calculus and fractional order differential equations

and (2.7). Here γn (α)=

n   π 2 2α  α2  n−α , r= (x1  2

61

− ξ1 )2 + (x2 − ξ2 )2 + ... + (xn − ξn )2 is a

Euclidean distance. In [475] it was shown that Iα+2 f (P ) = −Iα f (P )

and

α+2 α I f (P ) = I f (P ),

α were conproving the Green formula (2.8), and analytical continuations of Iα and I P structed. M. Riesz considered the potential (2.7) not only over DS but also over the + = {x : x12 ≥ x22 + ... + xn2 , x1 ≥ 0}. cone K+ As for the theory of elliptic Riesz potentials during the 1940s and 1950s, it has been the subject of many independent studies for a wide variety of generalizations (see [87,215]). Besides the studies of the Dirichlet problem, elliptic Riesz potentials were adapted to the study of the sign of the integral of energy and the principle of the maximum. Popularization of the theory of distributions presented in books of L. Schwartz [495] and I. M. Gelfand and G. E. Shilov [177] had a significant effect on the further development of the Riesz potential theory. In [177,495] generalized functions gener2 ated by quadratic forms r λ , (P ± i0)λ (x ∈ Rn , r = |x|, P = x12 + ... + xp2 − xp+1 − 2 ... − xn ) and their Fourier transforms were studied. It turned out that it is convenient to α as convolutions with functions r λ and (P ± i0)λ , present Riesz potentials Iα and I α respectively (p = 1 for I ). In addition, the Riesz ultrahyperbolic potential immediately appeared as a convolution with (P ± i0)λ , p > 1. Some fundamental problems were not solved by M. Riesz; among them are boundedness estimates, finding spaces invariant with respect to potentials, and inversion of α. potentials Iα and I Necessary and sufficient conditions for boundedness of Iα from Lp (Rn ) to Lq (Rn ) (0 < α < n, 1 < p < n/α, 1/g = 1/p − α/n) were given by Sobolev [544] in 1938. As for the elliptic potential Iα the space invariant with respect to this potential is a Lizorkin space  = {F ψ ∈ S(Rn ), ψ ∈ }, where S(Rn ) is a Schwartz space, F is a Fourier transform, and  = {ψ ∈ S(Rn ), (D j ψ)(0) = 0, |j | = 0, 1, 2, ...}. This fact was first noted by Semyanistyi [496] in 1960 and Lizorkin [331] in 1963. The inversion of the elliptic Riesz potential was constructed by S. G. Samko in the form of a hypersingular integral [493] in 1976 and by the method of approximate operators [492] in 1998. In [173] and [194] kernels of fractional powers which are the set of all positive powers of the operator generated by the Green function for the Laplace equation were studied. Riesz potentials with Euclidean distance was also studied in [15,162,241,431, 475,485,494,495,543,575]. Generalization of the Riesz hyperbolic potential or potential with Lorentz distance to the ultrahyperbolic case was considered by Nozaki [428] in 1964. I. A. Kipriyanov and L. A. Ivanov [244,245] in 1986–1987 introduced the following modification of the hyperbolic Riesz potential:  1 α I f (x) = f (y)shα−n (rxy )dz. (2.9)  Hn (α) Dx

62

Transmutations, Singular and Fractional Differential Equations

Such modification is connected with the Lorentz space. In [244,245] the Fourier transform was obtained, an analogue of the Hardy–Littlewood–Sobolev theorem on potential estimates was proved, and an application to the Cauchy problem for the Euler–Poisson–Darboux equation on the Lorentz space was given. Potential studied by I. A. Kipriyanov and L. A. Ivanov was used by S. Helgason [163] to represent a function through its orbital integrals in isotropic Lorentzian manifolds. V. A. Nogin and E. V. Sukhinin [425,426] in 1992–1993 completely solved the problems of the boundedness and inversion of the Riesz hyperbolic and ultrahyperbolic Riesz potentials of the form  α−n 1 α I f (x) = f (x − y)[y12 − y22 − ... − yn2 ] 2 dy, n − 2 < α < n, Hn (α) + K+

where f ∈ Lp (Rn ), 1 < p < αn , and  IPα ±i0 f (x) = C

f (x − y)(P ± i0)

α−n 2

dy,

n − 2 < α < n,

Rn ± n−s πi

2 2 where P (y) = y12 + ... + ys2 − ys+1 − ... − yn2 , C = e γn (α) , respectively. Boundedness was proved using the Marcinkiewicz interpolation theorem and inversion was obtained by the method of approximate operators. Another approach to inversion of Riesz ultra-hyperbolic was given in [58]. Another widely studied potential is the Bessel potential  Gα f (x) = Gα (x − y)f (y)dy, α > 0,

Rn

where Gα (x) =

2

2−n−α 2

n 2

π 

α 2

K n−α (|x|) 2

|x|

n−α 2

, α

which realizes the fractional powers of the operator (I − )− 2 . Here Kν is the modified Bessel functions of the second kind (1.17). Such potential appeared in the papers of N. Aronzajn and K. T. Smith [10] in 1961 and Calderon [44] in 1961. The space of Bessel potentials is sometimes called the Liouville space of fractional smoothness α. n This space is an extension of the Sobolev spaces Lm p (R ) to the case of fractional order α; that is why it is also called Sobolev space of fractional order. Results about the space of Bessel potentials were obtained by I. Stein [542] in 1961 in the case 0 < α < 2 and by Lizorkin [332] in 1970 in the general case. The inversion of Bessel potentials using hypersingular integrals was given by V. A. Nogin [422–424] in 1981–1985. Necessary and sufficient conditions for the strong and weak boundedness of the Riesz potential on Orlicz spaces were given in [209].

Basics of fractional calculus and fractional order differential equations

63

The potential theory comes from mathematical physics. The most well-known areas of its application are electrostatic and gravitational theory, probability theory, scattering theory, and biological systems. The first application of Riesz potentials was given by M. Riesz himself, and it was a solution of the Maxwell equations for the electromagnetic field (see [475], p. 146, and [152]). The Maxwell equations are fundamental equations of classical electrodynamics and optics. The equations completely describe all electromagnetic phenomena in an arbitrary environment and give a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar, etc. So the Riesz potential can be used for studying realistic singleparticle energy levels. An interesting fact was noted in [43]. Namely, in this paper it was shown that the elliptic Riesz potential can be interpreted as a transmutation operator. More precisely, the operator square root of the Laplacian was obtained from the harmonic extension problem to the upper half-space as the operator that maps the Dirichlet boundary condition to the Neumann condition. The same result but for hyperbolic Riesz potentials was obtained in [120]. Spaces of Riesz and Bessel potentials are used in connection with problems which arise in the theory of integral equations of the first kind with a potential type or oscillating kernel. P. I. Lizorkin in [333] showed that the space of elliptic Riesz potentials is the functional completion of infinitely differentiable functions finite in Rn in lpr (Rn ), where lpr (Rn ) is the class of such functions that are Liouville derivatives of order r belonging to Lp (Rn ). In [195–201] optimal embedding of spaces of Bessel and Riesz type potentials are obtained. S. Helgason (see [163], p. 137) solved the problem of determining a function by its orbital integrals over Lorentzian spheres. In other words he built the inversion of the generalized Radon transform in the isotropic Lorentz spaces. It turned out that the inverse operator to such a Radon transform would be the operator (2.9).

2.1.4 Differential equations of fractional order Here we give brief overview of the results for differential equations of fractional order on a finite interval of the real axis following [241,494]. The differential equations of fractional order α ∈ / N have the following general form: F [x, y(x), D α1 y(x), ..., D αm y(x)] = f (x).

(2.10)

Here x = (x1 , ..., xn ) is a point of n-dimensional Euclidean space Rn , F, f are given functions, and D αk are fractional differentiation operators of real αk > 0 or complex Re αk > 0 numbers, k = 1, 2, ..., m. The most studied equation of fractional order is α (Da+ y)(x) = f [x, y(x)],

Re α > 0,

x > a,

a ∈ R,

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Transmutations, Singular and Fractional Differential Equations

α is the left-sided Riemann–Liouville fractional derivative on a segment where Da+ (2.14) or on semiaxes (2.28) (in this case a = 0). To this equation initial conditions α−k (Da+ y)(a+) = bk ,

Re k = 1, ..., n,

n = [Re α] + 1,

x > a,

bk ∈ C,

α−k y)(a+) means that the limit is taken at almost all points are added. The notation (Da+ of the right-sided neighborhood (a, a + ε) (ε > 0). Let us note that α−n n−α y)(a+) = (Ia+ y)(a+), (Da+

0 (Da+ y)(a+) = y(a),

n−α is the Riemann–Liouville fractional integration operator defined by (2.12) where Ia+ or (2.26) (in this case a = 0), accordingly. Such problems are called Cauchy type problems. When 0 < Re α < 1 the weighted Cauchy type problem is α (Da+ y)(x) = f [x, y(x)],

lim (x − a)1−α y(x) = c,

x→a+0

c ∈ C.

In [17] it was shown that if f (x) ∈ L(a, b), the Cauchy type problem for the linear differential equation α−k α (Da+ y)(x) − λy(x) = f (x), (Da+ y)(a+) = bk ,

k = 1, ..., n, n = [Re α] + 1, bk ∈ C, has the unique solution y(x) in some subspace of L(a, b) given by y(x) =

n 

bk x α−k Eα,α−k+1 (λ(x − a)α )

k=1

x +

(x − t)α−1 Eα,α (λ(x − t)α )f (t)dt, a

where Eα,α is the Mittag-Leffler function defined by (1.39). In particular for 0 0, λ > 0, ( GC D0+

y (k) (0) = bk , k = 1, ..., n, n = [Re α] + 1, bk ∈ C,

Basics of fractional calculus and fractional order differential equations

65

has a unique solution of the form y(x) =

n 

x bk x

α−k

Eα,k (λx ) +

k=1

2.2

(x − t)α−1 Eα,α (λ(x − t)α )f (t)dt.

α

0

Standard fractional order integro-differential operators

Fractional integro-differential operators are studied in [17,78,107,202,381,413,414, 429,431,446,494].

2.2.1 Riemann–Liouville fractional integrals and derivatives on a segment In this subsection we present definitions of the Riemann–Liouville fractional integrals and fractional derivatives on a finite segment of the real line. Also we give some of their properties in spaces of summable and continuous functions. Definition 15. Let 0 < α, f ∈ L1 (a, b), a, b ∈ R. Then integrals α f )(x) = (Ib−

1 (α)

b x

f (t) dt, (t − x)1−α

x < b,

(2.11)

f (t) dt, (x − t)1−α

x > a,

(2.12)

and α (Ia+ f )(x) =

1 (α)

x a

are called right-sided (2.11) and left-sided (2.12) Riemann–Liouville fractional integrals of the order α on a segment [a, b]. Let α > 0 and not integer, n = [α] + 1. Right-sided and left-sided Riemann– Liouville fractional derivatives of the order α on a segment [a, b] for a function f ∈ L1 (a, b) are introduced by the relations α f )(x) = (Db−

d − dx

n n−α (Ib− f )(x)



b 1 f (t)dt d n = , − (n − α) dx (t − x)α−n+1 x

(2.13)

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Transmutations, Singular and Fractional Differential Equations

n−α where Ib− f ∈ C n (a, b) and

α (Da+ f )(x) =

d dx

n n−α (Ia+ f )(x)

1 = (n − α)



d dx

n x a

f (t)dt , (x − t)α−n+1

(2.14)

n−α f ∈ C n (a, b). where Ia+ When α = n ∈ N for x ∈ [a, b], f ∈ C n (a, b), n f )(x) = (Db−



d n − f (x), dx

n (Da+ f )(x) =

d dx

n f (x).

Next, following [241,494] we present some properties of the Riemann–Liouville fractional integrals and fractional derivatives on the segment. Theorem 13. (see [241], p. 73) Let 0 ≤ α and n = [α] + 1. If f ∈ AC n [a, b], then the α and D α exist almost everywhere on a segment [a, b] and fractional derivatives Db− a+ can be represented in the forms

α f )(x) = (Db−

(−1)n (n − α)

b

 (−1)k f (k) (b) f (n) (t)dt + (b − x)k−α α−n+1 (1 + k − α) (t − x) n−1 k=0

x

and α (Da+ f )(x) =

1 (n − α)

x a

 f (k) (a) f (n) (t)dt + (x − a)k−α , (1 + k − α) (x − t)α−n+1 n−1 k=0

respectively. α Theorem 14. (see [241], p. 72) (1) The fractional Riemann–Liouville integrals Ib− α and Ia+ with α > 0 are bounded in Lp (a.b), 1 ≤ p ≤ ∞: α ||Ib− f ||p ≤ K||f ||p ,

α ||Ia+ f ||p ≤ K||f ||p ,

K=

(b − a)α . (α + 1)

α and I α are bounded from (2) If 0 < α < 1 and 1 < p < α1 , then the operators Ib− a+ p Lp (0, ∞) into Lq (0, ∞), where q = 1−αp . α and I α is given The semigroup property of the fractional integration operators Ib− a+ by the following result. Often the semigroup property is called index law.

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67

Theorem 15. (see [494], Sections 2.3 and 2.5) If α > 0 and β > 0, then equalities β

α+β

α (Ib− Ib− f )(x) = (Ib− f )(x),

β

α+β

α (Ia+ Ia+ f )(x) = (Ia+ f )(x)

are satisfied at almost every point x ∈ [a, b] for f ∈ Lp (a, b), 1 ≤ p ≤ ∞. If α + β > 1, then these relations hold at any point of [a, b]. It is well known that the usual differentiation

d dx

x and integration ...dt are mutually a

inverse operations if the differentiation is applied on the left, i.e.,

d dx

x

f (t)dt = f (x).

a

x However, generally speaking, f  (t)dt = f (x) (as the constant −f (a) is added). a  d n n n f (n) does not coincide with a function f and differs Similarly dx Ia+ f = f but Ia+ from it by a polynomial of order (n − 1). In the same way for fractional differentiation α I α f = f is valid but I α D α f does not coincide with a function the equality Da+ a+ a+ a+ f (x) and differs from it by a sum of functions (x − a)α−k , k = 1, 2, ..., [α] − 1. Lemma 8. (see [241], p. 74) If α > 0 and f ∈ Lp (a, b), 1 ≤ p ≤ ∞, then the equalities β

α Ib− f )(x) = f (x), (Db−

β

α (Da+ Ia+ f )(x) = f (x)

hold almost everywhere on [a, b]. We need the following class of functions. α (L ), α > 0, be the class of functions f (x) Definition 16. (see [494], p. 43) Let Ib− p such that α (Lp ), f ∈ Ib−

α>0

⇔

α f = Ib− ϕ,

ϕ ∈ Lp (a, b),

1 ≤ p ≤ ∞.

α (L ), α > 0, be the class of functions f (x) such that Let Ia+ p α (Lp ), f ∈ Ia+

α>0

⇔

α f = Ia+ ϕ,

ϕ ∈ Lp (a, b),

1 ≤ p ≤ ∞.

α (L ) and I α (L ). The next theorem gives the description of the classes Ib− 1 a+ 1

Theorem 16. (see [494], p. 43) (1) The necessary and sufficient conditions required α (L ), α > 0, are for f ∈ Ib− 1 n−α f ∈ AC n ([a, b]), fn−α (x) = Ib−

where n = [α] + 1 and (k)

fn−α (b) = 0,

k = 0, 1, 2, ..., n − 1.

α (L ), α > 0, are (2) The necessary and sufficient conditions required for f ∈ Ia+ 1 n−α f ∈ AC n ([a, b]), fn−α (x) = Ia+

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Transmutations, Singular and Fractional Differential Equations

where n = [α] + 1 and (k)

fn−α (a) = 0,

k = 0, 1, 2, ..., n − 1.

We note that the representability of a function f by fractional integral of the order α and the existence of a fractional derivative of f are two different things. The hypothesis “the fractional derivative exists almost everywhere and is summable” is not enough to produce satisfactory theory. Therefore we give the next definition. Definition 17. (see [494], p. 44) Let α > 0, n = [α] + 1. α f if I n−α f ∈ Function f (x)∈L1 (a, b) has a summable fractional derivative Db− b− AC n ([a, b]). α f if I n−α f ∈ Function f (x)∈L1 (a, b) has a summable fractional derivative Da+ a+ AC n ([a, b]).  α f = − d n I n−α f exists in the usual sense, i.e., Remark 4. (see [494], p. 44) If Db− b− dx n−α Ib− f is differentiable by the order n at each point of [a, b], then f has a summable α f in the sense of Definition 17. fractional derivative Db−  n n−α n−α α f = d If Da+ Ia+ f exists in the usual sense, i.e., Ia+ f is differentiable by dx α f the order n at each point of [a, b], then f has a summable fractional derivative Da+ in the sense of Definition 17. The next theorems give conditions when fractional Riemann–Liouville integration and differentiation are used as reciprocal operations. The following statement characterizes the composition of the fractional integration α with the fractional differentiation operator D α . operator Ib− b− Theorem 17. (see [241], p. 75) Let α > 0. Then the equality α α Ib− f )(x) = f (x) (Db−

(2.15)

is valid for any summable function f (x). The equality α α Db− f )(x) = f (x) (Ib−

(2.16)

is satisfied for α (L1 ). f (x) ∈ Ia+

(2.17)

If we assume that instead of (2.17) a function f ∈ L1 (a, b) has a summable fractional α f (in the sense of Definition 17), then (2.15) is not true in general and derivative Db− has to be replaced by the result α α (Ib− Db− f )(x) = f (x) −

n−1  (−1)n−k−1 (b − x)α−k−1 k=0

(α − k)

(n−k−1)

fn−α

(b),

(2.18)

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69

n−α where n = [α] + 1 and fn−α (x) = Ib− f . In particular, for 0 < α < 1 we have α α (Ib− Db− f )(x) = f (x) −

f1−α (b) (b − x)α−1 . (α)

(2.19)

Theorem 18. (see [494], p. 44) Let α > 0. Then the equality α α Ia+ f )(x) = f (x) (Da+

(2.20)

is valid for any summable function f (x). The equality α α Da+ f )(x) = f (x) (Ia+

(2.21)

is satisfied for α f (x) ∈ Ia+ (L1 ).

(2.22)

If we assume that instead of (2.22) a function f ∈ L1 (a, b) has a summable fractional α f (in the sense of Definition 17), then (2.20) is not true in general and derivative Da+ is to be replaced by the result α α Da+ f )(x) = f (x) − (Ia+

n−1  (x − a)α−k−1 k=0

(α − k)

(n−k−1)

fn−α

(a),

(2.23)

n−α f . In particular, for 0 < α < 1 we have where n = [α] + 1 and fn−α (x) = Ia+ α α (Ia+ Da+ f )(x) = f (x) −

f1−α (a) (x − a)α−1 . (α)

(2.24)

2.2.2 Riemann–Liouville fractional integrals and derivatives on a semiaxis Definition 18. Let 0 < α, f ∈ L1 (0, ∞). Then integrals (I−α f )(x) =

1 (α)

∞ x

f (t) dt, (t − x)1−α

x > 0,

(2.25)

and α f )(x) = (I0+

1 (α)

x 0

f (t) dt, (x − t)1−α

x > 0,

(2.26)

are called right-sided (2.26) and left-sided (2.25) Riemann–Liouville fractional integrals of the order α on a semiaxis (0, ∞).

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Transmutations, Singular and Fractional Differential Equations

Let α > 0 and not integer, n = [α] + 1. Right-sided and left-sided Riemann– Liouville fractional derivatives of the order α on a semiaxis (0, ∞) for function f ∈ L1 (0, ∞) are introduced by the relations

d n n−α α (I− f )(x) (D− f )(x) = − dx

∞ f (t)dt d n 1 , (2.27) − = (n − α) dx (t − x)α−n+1 x

where

I−n−α f

∈ C n (0, ∞) and n d n−α α (D0+ f )(x) = (I0+ f )(x) dx n x f (t)dt d 1 , = (n − α) dx (x − t)α−n+1

(2.28)

0

n−α f I0+

∈ C n (0, ∞). where When α = n ∈ N for x ∈ (0, ∞), f ∈ C n (0, ∞), n

d n d n n f (x), (D0+ f )(x) = f (x). (D− f )(x) = − dx dx Theorem 19. (see [241], p. 83) If α > 0, β > 0, p ≥ 1, α + β < 1/p, and f ∈ Lp (0, ∞), then the equalities β

α+β

(I−α I− f )(x) = (I−

f )(x),

β

α+β

α (I0+ I0+ f )(x) = (I0+ f )(x)

hold. Lemma 9. (see [241], p. 83) If α > 0 and f ∈ L1 (0, ∞), then the equalities β

β

α I− f )(x) = f (x), (D−

α (D0+ I0+ f )(x) = f (x)

hold. Theorem 20. (Hardy–Littlewood–Polya theorem, see [241], p. 82) Let 1 ≤ p ≤ ∞, α are bounded from L (0, ∞) 1 ≤ q ≤ ∞, and α > 0. Then the operators I−α and I0+ p into Lq (0, ∞) if and only if 0 < α < 1,

1 0, then the relation ∞

∞ α ϕ(x)(I0+ ψ)(x)dx

0

=

ψ(x)(I−α ϕ)(x)dx 0

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71

holds for functions ϕ ∈ Lp (0, ∞) and ψ ∈ Lq (0, ∞). The relation ∞

∞ α f (x)(D0+ g)(x)dx

=

0

g(x)(I−α f )(x)dx 0

α (L (0, ∞)), where p > 1, q > 1, holds for functions f ∈ I−α (Lp (0, ∞)) and g ∈ I0+ q 1 1 + = 1 + α. p q

2.2.3 Gerasimov–Caputo fractional derivatives In 1948 (see [179], submitted in 1947) the Soviet mechanic A. N. Gerasimov introduced the fractional derivative of the form (

G

1 (α)

α D−,t u)(x, t) =

t −∞

uy (x, y)dy , (t − y)α

t > 0,

x ∈ R,

0 < α < 1.

In the same work, A. N. Gerasimov studied two new problems in viscoelasticity theory. He reduced this problem to differential equations with partial fractional derivative. In the same way it is possible to introduce fractional derivatives of Gerasimov type for other fractional derivatives analogous to Riemann–Liouville ones (cf. the paper by the author [517]). After 20 years the same construction with higher order derivatives was introduced by the Italian mechanic M. Caputo in 1967 in the paper [49] and studied in the monograph [50]. Therefore, in many books and papers, the fractional derivative of the type GC D α is called the Caputo derivative. −,t An obvious modification of the fractional derivative of Gerasimov to the case of a higher order leads to fractional derivatives and integrals of Gerasimov–Caputo. Obvious generalization of Gerasimov’s fractional derivative for higher order derivatives leads to Gerasimov–Caputo operators. The general form of the Gerasimov– Caputo derivative is ([179], [241], p. 97, formula (2.4.47)) (

GC

α Da+ g)(x) =

1 (n − α)

x a

g (n) (t)dt , (x − t)α−n+1

n = [α] + 1,

x ∈ (a, b), (2.29)

where α > 0, α ∈ / N0 , and y ∈ C n (a, b). For α = n = 0, 1, 2, ... n ( GC Da+ f )(x) = f (n) (x).

For noninteger α > 0, (

GC

α D0+ f )(x) =

1 (n − α)

x 0

f (n) (t)dt , (x − t)α+1−n

x ∈ [0, ∞),

(2.30)

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Transmutations, Singular and Fractional Differential Equations

is the left-sided Gerasimov–Caputo fractional derivative on semiaxes ([179], [241], p. 97, formula (2.4.47)) and for α = n = 0, 1, 2, ... n ( GC D0+ f )(x) = f (n) (x).

The paper of A. N. Gerasimov is the first in mathematical literature in which fractional derivatives were used for studying viscoelastic materials. Let us mention the book of O. Novozhenova [427], who gathered many biographical facts and papers of A. N. Gerasimov, including his pioneering paper [179]. The priority of A. N. Gerasimov was first pointed out by A. Kilbas in his lectures [238]. Note that M. Caputo never insisted on his priority, as he worked in the field of mechanics; it was propagandized by his disciple F. Mainardi. The importance of Gerasimov–Caputo fractional derivatives is illustrated by the simple example of its application to fractional differential equations. If we consider the fractional differential equation with Riemann–Liouville fractional derivative of the form α (D0+ y)(x) = λy(x),

x > 0,

0 < α ≤ 1,

λ ∈ R,

we should add the initial conditions here, α−1 (D0+ y)(0+) = 1,

and the solution can be found explicitly and equals y(x) = x α−1 Eα,α (λx α ),

(2.31)

via the Mittag-Leffler function Eα,α (λx ) = α

∞  n=0

(λx α )n (αn + α)

(cf. (1.39) for its definition). Note a singularity at zero in (2.31) when α < 1. It is not possible to consider the classical Cauchy problem at zero initial point with Riemann– Liouville fractional derivative α f )(x) = (D0+

1 d (1 − α) dx

x 0

f (t)dt , (x − t)α

0 < α ≤ 1.

α−1 y)(0+) = 1 instead of classical initial Of course we can use the conditions (D0+ conditions, but repeat once more – in this case the classical Cauchy problem is incorrect. Opposite to it the general solution to the fractional differential equation with Gerasimov–Caputo fractional derivative α ( GC D0+ y)(x) = λy(x),

x > 0,

0 < α ≤ 1,

λ ∈ R,

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73

y(0+) = 1, is also found explicitly and equals y(x) = Eα,1 (λx α ),

(2.32)

via the Mittag-Leffler function Eα,1 (λx α ) =

∞  (λx α )n . (αn + 1) n=0

Now it is bounded at zero so the classical Cauchy problem is correct. As a conclusion we may formulate that different problems need different fractional derivatives to be involved; some of them need Riemann–Liouville fractional derivatives, some Gerasimov–Caputo fractional derivatives, and some other types of fractional operators.

2.2.4 Dzrbashian–Nersesyan fractional operators and sequential order fractional operators Dzrbashian–Nersesyan fractional derivatives, associated with a sequence {γ0 , γ1 , . . . , γm } of order σ , where σ = γ0 + γ1 + ... + γm , are defined by σ = D γ0 D γ1 · · · D γm , DDN

(2.33)

where D γk are fractional integrals and derivatives of Riemann–Liouville with some endpoint. These operators were introduced in [108–110] and then studied and applied in [102,105,111,112]. The original definitions demand −1 ≤ γ0 ≤ 0, 0 ≤ γk ≤ 1, 1 ≤ k ≤ m, as in the above papers integro-differential equations under such conditions were studied for operators (2.33). But Dzrbashian–Nersesyan fractional operators may be defined and considered for any parameter γk if appropriate definitions of Riemann– Liouville operators are used. Riemann–Liouville, Gerasimov, and Gerasimov–Caputo fractional operators are special cases of Dzrbashian–Nersesyan fractional operators defined in the above-mentioned generalized sense. Operators of Gerasimov, Gerasimov–Caputo, and Dzrbashian–Nersesyan were patterns for introducing in the book of Miller and Ross [381] more general sequential operators of fractional integro-differentiation for which compositions in definitions of the form (2.33) consist of any fractional operators (cf. a useful discussion in [446]).

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Transmutations, Singular and Fractional Differential Equations

2.3 Some more fractional order integro-differential operators 2.3.1 The Erdélyi–Kober operators Definition 19. Let α > 0. The Erdélyi–Kober operators are defined by the following formulas:

α I0+; 2, y f =

2 −2(α+y) x (α)

x (x 2 − t 2 )α−1 t 2y+1 f (t) dt,

(2.34)

∞ 2 2y α (t 2 − x 2 )α−1 t 2(1−α−y)−1 f (t) dt. I−; 2, y f = x (α)

(2.35)

0

x

For α > −n, n ∈ N they are defined by

d n 2(α+y+n) α+n α −2(α+y) I0+; f = x x I0+; 2, y f, 2,y dx 2

d n 2(α−y) α+n α 2y f = x x I−; 2, y−n f. I−; − 2,y dx 2

(2.36) (2.37)

Let us denote that in the classical Russian version of the monograph [494] cases of integral limits 0 and ∞ were not considered. In the English version of this book [494] these limits are considered, but definitions are given with inaccuracies, and in particular cases they lead to complex values in integrals. The Erdélyi–Kober operators are essential and important in transmutation theory because the most well-known transmutations of Sonine and Poisson are of this class; they are discussed in monographs [234,537], where a more historically exact term is used for them: Sonine–Poisson–Delsarte transmutations. Important properties of Erdélyi–Kober operators were studied in the monograph [234] and in papers of Yu. Luchko (cf. [340] and more references therein).

2.3.2 Fractional integrals and fractional derivatives of a function with respect to another function Definition 20. Let Re α > 0. The left- and right-sided fractional integrals of a function f with respect to another function g are

α I0+,g f

1 = (α)

x 0

(g(x) − g(t))α−1 g  (t)f (t)d t,

(2.38)

Basics of fractional calculus and fractional order differential equations

α I−,g f

1 = (α)

∞

(g(t) − g(x))α−1 g  (t)f (t)d t.

75

(2.39)

x

Moreover, Riemann–Liouville fractional integrals on a semiaxis (2.26) and (2.25) are obtained by choosing g(x) = x in (2.38) and (2.39), respectively. If we take in (2.38) and (2.39) the function g(x) = x 2 we obtain Erdélyi–Kober operators (2.34) and (2.35); if we take g(x) = ln x in (2.38) and (2.39) we get Hadamard fractional integrals, and the choice g(x) = exp(−x) with its applications was considered in [107]. As A. M. Dzrbashian pointed out, operators of fractional derivatives of a function with respect to another function (2.38) even in some more general setting were introduced and studied by his father M. M. Dzrbashian (cf. [99–101,103,104] and the monograph [494]). In these papers integral representations of this operator class, their inversion, and corresponding integro-differential equations of fractional order were studied.

2.3.3 Averaged or distributed order fractional operators Further generalizations of fractional integro-differential operators are connected with combinations and compositions of more standard fractional operators defined above. The averaged or distributed order fractional operator, associated with any given fractional operator R t , is introduced by the following formula:

(a,b) IMR f

b =

R t f (t)d t,

(2.40)

a

where R t is a given fractional operator of order t of any kind. In the case R t is in particular the fractional Riemann–Liouville operator, the names continued or distributed fractional integrals or derivatives are often used. Such operators were studied by A. Pskhu and his disciples [459,460]. Note that one of the authors proposed the following modification of an averaged expression in (2.40) to be more convenient and also similar to an integral mean value:

(a,b) I MR f

1 = b−a

b R t f (t)d t. a

This variant of modified definition interested and was approved by A. Nakhushev, but it is still not used unfortunately.

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Transmutations, Singular and Fractional Differential Equations

2.3.4 Saigo, Love, and other fractional operators with special function kernels Saigo fractional integrals (see [487] and [468]) are γ ,β,η Jx f (x) =

1 (γ )

∞  x (t − x)γ −1 t −γ −β 2 F1 γ + β, −η; γ ; 1 − f (t)dt t x

(2.41) and γ ,β,η Ix f (x) =

x −γ −β (γ )

x

γ −1

(x − t)

2 F1

t f (t)dt, γ + β, −η; γ ; 1 − x

0

(2.42) where γ > 0, β, θ are real numbers. Another similar class of generalizations introduced by Love and more generalizations with special function kernels are mentioned in [494]. Also important generalizations of classical fractional operators were studied, namely, Buschman–Erdélyi operators and fractional Bessel operators, which we consider in detail in this book.

2.4 Integral transforms and basic differential equations of fractional order One of the most popular methods to obtain explicit solutions to fractional differential equations is the integral transforms method. Usually Laplace, Mellin, and Fourier transforms are used.

2.4.1 Integral transforms of fractional integrals and derivatives 2.4.1.1

Laplace transform of Riemann–Liouville fractional integrals and derivatives on semiaxes

Theorem 22. (see [241], p. 84) Let α > 0 and f ∈ L1 (0, b) for any b > 0. Also let the estimate |f (x)| ≤ Aep0 x ,

x > b > 0,

hold for some constants A > 0 and p0 > 0. (1) If f ∈ L1 (0, b) for any b > 0, then the relation α (LI0+ f )(s) = s −α (Lf )(p)

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77

is valid for Re s > p0 . (2) If n = [α] + 1, g ∈ AC n [0, b] for any b > 0, the estimate |g(x)| ≤ Beq0 x ,

x > b > 0,

holds for constants B > 0 and q0 > 0, and g (k) (0) = 0, k = 0, 1, ..., n − 1, then the relation α g)(s) = s α (Lg)(s) (LD0+

is valid for Re s > q0 . Remark 5. (see [241], p. 84) If α > 0, n = [α] + 1, g ∈ AC n [0, b] for any b > 0, the condition |g(x)| ≤ Beq0 x , x > b > 0, holds for constants B > 0 and q0 > 0, and there exist the finite limits n−α lim [D k I0+ ]

x→+0

and n−α lim [D k I0+ ] = 0,

x→∞

D=

d , dx

k = 0, 1, ..., n − 1,

then α g)(s) = s α (Lg)(s) − (LD0+

n−1 

n−α s n−k−1 D k (I0+ g)(+0),

Re s > q0 .

k=0

(2.43) In particular, when 0 < α < 1 and g ∈ AC[0, b] for any b > 0, then 1−α α g)(s) = s α (Lg)(s) − (I0+ g)(+0). (LD0+

2.4.1.2

Mellin transform of Riemann–Liouville fractional integrals and derivatives on semiaxes

α and I α and The Mellin transform of the Riemann–Liouville fractional integrals I0+ − α α fractional derivatives D0+ and D− are given by the following statements. p We need here the space Xc (a, b), c ∈ R, 1 ≤ p ≤ ∞, consisting of those complex valued Lebesgue measurable functions f on (a, b) for which ||f ||Xcp < ∞, with

||f ||Xcp

⎛ b ⎞ p1  dt = ⎝ |t c f (t)|p ⎠ , t a

and ||f ||Xc∞ = ess sup[x c |f (x)|]. a≤x≤b

1 ≤ p < ∞,

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Transmutations, Singular and Fractional Differential Equations

1 (0, ∞). Theorem 23. Let α > 0, s ∈ C, and f ∈ Xs+α (1) If Re s > 0, then

(MI−α f )(s) =

(s) (Mf )(s + α). (s + α)

(2) If Re s < 1 − α, then (1 − α − s) (Mf )(s + α). (1 − s)

α (MI0+ f )(s) =

1 (0, ∞). Theorem 24. Let α > 0, n = [α] + 1, s ∈ C, and g ∈ Xs−α (1) If Re s > 0 and the conditions

lim [x s−k−1 (I−n−α g)(x)] = 0,

k = 0, 1, ..., n − 1,

lim [x s−k−1 (I−n−α g)(x)] = 0,

k = 0, 1, ..., n − 1,

x→+0

and x→∞

hold, then α (MD− g)(s) =

(s) (Mf )(s − α). (s − α)

(2) If Re s < 1 + α and the conditions n−α g)(x)] = 0, lim [x s−k−1 (I0+

k = 0, 1, ..., n − 1,

n−α lim [x s−k−1 (I0+ g)(x)] = 0,

k = 0, 1, ..., n − 1,

x→+0

and x→∞

hold, then α (MD0+ g)(s) =

(1 + α − s) (Mf )(s − α). (1 − s)

1 (0, ∞). Remark 6. Let α > 0, n = [α] + 1, s ∈ C, and g ∈ Xs−α (1) If Re s > 0, then α (MD− g)(s) =

(s) (Mg)(s − α) (s − α) +

n−1  (−1)n−k (s) k=0

(s − k)

[x s−k−1 (I−n−α g)(x)]∞ 0 .

Basics of fractional calculus and fractional order differential equations

79

(2) If Re s < 1 + α, then α (MD0+ g)(s) =

(1 + α − s) (Mg)(s − α) (1 − s) +

n−1  (1 + k − s) k=0

(1 − s)

n−α g)(x)]∞ [x s−k−1 (I0+ 0 .

In particular, when 0 < α < 1, then α (MD− g)(s) =

(s) (Mg)(s − α) + [x s−1 (I−1−α g)(x)]∞ 0 (s − α)

and α g)(s) = (MD0+

2.4.1.3

(1 + α − s) 1−α g)(x)]∞ (Mg)(s − α) + [x s−1 (I0+ 0 . (1 − s)

Laplace transform of Gerasimov–Caputo fractional derivatives on semiaxes

Theorem 25. Let α > 0, n − 1 < α ≤ n, n ∈ N, such that y(x) ∈ C n (0, ∞), g (n) (x) ∈ L1 (0, b) for any b > 0 and the estimate |g(x)| ≤ Beq0 x ,

b > 0,

q0 > 0,

is valid. Let the Laplace transforms Ly and LD n y exist and lim (D k g)(x) = 0,

x→+∞

k = 0, 1, ..., n − 1.

Then α g)(s) = s α (Ly)(s) − (L GC D0+

n−1 

s α−k−1 (D k g)(0).

(2.44)

k=0

In particular, for 0 < α ≤ 1, α (L GC D0+ g)(s) = s α (Ly)(s) − s α−1 g(0).

(2.45)

2.4.2 Laplace transform method for the homogeneous equations with constant coefficients with the left-sided Riemann–Liouville fractional derivatives of the order α on a semiaxis (0, ∞) We first present a result from [241] which demonstrates how to use Laplace transform for solving the one-dimensional fractional nonhomogeneous differential equation with

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Transmutations, Singular and Fractional Differential Equations

constant coefficients of the form m 

αk Ak (D0+ y)(x) + A0 y(x) = f (x),

x > 0,

m ∈ N,

(2.46)

k=1

0 < α1 < ... < αm ,

A0 , A1 , ..., Am ∈ R,

with the left-sided Riemann–Liouville fractional derivatives of the order α on a semiaxis (0, ∞) (2.28). Let y1 (x), ..., yl (x), l − 1 < αm ≤ l, l ∈ N, are linearly independent solutions of (2.46) such that αm −k (D0+ yk )(0+) = 1,

k = 1, ..., l,

αm −k (D0+ yj )(0+) = 0,

k, j = 1, ..., l;

k = j,

where β

β

(D0+ g)(0+) = lim (D0+ g)(x). x→+0

Such system y1 (x), ..., yl (x) is the fundamental system of solutions of (2.46). Following [241] first we consider Eq. (2.46) when m = 1: α (D0+ y)(x) − λy(x) = 0,

x > 0,

λ ∈ R,

with conditions α−j

(D0+ y)(0+) = dj ,

j = 1, ..., l,

j = 1, ..., l,

l ∈ N,

dj ∈ R.

α y)(x) − λy(x) = 0, taking into account Applying the Laplace transform (1.54) to (D0+ (2.43), we obtain

s α L[y](s) −

l 

dj s j −1 = λL[y](s)

j =1

or L[y](s) =

l  j =1

dj

s j −1 , sα − λ

and (1.55) gives y(x) =

l  j =1

dj x α−j Eα,α+1−j (λx α ).

Basics of fractional calculus and fractional order differential equations

81

It is easily verified that the functions yj (x) = x α−j Eα,α+1−j (λx α ) are solutions to the α y)(x) − λy(x) = 0: equation (D0+ (D0α [t α−j Eα,α+1−j (λt α )])(x) = λx α−j Eα,α+1−j (λx α ),

j = 1, ..., l,

and by (1.39) α−k (D0+ yj )(x) =

∞  n=0

λn x αn+k−j . (αn + k + 1 − j )

Then α−k yk )(0+) = 1, (D0+

k = 1, ..., l,

α−k (D0+ yj )(0+) = 0,

k, j = 1, ..., l,

and k = j.

Thus the following result is valid. Theorem 26. [241] Let l − 1 < α ≤ l, l ∈ N, λ ∈ R. Then the functions yj (x) = x α−j Eα,α+1−j (λx α ),

j = 1, ..., l,

yield the fundamental system of solutions to the equation α (D0+ y)(x) − λy(x) = 0,

x > 0,

and the solution to this equation satisfying conditions α−j

(D0+ y)(0+) = dj ,

j = 1, ..., l,

l ∈ N,

dj ∈ R

is y(x) =

l 

dj x α−j Eα,α+1−j (λx α ).

j =1

Example 1. Let us consider the case when 2 < α ≤ 3. Then l = 3 and the solution to the problem α (D0+ y)(x) − λy(x) = 0, α−1 y)(0+) = d1 , (D0+

x > 0,

α−2 (D0+ y)(0+) = d2 ,

α−3 (D0+ y)(0+) = d2 ,

d1 , d2 , d3 ∈ R, is y(x) = d1 x α−1 Eα,α (λx α ) + d2 x α−2 Eα,α−1 (λx α ) + d3 x α−3 Eα,α−2 (λx α ).

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Transmutations, Singular and Fractional Differential Equations

In [17] it was shown that if f (x) ∈ L(a, b), bk ∈ C the Cauchy type problem for the linear differential equation α−k α y)(x) − λy(x) = f (x), (Da+ y)(a+) = bk , k = 1, ..., n, n = [Re α] + 1, (Da+

has the unique solution y(x) in some subspace of L(a, b) given by y(x) =

n 

bk x α−k Eα,α−k+1 (λ(x − a)α )

k=1

x +

(x − t)α−1 Eα,α (λ(x − t)α )f (t)dt,

(2.47)

a

where Eα,α is the Mittag-Leffler function defined by (1.39). In particular for 0 < Re α < 1 and f (x) = 0 the function y(x) = b1 x α−1 Eα,α (λ(x − a)α ) is the solution to α (Da+ y)(x) = λy(x),

1−α (Ia+ y)(a+) = b1 ,

b1 ∈ C.

2.4.3 Laplace transform method for homogeneous equations with constant coefficients with the left-sided Gerasimov–Caputo fractional derivatives of the order α on a semiaxis [0, ∞) In [241], p. 312, the Laplace transform method was applied to derive explicit solutions to homogeneous equations of the form α ( GC D0+ f )(x) = λf (x),

x > 0,

l − 1 < α ≤ l,

l ∈ N,

λ ∈ R, (2.48)

α f is the left-sided Gerasimov–Caputo fractional derivative on semiwhere GC D0+ axes (2.30). Gerasimov in [179] derived and solved fractional order partial differential equations with the derivative (2.30) for mechanical applied problems in 1948. The conditions

f k (0+) = dk ,

k = 0, 1, ..., l − 1,

dk ∈ R

(2.49)

were added to Eq. (2.48). The solution to the problem (2.48)–(2.49) is (see [241], p. 312) f (x) =

l−1 

dk x k Eα,k+1 (λx α ),

k=0

where Eα,β is the Mittag-Leffler function (1.39).

(2.50)

Basics of fractional calculus and fractional order differential equations

83

2.4.4 Mellin integral transform and nonhomogeneous linear differential equations of fractional order Nonhomogeneous linear differential equations of fractional order with given functions Ak (x), k = 0, 1, ..., m, and f (x) have the form A0 y(x) +

m 

Ak (x)(D αk y)(x) = f (x).

(2.51)

k=1

Differential fractional order operators in (2.51) can have various forms. For example we can take in (2.51) the right- and left-sided Riemann–Liouville fractional derivatives on semiaxes given by (2.27) and (2.28) and constant coefficients Ak , Bk ∈ R, k = 0, ..., m: m 

α+k Bk (D− y)(x) = f (x),

x > 0,

α > 0,

(2.52)

α+k Ak (D0+ y)(x) = f (x),

x > 0,

α > 0.

(2.53)

k=0

and m  k=0

The Mellin transform method for solving Eqs. (2.52) and (2.53) is based on the relations following from Theorem 24: α+k y)(s) = (Mx α+k D−

(s + α + k) (My)(s) (s)

α+k (Mx α+k D0+ y)(s) =

(1 − s) (My)(s). (1 − s − α − k)

and

Applying the Mellin transform to (2.52) and (2.53) we obtain  m   (s + α + k) Bk (My)(s) = (Mf )(s) (s) k=0

and



m  k=0

 (1 − s) Ak (My)(s) = (Mf )(s), (1 − s − α − k)

respectively. Using formula (1.64), the solution to (2.52) is ∞ y(x) =

G1α 0

x  t

f (t)

dt , t

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Transmutations, Singular and Fractional Differential Equations

where  G1α (x) = M−1

1 1 Pα (s)



Pα1 (s) =

(x),

m  k=0

Bk

(s + α + k) . (s)

Using the same formula (1.64), the solution to (2.53) is ∞ y(x) =

G2α (t) f (xt)dt, 0

where G2α (x) =

M

−1



1 2 Pα (1 − s)



(x),

Pα2 (s) =

m  k=0

Ak

(s) . (s − α − k)

Essentials of transmutations 3.1

3

Definition of the transmutation operator, some examples of classical transmutations

3.1.1 Introduction to transmutation theory Following [229], we give a definition of the transmutation operator. Definition 21. Let us have two operators (A, B). The nonzero operator T is called the transmutation operator if the following relation is satisfied: T A = B T.

(3.1)

The relation (3.1) is also called intertwining property, as they say that the transmutation operator T intertwines the operators A and B or is an intertwining operator. To transform (3.1) into a strict definition, it is necessary to specify spaces or sets of functions on which the operators A, B, and, therefore, T act. The method of solving problems based on the use of the operator T with the property (3.1) is called the method of transmutation operators. It is obvious that the notion of transmutation is a direct and far reaching generalization of the matrix similarity from linear algebra. But the transmutations do not reduce to similar operators because intertwining operators often are not bounded in classical spaces and the inverse operator may not exist or not be bounded in the same space. As a consequence, spectra of intertwining operators are not the same as a rule. Moreover, transmutations may be unbounded. This is the case for the Darboux transformations, which are defined for a pair of differential operators and are differential operators themselves; in this case all three operators are unbounded in classical spaces. But the theory of Darboux transformations is included in transmutation theory too. A pair of intertwining operators may not be differential ones. In transmutation theory there are problems for the following various types of operators: integral, integro-differential, difference-differential (e.g., the Dunkl operator), differential or integro-differential of infinite order (e.g., in connection with Schur’s lemma), general linear operators in functional spaces, and pseudodifferential and abstract differential operators. All classical integral transforms due to Definition 1 are also special cases of transmutations, including the Fourier, Petzval (Laplace), Mellin, Hankel, Weierstrass, Kontorovich–Lebedev, Meyer, Stankovic, Obrechkoff, finite Grinberg, and other transforms. In quantum physics, the study of the Shrödinger equation, and inverse scattering theory, the underlying transmutations are called wave operators. The commuting operators are also a special class of transmutations. The most important class consists of operators commuting with derivatives. In this case transmutations as commutants are usually in the form of formal series, pseudodifferential, Transmutations, Singular and Fractional Differential Equations With Applications to Mathematical Physics https://doi.org/10.1016/B978-0-12-819781-3.00009-4 Copyright © 2020 Elsevier Inc. All rights reserved.

86

Transmutations, Singular and Fractional Differential Equations

or infinite order differential operators. Finding commutants is directly connected with finding all transmutations in the given functional space. For these problems works a theory of operator convolutions, including the Berg–Dimovski convolutions [89]. Also, more and more applications are developed in connection with transmutation theory for commuting differential operators; such problems are based on classical results of J. L. Burchnall and T. W. Chaundy. The transmutations are also connected with factorization problems for integral and differential operators. Special class of transmutations are the so-called Dirichlet-to-Neumann and Neumann-to-Dirichlet operators, which link together solutions of the same equation but with different kinds of boundary conditions. And how do the transmutations usually work? Suppose we study properties for a rather complicated operator A. But suppose also that we know the corresponding properties for a more simple model operator B and transmutation (3.1) readily exists. Then we usually may copy results for the model operator B to corresponding ones for the more complicated operator A. This is shortly the main idea of transmutations. Let us consider for example an equation Au = f . Then applying to it a transmutation with property (3.1) we consider a new equation Bv = g, with v = T u, g = Tf . So if we can solve the simpler equation Bv = g, then the initial one is also solved and has solution u = T −1 v. Of course, it is supposed that the inverse operator exists and its explicit form is known. This is a simple application of the transmutation technique for finding and proving formulas for solutions of ordinary and partial differential equations. The monographs [22,51–53,139,571] are completely devoted to transmutation theory and its applications (note also surveys [234,532] and [377]). Moreover, essential parts of monographs [56,89,242,252,259,277,316,321,330,373,376,497,580], etc., include material on transmutations; the complete list of books which investigate some transmutational problems is now near 100 items. We use the term “transmutation” due to [53]: “Such operators are often called transformation operators by the Russian school (Levitan, Naimark, Marchenko, etc.), but transformation seems a too broad term, and, since some of the machinery seems ‘magical’ at times, we have followed Lions and Delsarte in using the word transmutation.” Now transmutation theory is a completely formed part of the mathematical world in which methods and ideas from different areas are used: differential and integral equations, functional analysis, function theory, complex analysis, special functions, and fractional integro-differentiation. The transmutation theory is deeply connected with many applications in different fields of mathematics. The transmutation operators are applied in inverse problems via the generalized Fourier transform, spectral functions, and the famous Levitan equation; in scattering theory the Marchenko equation is formulated in terms of transmutations; in spectral theory transmutations help to prove trace formulas and asymptotics for spectral functions; estimates for transmutational kernels control stability in inverse and scattering problems; and for nonlinear equations via the Lax method transmutations for Sturm–Liouville problems lead to proving existence and explicit formulas for soliton solutions. Special kinds of transmutations are the generalized analytic functions, generalized translations and convolutions, and Darboux transformations. In the

Basic definitions and propositions

87

theory of partial differential equations the transmutations work for proving explicit correspondence formulas among solutions of perturbed and nonperturbed equations, for singular and degenerate equations, pseudodifferential operators, problems with essential singularities at inner or corner points, and estimates of solution decay for elliptic and ultraelliptic equations. In function theory transmutations are applied to embedding theorems and generalizations of Hardy operators, Paley–Wiener theory, and generalizations of harmonic analysis based on generalized translations. Methods of transmutations are used in many applied problems: investigation of Jost solutions in scattering theory, inverse problems, Dirac and other matrix systems of differential equations, integral equations with special function kernels, probability theory and random processes, stochastic random equations, linear stochastic estimation, inverse problems of geophysics, and transsound gas dynamics. The number of applications of transmutations to nonlinear equations is constantly increasing. In fact, the modern transmutation theory originated from two basic examples (see [532]). The first is the transmutation T for Sturm–Liouville problems with some potential q(x) and natural boundary conditions T (D 2 y(x) + q(x)y(x)) = D 2 (T y(x)), D 2 y(x) = y  (x).

(3.2)

The second example is a problem of studying transmutations intertwining the Bessel operator Bν and the second derivative: T Bν f = D 2 Tf, Bν = D 2 +

2ν + 1 d2 D, D 2 = 2 , ν ∈ C. x dx

(3.3)

This class of transmutations includes the Sonine–Poisson–Delsarte, Buschman– Erdélyi operators and their generalizations. Such transmutations found many applications for a special class of partial differential equations with singular coefficients. A typical equation of this class is the B-elliptic equation with the Bessel operator in some variables of the form n 

Bν,xk u(x1 , . . . , xn ) = f.

(3.4)

k=1

Analogously, B-hyperbolic and B-parabolic equations are considered; this terminology was proposed by I. Kipriyanov. This class of equations was first studied by Euler, Poisson, and Darboux and continued in Weinstein’s theory of generalized axially symmetric potential (GASPT). These problems were further investigated by Zhitomirslii, Kudryavtsev, Lizorkin, Matiychuk, Mikhailov, Olevskii, Smirnov, Tersenov, He Kan Cher, Yanushauskas, Egorov, and others. In the most detailed and complete way, equations with Bessel operators were studied by the Voronezh mathematician Kipriyanov and his disciples Ivanov, Ryzhkov, Katrakhov, Arhipov, Baidakov, Bogachov, Brodskii, Vinogradova, Zaitsev, Zasorin, Kagan, Katrakhova, Kipriyanova, Kononenko, Kluchantsev, Kulikov, Larin, Leizin, Lyakhov, Muravnik, Polovinkin, Sazonov, Sitnik, Shatskii, and Yaroslavtseva. The

88

Transmutations, Singular and Fractional Differential Equations

essence of Kipriyanov’s school results was published in [242]. For classes of equations with Bessel operators, Kipriyanov introduced special functional spaces which were named after him [243]. In this field interesting results were investigated by Katrakhov and his disciples; now these problems are considered by Gadjiev, Guliev, Glushak, and Lyakhov with their coauthors and students. Abstract equations of the form (3.4) originating from the monograph [56] were considered by Egorov, Repnikov, Kononenko, Glushak, Shmulevich, and others. Transmutations are one of the basic tools for equations with Bessel operators, and they are applied to the construction of solutions, fundamental solutions, the study of singularities, and new boundary value and other problems. We must note that the term “operator” is used in this chapter for brevity in the broad and sometimes not exact meaning, so appropriate domains and function classes are not always specified. It is easy to complete and make strict every special result.

3.1.2 Some examples of classical transmutations Let give some examples of classical transmutations. Example 1. Transmutation operator intertwining the second order and first order derivatives. If u(t, x) is a function, satisfying the abstract Cauchy problem utt = Au,

u(0, x) = f (x),

ut (0, x) = 0,

then 1  v(t, x) = Lu(t, x) = √ πt

∞

s2

u(s, x)e− 4t ds

(3.5)

0

satisfies the abstract Cauchy problem vt = Av,

v(0, x) = f (x).

This example was given in lecture notes of R. Hersh [165] among the five examples  has been of transmutations. R. Hersh mentioned that the transmutation operator L rediscovered repeatedly in [14,37,205,476,576]. Let consider how we can use (3.5) for obtaining the solution to the Cauchy problem for the diffusion equation if we know the solution to the Cauchy problem for the wave equation. We have utt = uxx , u(0, x) − f (x),

ut (0, x) = 0.

Then u=

f (x + t) + f (x − t) 2

Basic definitions and propositions

89

and the solution to vt = vxx ,

v(0, x) = f (x)

is 1 v(t, x) = √ πt

∞

s2

f (s + x)e− 4t ds.

−∞

Example 2. Let us consider the Poisson operator (see [317]) γ

Px f (x) =

2C(γ ) x γ −1

x 

x2 − t 2

 γ −1 2

f (t) dt,

0

   γ +1 2  . C(γ ) = √ π  γ2

2 the Poisson operator acts as a transmutation operator by the formula For f ∈ Cev γ

γ

Px D 2 f = Bγ Px f,

D2 =

d2 , dx 2

Bγ =

d2 γ d + . x dx dx 2

The Poisson operator will be considered in detail in Section 3.4.1. Using the previous example we obtain that for the problem 2 wt = wrr + wr r w(0, r) = f (r),

0 ≤ r < ∞,

the bounded solution has the form 1 1 w(t, r) = 2C(2) √ πt r

r ∞

s2

f (s + p)e− 4t dsdp.

0 −∞

Example 3. The Radon transform (see Definition 14) intertwines a partial derivative with a univariate derivative:

∂ ∂ R f (x) = ωi Rf (ω, s). ∂xi ∂xi Let  denote the Laplacian on Rn : =

∂2 ∂2 + ··· + 2 2 ∂xn ∂x1

and let L denote the “radial” second derivative: L=

∂2 . ∂s 2

90

Transmutations, Singular and Fractional Differential Equations

The Radon transform and its dual are intertwining operators for  and L in the sense that R(f ) = L(Rf ),

R∗ (Lg) = (R∗ g).

Example 4. In [355] the transmutation operator 1 (Tl g)(r) =

n−2 2

g(rt)Cl

(t)(1 − t 2 )

n−3 2

dt

−1

was presented. For operators D1 and D2 acting by formulas (D1 g)(t) = (1 − t 2 )g  (t) + αtg  (t) + βg(t) and





n−1  l(l + n − 2) g (t) + β − g(t), (D2 g)(t) = (1 − t )g (t) + αt + t t2 2



the intertwining relation D2 T l = T l D 1 holds.

3.2 Transmutations for Sturm–Liouville operator 3.2.1 Description of the problem and terminology The main problem of this subsection is the construction of different transmutation operators intertwining the simplest Sturm–Liouville operator y  (x) + λ2 y(x) = (L0 y)(x)

(3.6)

with the Sturm–Liouville operator of the general kind y  (x) + q(x)y(x) + λ2 y(x) = (Ly)(x).

(3.7)

Function q(x) in (3.7) is called a potential function. Appropriate functions can be complex, λ ∈ C, x ∈ R. We are looking for a transformation operator satisfying the identity SLf = L0 Sf

(3.8)

on suitable functions f (x). A natural requirement for the transmutation operator S is linearity. It is very convenient if there is the invert to the S operator in some space.

Basic definitions and propositions

91

The linearity requirement of S after substitution of (3.6) and (3.7) in (3.8) leads to a relation that is independent of λ:   S D 2 + q(x) f = D 2 Sf, (3.9) where D = d/dx. Invertibility of S naturally leads to the search of it as an integral operator b(x) (Sf )(x) = K(x, t)f (t) dt.

(3.10)

a(x)

Here the kernel K(x, t) in the general case can be distribution (for example, K(x, t) = δ(x − t) + G(x, t), G is a smooth function). In (3.10) a(x), b(x) are some functions R → R. There are two different approaches to constructing transmutations. In the first, transmutations are built on arbitrary functions, possibly with some growth restrictions at various points. We will detail this particular approach but it is not generally accepted. Usually, the second approach is taken, which is based only on solutions of the equations L0 y = 0, Ly = 0 with the operators (3.6) and (3.7). Such a method was adopted in pioneering classical works on transmutation operators. We will present it briefly. Essentially both of these approaches are equivalent. Operators constructed on arbitrary functions are also defined on solutions with suitable boundary conditions. On the other hand, an operator built on eigenfunctions for any λ can be extended to fairly wide classes of functions. Let us make some comments. We will usually call operators differential or integral expressions. When constructing transmutations it is assumed that functions f (x) belong to some class . In calculations, it is assumed that the kernels K(x, t) have a certain smoothness in both variables. Exact definitions are given in each case. Now we consider transmutation operators S of the forms b (Sf )(x) =

K(x, t)f (t) dt,

(3.11)

a

x (Sf )(x) = f (x) +

K(x, t)f (t) dt,

(3.12)

K(x, t)f (t) dt,

(3.13)

K(x, t)f (t) dt.

(3.14)

−x x

(Sf )(x) = f (x) + c

d (Sf )(x) = f (x) + x

92

Transmutations, Singular and Fractional Differential Equations

Here K(x, t) is smooth in both variables, numbers a, b, and c belong to the extended numerical axis R. Fredholm operators (3.11) are the simplest. Volterra operators (3.12)–(3.14) are easily invertible in standard spaces. The freedom to choose the limits of integration allows in each case to choose those transmutations that are best suited for a concrete problem. In particular (3.12) preserves the asymptotic behavior of the function being converted and its derivatives for x → 0 and the transmutation of the form (3.14) for x → d. For example, transmutation (3.14) with d = +∞ preserves the asymptotic behavior at infinity; this is a transmutation of the B. Ya. Levin type. Such operator is used to solve inverse problems of the quantum theory of scattering and problems of estimating the rate of decrease at infinity of solutions of differential equations, including partial differential equations. Operator (3.13) for c = 0 is called transmutation of the A. Povzner type (see [454]). It is also possible to consider transmutations of other types, not like (3.11)–(3.14), for example, Fredholm and Volterra operators of the third kind. We show that in the case of a smooth potential q(x) transmutations (3.11)–(3.14) are constructed most simply and “naturally.” The construction of the transmutations of other types is difficult.

3.2.2 Transmutations in the form of the second kind Fredholm operators Let consider here the construction of transmutation operators as the second kind Fredholm operators b (Sf )(x) =

K(x, t)f (t) dt.

(3.15)

a

If such operator exists, then equality (3.9) is true. This equality in certain cases has the form b



f (x) + q(x)f (x) +

K(x, t)[f  (t) + q(t)f (t)] dt

a

= f  (x) +

b

∂ 2K f (t) dt. ∂x 2

a

Integrating by parts we obtain b b ∂K   q(x)f (x) + K(x, t)f (t) a − f (t) ∂t a

b 2 ∂ K ∂ 2K = − 2 − q(t)K f (t) dt. ∂x 2 ∂t 

a

(3.16)

Basic definitions and propositions

93

Therefore at least for finite functions f ∈ C0∞ (a, b), the equality b G(x, t)f (t) dt = q(x)f (x)

(3.17)

a

must be fulfilled. In (3.17) ∂ 2K ∂ 2K − 2 − q(t)K. ∂x 2 ∂t   If we require that G ∈ C [a, b]2 , then it is easy to see that equality (3.17) is impossible in the general case, which follows from a comparison of the spectra of the operators on the left and right sides of (3.17) (for example, in L2 (a, b)). Therefore, in the general case, it is impossible to construct transmutation of the form (3.15) satisfying the identity (3.9) with a smooth kernel. Of course, the identities (3.16)–(3.17) can hold in the case of q(x) ≡ 0. In this case, the item weaves D 2 → D 2 , that is, commutes with the second derivative. Such operators exist. For this, it is enough, for example, that the core satisfies the wave equation G(x, t) =

∂ 2K ∂ 2K = ∂x 2 ∂t 2 and boundary conditions b b ∂K   H (x) = K(x, t)f (t) a − f (t) = 0. ∂t a 

(3.18)

Here after choosing the class values f (x), f  (x) at x = a, x = b are fixed. Usually, condition (3.18) gives  

= f (x) ∈ C 2 (a, b) | f (a) = f (b) = f  (a) = f  (b) = 0 . Remark 7. (1) The kernel K(x, t) independent of f (x) does not exist. (2) When q(x) = μ = const, the spectrum Af = q · f = muf (x) consists of one point. The above reasoning justifies the impossibility of constructing a transmutation operator of the form (3.15) for sufficiently smooth potentials q(x). If q(x) has singularities for x ∈ [a, b], then the comparison of the spectra on which we were based does not work. In this case, the operator Af = q · f is not defined in standard spaces. Therefore, the question of constructing a transmutation operator for potentials q(x) with singularities remains open. Therefore, it is interesting to note that for the potentials q(x) ∼ 1/x 2 for x → 0 there is an interesting class of transmutation operators of the form (3.15). These are

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Transmutations, Singular and Fractional Differential Equations

Kram–Krein transmutations. They are defined only on solutions of the equations Ly = 0 with regular and singular potentials and relate such solutions to each other. Note that the condition (3.18) of the form    K(x, t) f (t)    Wt (K, f ) =  ∂K ∂f  ∂t

∂t

at points x = a and x = b arises naturally in the theory of Kram–Krein operators. It is these conditions that allow us to write the Kram–Krein transmutations in the integral form (3.15) (see [4]). In the general case, these operators are differential [376]. Now let us consider the problem of constructing a transmutation operator in the form (3.11): b (Sf )(x) =

K(x, t)f (t) dt. a

Substitution of this expression in (3.9) gives b

d2 K(x, t)[f (t) + q(t)f (t)] dt = 2 dx 

a

b K(x, t)f (t) dt. a

We assume that integration by parts and under the sign of the integral is possible. This leads to the equality b a

b

b ∂K  ∂ 2K ∂ 2K   −  . − − q(t)K f (t)] dt = K(x, t)f (t) f (t)  a 2 2 ∂t ∂x ∂t a

Let f (t) ∈ and let be dense in L2 (a, b). Then it is enough to demand ∂ 2K ∂ 2K = 2 + q(t)K, 2 ∂x ∂t b b ∂K  f (t) = Wt (K, f )(x) = W (x) = 0. K(x, t)f  (t)a − ∂t a Here we denote by Wt (K, f ) the Wronsky determinant    K(x, t) f (t)   Wt (K, f ) =  ∂K(x,t) ∂f  . ∂t

(3.19) (3.20)

(3.21)

∂t

Next, we consider solutions to Eq. (3.21). It is known that for the existence of a classical C 2 -solution it is necessary to require q(t) ∈ C 1 (a, b). In this case, the Riemann function of Eq. (3.19) R ∈ C 2 . If we restrict ourselves to the condition q(t) ∈ C(a, b), then there is only a generalized solution to (3.19) from the class C 1 (in this

Basic definitions and propositions

95

case, the Riemann function R ∈ C 1 ). Therefore, until the end of this subsection we will consider the potential q ∈ C 1 in the domain of our definition. That gives us the next theorem. Theorem 27. Let a, b, c, d ∈ R and K(x, t) ∈ C 2 ([c, d], [a, b]). Class is the set of functions f (t) such that (1) (2)

f (t) ∈ C 2 (a, b),

b b ∂K  f (t) = 0. Wt (K, f ) = K(x, t)f  (t)a − ∂t a

(3.22)

Then for existence of the transmutation operator of the form (3.11) for f ∈ it is enough that the kernel K(x, t) satisfies the hyperbolic Eq. (3.19). Note that practically, except in very special cases, the condition (3.22) forces us to accept  

= f ∈ C 2 (a, b) | f (a) = f (b) = f  (a) = f  (b) = 0 . It follows from (3.22) that at least always contains functions that are compactly supported on (a, b). Therefore, the assumption made about the density in L2 (a, b) is always fulfilled. The authors did not find in the literature constructions of the type (3.11). Nevertheless, we will show further that in practice this kind exists. For some simple potentials, they will be constructed explicitly.

3.2.3 Transmutations in the form of the second kind Volterra operators We proceed to the construction of one of the most important classes of transmutation operators. These are Volterra operators of the second kind, having the form (3.13) x (Sf )(x) = f (x) +

K(x, t)f (t) dt. c

Such operators in the case c = 0 were first obtained by Povzner (see [325,454]). The ∞ case (Sf )(x) = f (x) + K(x, t)f (t) dt was introduced by Levin (see [313,314]). x

We will consider them in the form x (Sf )(x) = f (x) +

K(x, t)f (t) dt, c

where c ∈ R, which allows us to combine these cases and unify the text.

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Transmutations, Singular and Fractional Differential Equations

Substitution in formula (3.9) leads to the relation f  (x) + q(x)f (x) +

x

K(x, t)[f  (t) + q(t)f (t)] dt

c

= f  (x) +

d2 dx 2

x K(x, t)f (t) dt. c

Transforming the first part of this formula, we obtain

x ∂ 2K + q(t)K f (t) dt + f  (x) + q(x)f (x) + K(x, t)f  (t)c 2 ∂t x  ∂K − f (t) . ∂t c

x c

Transforming the remaining part, d2 dx 2

x K(x, t)f (t) dt c

x =

∂ 2K ∂K(x, x) f (t) dt + f (x) + K  (x, x)f (x) + K(x, x)f  (x). 2 ∂x ∂x

c

Equating the corresponding terms, we obtain ∂2 ∂ 2K + q(t)K = , 2 ∂x ∂t

d ∂K ∂K K(x, x) + lim + = q(x), t→x ∂x dx ∂t lim Wt (f, K(x, t)) = lim W (x) = 0. t→c

t→c

(3.23) (x, t) ∈  ⊂ R2 ,

(3.24) (3.25)

By  we denote the domain of the function K(x, t), whose closure contains part of the diagonal t = x. Eq. (3.23) is standard in the problem we are considering. The condition (3.25) highlights the point c. This condition reduces to weighted boundary conditions on the function f (x) and its first derivative f  (x) for x → c. The relation (3.25) dictates the choice of the class . Now we show that the condition (3.24) can be simplified if we assume that K ∈ C 1 . For this, we prove two technical lemmas.

Basic definitions and propositions

97

Lemma 10. Let K(x, t) ∈ C 1 (),  ∩ {(x, t) | x = t} = ∅. Then for (t, x) ∈  the equality

∂K ∂K d K(x, x) = lim (x, t) + (x, t) t→x ∂x dx ∂t

(3.26)

is valid. Proof. Consider the function of two variables K(x, y), the variables themselves depend on the parameter t: x = φ(t), y = ψ(t). Then using the formula ∂K dx ∂K dy d K(x, y) = + dt ∂x dt ∂y dt

(3.27)

from [143] and putting x = y = t in (3.27) we obtain d K(t, t) = dt



 ∂K ∂K  . + ∂x ∂y x=y=t

This formula is equivalent to (3.26). The second lemma is proved similarly. Lemma 11. Under the conditions of Lemma 10 we have

∂K ∂K d K(x, −x) = lim − . t→−x ∂x dx ∂t

(3.28)

So, by Lemma 10 the relation (3.24) takes the form 1 d K(x, x) = q(x). dx 2

(3.29)

Equality (3.29) shows that the value of the core K(x, t) on the diagonal t = x allows one to reconstruct the potential q(x). This fact is fundamental in the theory of inverse problems. Therefore, the most common methods for solving inverse problems come down to finding the kernel of the transformation operator by the spectral function (as in the Gelfand–Levitan equation) or by scattering data (as in the Marchenko equation). Assume that the core K ∈ C 2 (), where  = {(x, t) | x ∈ [a, b], t is between c and x} . Define ∈ C 2 (a, b) as a set of functions f , satisfying the condition (3.25). Let c ∈ R and let kernel K and class satisfy the conditions stated above. Then for existing transmutation operators of the form (3.14) for functions f ∈ it is sufficient that the kernel K satisfies Eq. (3.23) and condition (3.29).

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Transmutations, Singular and Fractional Differential Equations

3.2.4 Transmutations in the form of the first kind Volterra operators Let us consider the possibility of construction of transmutations in the form of the first kind Volterra operators, i.e., in the form x (Sf )(x) =

(3.30)

K(x, t)f (t) dt. c

Practically repeating the calculations of the previous subsections, Theorem 27 is valid and relations (3.23) and (3.25) are the same but instead of (3.24) we get

d ∂K ∂K K(x, x) + lim + = 0. t→x dx ∂t ∂x

(3.31)

Of course, for kernels smooth up to the diagonal t = x this condition is replaced by d K(x, x) = 0, dx

K(x, x) = const.

(3.32)

However, we prefer a more accurate notation (3.31), since the kernel may have a gap at t → x. This is permissible. An example of a transmutation operator of the form (3.30) is the fractional integral

(Sf )(x) =



α Ic+



1 (x) = (α)

x (x − t)α−1 f (t) dt,

α > 0.

c

This operator intertwines D 2 and D 2 , i.e., commutes with D 2 . Condition (3.31) obviously is valid for the kernel K(x, t) =

1 (x − t)α−1 . (α)

Note that for 0 < α < 1, it is precisely (3.31) that is valid, not (3.32). For α > 1, both of these relations hold. The condition (3.25) reduces to f (c) = f  (c) = 0. These conditions are rougher than the minimum sufficient. The latter are given, for example, in [494]. Instead of the introduced Riemann–Liouville fractional integration operators, we can consider the Weil operators or fractional integrals over an arbitrary function (2.38) and (2.39).

Basic definitions and propositions

99

Let us consider an important case of transmutations of the form (3.14). Using integration by parts, we calculate the left side in (3.9): x



f (x) + q(x)f (x) + x = −x

K(f  (t) + q(t)f (t)) dt

−x

x x  ∂K ∂ 2K    + q f dt+f (x)+ q(x)f (x) + K(x, t)f (t) −x − f (t) . 2 ∂t ∂t −x

Calculating the right side in (3.9) we obtain d2 f (x) + 2 dx 

x K(x, t)f (t) dt

−x x



= f (x) + −x x

= f  (x) +

−x

x    ∂ 2K ∂K f dt + f (t) + K(x, t)f (t)|x−x 2 ∂x ∂x −x ∂ 2K ∂K ∂K f dt + lim f (t) + lim f (t) 2 t→x ∂x t→−x ∂x ∂x

+ K  (x, x)f (x) + K(x, x)f  (x) + K  (x, −x)f (−x) + K(x, −x)[f (−x)] . Therefore x −x

∂ 2K + q(t)K f dt + q(x)f (x) + K(x, x)f  (x) − lim K(x, t)f  (t) t→−x ∂t 2

− x = −x

∂K ∂K (x, x)f (x) + lim f (−x) t→−x ∂t ∂t d 2K ∂K f dt + lim f (x) + K  (x, x)f (x) 2 t→x ∂x dx

+ K(x, x)f  (x) + lim

t→−x

+ K(x, −x)

∂K d f (t) + (K(x, −x)) f (−x) ∂x dx

d lim f (t). dx t→−x

In order for the calculations to make sense, we assume that f (x) ∈ ⊂ C 2 (a, −a) for some a > 0. Then it is enough that again the kernel satisfies the equation ∂ 2K ∂ 2K = + q(t)K ∂x 2 ∂t 2

(3.33)

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Transmutations, Singular and Fractional Differential Equations

and the additional condition

d  K(x, −x) lim f (t) + K(x, −x) lim f (t) t→−x dx t→−x

d ∂K ∂K + f (x) K(x, −x) + lim + − q(x) t→x ∂x dx ∂t

d ∂K ∂K − f (−x) lim − − K(x, −x) = 0. t→x ∂t ∂x dx In the last expression, the term in parentheses is zero. Therefore, the equality

d ∂K ∂K f (x) K(x, −x) + lim + − q(x) t→x ∂x dx ∂t

d ∂K ∂K = f (−x) lim − − K(x, −x) (3.34) t→x ∂t ∂x dx is true in the case of the smooth down to the line t = x kernel K(x, t). Expression (3.34) according to (3.31) can be simplified to     f (x) 2K  (x, x) − q(x) + f (−x) 2K  (x, −x) = 0. (3.35) A simple analysis of the relation (3.35) shows that if f (x) and f (−x) are independent, then both equalities should be satisfied in the case of smooth down to the line  K  (x, x) = 12 q(x), (3.36) K  (x, −x) = 0. For even functions we get the equality 1 K  (x, x) + K  (x, −x) = q(x), 2

(3.37)

and for odd functions we get 1 K  (x, x) − K  (x, −x) = q(x). 2

(3.38)

Let the kernel K(x, t) ∈ C 2 in the considered domain for some a > 0. We define the class of functions = C 2 (−a, a) (without any boundary conditions!). Then, in order for the operator (3.14) to be transmutation operator on functions f ∈ , it is enough that the kernel satisfies Eq. (3.33) and the system (3.36). If the class consists of even functions on (−a, a), then (3.36) is replaced by (3.37). If the class consists of odd functions on (−a, a), then (3.36) is replaced by (3.38). It is important to note that transmutations of the form (3.14) considered in this subsection essentially do not require any conditions on the function (of course, except

Basic definitions and propositions

101

for smoothness). Apparently, it is precisely these operators that are the most general and, in a certain sense, “natural.” Now let us show that every transmutation of the form (3.14) generates transmutation (3.15) with c = 0. Moreover, the relations for kernels (3.33)–(3.36) give relations (3.23)–(3.25) for new kernels. In this case, the potential q(x) will be assumed to be even. So let the transmutation of the form (3.15) be defined on the set of even functions . Then for x > 0 x (Sf )(x) = f (x) +

x K(x, t)f (t) dt = f (x) +

−x x

= f (x) +

[K(x, t) + K(x, −t)]f (t) dt 0

G(x, t)f (t) dt.

(3.39)

0

We obtained transmutation of the form (3.15) for c = 0. Let us show that the new kernel G(x, t) satisfies the conditions of Theorem 27. It is clear that the smoothness conditions of G are fulfilled (we assume that they were satisfied for the original kernel K). Let us verify that for x ∈ [0, a] and the functions f (x) ∈ , the class is defined by equality (3.25). Actually,

∂ ∂ Wt (f, G) = f (t) K(x, t) − K(x, −t) − (K(x, t) + K(x, −t)) [f  (t)]. ∂t ∂t For t → 0 both expressions in square brackets tend to zero (since f (t) is even we have f  (0) = 0). Therefore condition (3.25) is satisfied. Condition (3.23) follows from (3.33) and (3.24) or in our case (3.29) follows from (3.37). Thus, by Theorem 27, the operator (3.39) is a transmutation operator on functions from the class defined on [0, a] and admitting smooth even extension. The question whether it is possible to construct a transmutation operator of the form (3.14) with c = 0 using an operator of the form (3.15) is more complicated. It is connected to the question whether it is possible to smoothly extend the solutions of the hyperbolic Eq. (3.23) from the segment [0, a] to the segment [−a, a]. Obviously, an operator of the form (3.39) can be determined by the operator (3.14) and on functions that admit an odd continuation. Finally, we consider the question of necessary and sufficient conditions of existing transmutation operators of the form (3.14) with the kernel K ∈ C 2 (). Theorem 28. We assume that the kernel K(x, t) in (3.14) belongs to the class C 2 () and f ∈ . Then in order that the transmutation operator of the form (3.14) exists it is necessary and sufficient that the relations (3.23) and (3.29) are satisfied. Proof. Sufficiency was proved in Theorem 27. We turn our attention to necessity. Let the transmutation operator (3.14) with a kernel K ∈ C 2 exist on . Let us show that (3.23) and (3.29) hold.

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Transmutations, Singular and Fractional Differential Equations

Calculations in (3.9) show that since f ∈ , we have x

∂ 2K ∂ 2K + q(t)K − 2 ∂t ∂x 2



  f (t) dt + q(x) − 2K  (x, x) f (x) = 0

c

or x (Af )(x) =

∂ 2K ∂ 2K + q(t)K − ∂t 2 ∂x 2

f (t) dt = (Bf )(x) = g(x)f (x),

c

g(x) = 2K  (x, x) − q(x) ∈ C 1 (a, b). Both operators A and B act from L2 (a, b) to L2 (a, b), where (a, b) is an arbitrary segment on which q(x), f (x), and K(x, t) are defined. We compare their spectra. Operator A is the Volterra operator. Operator A has only one point in the spectrum λ = 0, λ ∈ C. The operator B has a continuous spectrum consisting of the set of values of the function g(x) defined on the closed segment [a, b]. If g(x) is defined on the open interval (a, b), then the spectrum coincides with the closure of its set of values. Therefore, the set of values of the function g(x) consists of a single point, that is, g(x) ≡ 0. Therefore, we have (3.29) and the equality x

∂ 2K ∂ 2K + q(t)K − ∂t 2 ∂x 2

f (t) dt = 0.

c

The kernel of this operator is from C(a, b) ⇒. Therefore, K ∈ L2 (a, b), since mes(a, b) < ∞. Function f ∈ , and is dense in L2 (a, b); therefore the kernel is equal to zero and (3.23) is true. For cases of nonsmooth potentials q(x) or kernels K(x, t) with singularities, for example, for x = t, this proof is not correct. In essence, they come down to a superposition of the operator of multiplication by the function p(x) and ordinary transmutations. Such a reduction is difficult if p(x) = 0 for some admissible x. In conclusion of this section, we show that linear operators in the general case admit a natural construction for linear differential operators. Theorem 29. There is no nonzero linear operator S satisfying the identity S(D 2 y + y 2 ) = D 2 Sy

(3.40)

for arbitrary f ∈ C 2 [a, b]. Proof. Let (3.40) be valid. Then we choose an arbitrary function y(x) and in addition to the relation (3.40) for y(x) we can write this relation for f (x)=y(x)+1: S(D 2 y + y 2 + 2y + 1) = D 2 S(y + 1).

Basic definitions and propositions

103

Since S is linear we get S(D 2 y + y 2 )(x) + 2S(y)(x) + S(1)(x) = D 2 S(y)(x) + D 2 S(1)(x). Now using (3.40) we can write  1 2 S(y)(x) = D S(1)(x) − S(1)(x) . 2 So operator S does not depend on y. Again due to linearity S(y)(x) = S(0)(x) = 0. This theorem holds for any class of the function ∈ C 2 , which, together with each f (x), also contains f (x) + 1. This condition is not satisfied for the class if, for example, we fix the boundary condition at zero f (0) = 0. The following theorem shows, even on a very narrow class of functions with maximally restrictive boundary conditions, that there is no linear transmutation operator D 2 + (·) → D 2 . Theorem 30. On a class of functions 

= f (x) ∈ C 2 (a, b) | (D n f )(c) = 0,

n ∈ N,

 c ∈ (a, b)

there is no linear transmutation operator satisfying the relation (3.40). Proof. Note that if f ∈ , then λf ∈ , ∀λ and f 2 ∈ . Let us write (3.40) for y(x) = λf (x), where λ ∈ C but λ = 0. We have S(D 2 λf + λ2 f 2 ) = D 2 S(λf ). By linearity, we obtain S(D 2 f + λf 2 ) = D 2 S(f ). Subtract from this equality (3.40) for y = f . We have (λ − 1)S(f 2 )(x) = 0,

∀f ∈ .

Therefore S(y) = 0 for y ≥ 0. Due to linearity we get S ≡ 0

∀y.

However, it should be noted that linear operators find the most important applications in the theory of nonlinear equations. For example, they are an important part of the methods of the inverse problem of integrating nonlinear evolution equations. Of course, the above considerations are applicable to S(D 2 y + y α ) = D 2 Sy. The following theorem holds for the general case. The meaning of this interesting result is that the existence of a good operator S : D 2 + H (y) → D 2 necessarily leads to linearity of the function itself H (y).

(3.41)

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Transmutations, Singular and Fractional Differential Equations

Theorem 31. Let be some linear, ⊂ C 2 (a, b), and H (y) : → . Then if the linear transmutation operator (3.41) S : → exists, then operator SH (y) is also a linear operator from to . Proof. It is obvious that SH (y) : → . Let us calculate SH (λy). Since λy ∈ , from (3.41) we get   S D 2 λy + H (λy) = D 2 S(λy), SH (λy) = λ(D 2 S(y) − SD 2 y) = λSH (y). Additivity is verified in the same way: SH (y1 + y2 ) = D 2 S(y1 ) − SD 2 y1 + D 2 S(y2 ) − SD 2 y2 = SHy1 + SHy2 . The theorem is proved similarly. Theorem 32. Suppose that the conditions of Theorem 31 are satisfied. Then if S is invertible in some space L containing , then the operator H (y) is linear in y.

3.3 Transmutations for different potentials In this section we construct transmutation operators intertwining operators of the Sturm–Liouville type from the previous section for different concrete potentials q.

3.3.1 Kernel of transmutation intertwining operators of the Sturm–Liouville type Note that in all the cases considered by us, the equations for the kernels of the transmutations coincide (see (3.19), (3.23), and (3.33)). Therefore, we should use some of the methods for solving this hyperbolic equation. The following action plan is usually implemented. At the first step, we pass from the partial differential equation to the integral one. These equations are not equivalent, but each solution of the integral equation satisfies the original hyperbolic. At this step, the existence of a certain kernel of the transmutation operator and its certain smoothness are proved. At the second step, some additional conditions for the kernel are checked and the appropriate class of functions is selected. This completes the construction of the transmutation operator. So, let us move on to solving the equation ∂ 2K ∂ 2K = + q(t)K ∂x 2 ∂t 2

(3.42)

with an additional condition on the diagonal x = t (see (3.24), (3.29), and (3.36)), 1 K(x, x) = q(x). 2

(3.43)

Basic definitions and propositions

105

We perform the standard change of variables by the formulas 1 u = (x + t), 2

1 v = (x − t). 2

(3.44)

The diagonal equality x = t in the new variables takes the form v = 0. We introduce the notation for the kernel in new variables H (u, v) = K(u + v, u − v) = K(x, t).

(3.45)

For function H we pass from the relations (3.42) and (3.43) to the new ∂ 2H = q(u − v)H, ∂u∂v u 1 q(s) ds. H (u, 0) = 2

(3.46) (3.47)

c

Here function q(u − v) should be defined, c is an arbitrary number, possible c = ±∞. An important point to make follows here. It follows from (3.44) that both variants, u > 0, v > 0 and u < 0, v < 0, are possible. The system (3.46)–(3.47) is a Cauchy problem with only one initial condition. So under our assumptions on q (q(x) ∈ C 1 ) this system has infinitely many solutions. Therefore, for every potential q(x) there are infinitely many transmutations, for example, of the form (3.14). This is extremely convenient in applications where it is possible to choose, with the same potential, different operations most suitable for each specific problem. One of the ways to construct kernels satisfying (3.33) and (3.34) is using the formula 1 H (u, v) = 2

u

u q(s) ds

c

v q(α − β)H (α, β) dβ.

dα d

(3.48)

0

We should check that each C 2 -function of the form (3.48) satisfies (3.46) and (3.47). Arbitrary numbers c, d are from R. We note again that u, v, α, and β in (3.48) can have any sign. Another way of solving (3.48) is using a Riemann function. The importance of studying Eq. (3.48) with different c, d is that we can simultaneously study the case c = d = 0, which arises when constructing operators of the Povzner–Levitan type and the case c = d = +∞, which arises when constructing Levin type operators, and the case of arbitrary different c, d. Usually, these types of operators were studied separately, and the case of arbitrary c (c = 0, c = ±∞) was not considered. The further content of this chapter essentially consists in studying Eq. (3.48) under various assumptions. The most important case for us will be c = d. In specific

106

Transmutations, Singular and Fractional Differential Equations

examples, it is usually convenient to select one of the following values: −∞, 0, or +∞. When constructing transmutations of the form (3.15) it is usually also convenient (although not necessary) if the values of c in formulas (3.15), (3.47), and (3.48) coincide. Next, we consider those simplest potentials q(x) that admit the construction of transmutations explicitly. Here, various solutions of Eq. (3.48) will be denoted identically by H (u, v). We will only be interested in deriving formulas for the nuclei themselves. From these formulas, their continuity and the existence of the desired number of continuous derivatives will automatically follow.

3.3.2 Cases when potential q(x) is an exponential function Consider the following problem. Find a solution to Eq. (3.48) under the conditions q(x) = ex ,

c = d = −∞.

Then Eq. (3.48) takes the form 1 H (u, v) = eu + 2

u

v α

e dα

−∞

e−β H (α, β) dβ.

(3.49)

0

Using the method of successive approximations, we set 1 H0 (u, v) = eu , 2 u v α Hn+1 (u, v) = e dα e−β Hn (α, β) dβ. −∞

0

We obtain the first interactions u H1 (u, v) =

v α

e dα

−∞

=

1 e2u 2 2!

e

−β 1 α

1 e dβ = 2 2

0

v

u

v e

2α

−∞

dα

e−β dβ

0

e−β dβ,

0

1 e3u H2 (u, v) = 2 3!

v e 0

−β2

β2

e−β1 dβ1 dβ2 .

0

Using mathematical induction, it is easy to show that Hn (u, v) =

1 e(n+1)u yn (v), 2 (n + 1)!

(3.50)

Basic definitions and propositions

107

where yn (v) is defined by v yn (v) =

e

−βn

0

βn e

−βn−1

β2 ...

0

The formula 

1 − e−v yn (v) = n!

e−β1 dβ1 .

(3.51)

0

n (3.52)

is valid. Using mathematical induction we get y0 (v) = 1,  n v v 1 − e−v yn+1 (v) = e−β yn (β) dβ = e−β dβ n! 0

0

1 = n!  =

v



1−e

 −β n

0

(−1) de

−β

(−1) = n!



e−v (1 − t)n dt 1

n+1 e−v  e = t, 1 − e−v 1 (1 − t)n+1  = . = n! (n + 1) 1 (n + 1)! − e−β dβ = dt −β

Substitution of (3.52) in (3.50) gives  n 1 e(n+1)u 1 − e−v 1 Hn (u, v) = = eu [q(u, v)]n , 2 (n + 1)! n! 2n!(n + 1)! where   q(u, v) = q = eu 1 − e−v . From here we obtain solutions of Eq. (3.49) in the form of the Neumann series: H (u, v) =

∞  n=0

Hn (u, v) =

∞ qn eu  . 2 n!(n + 1)! n=0

The identity ∞  n=0

 √  qn 1 = √ I1 2 q n!(n + 1)! q

is valid. Here q ∈ C, I (·) is a modified Bessel function of the first kind (1.16).

(3.53)

108

Transmutations, Singular and Fractional Differential Equations

By formula (1.16) we have I1 (x) =

∞  (x/2)2n+1 n=0

∞

x  (x 2 /4)n = . n!(n + 1)! 2 n!(n + 1)! n=0

√

Denoting q = x 2 /4, x = 2 q we can write ∞  √  √  I1 2 q = q n=0

qn . n!(n + 1)!

That gives (3.53). Now from (3.53) we obtain a solution to (3.49) in the form  √  eu 1 √ I1 2 q 2 q

   eu 1 u −v I1 2 e 1 − e =   2 eu 1 − e−v 

H (u, v) =

1 = 2



   eu u 1 − e−v I e . 2 1 1 − e−v

Now consider the same problem, but under the conditions q(x) = ex ,

c = d = 0.

Since all arguments are similar, we will outline them briefly. We have 1 H0 (u, v) = 2

u es ds, 0

u Hn+1 (u, v) =

v α

e dα 0

1 H1 (u, v) = 2

0

u

α2 α2

v α1

e dα2 0

e−β Hn (α, β) dβ,

e dα1 0

e−β1 dβ1 .

0

So 1 Hn (u, v) = zn (u)yn (v), 2 where yn (·) is defined by (3.52). Let us find a formula for zn (u): z0 (u) = eu − 1,

(3.54)

(3.55)

Basic definitions and propositions

u z1 (u) =

109

  (eu − 1)2 , eα eα − 1 dα = 2!

0

u zn+1 (u) =

u e zn (u) = α

0

αn+1

e

αn+1

dαn+1

0

α2 eα1 dα1 .

... 0

(3.56)

0

We obtain the statement proved by induction: zn (u) =

(eu − 1)n+1 . (n + 1)!

(3.57)

Therefore,  √   1 1 u e − 1 √ I1 2 q 2 q  

u   1 e −1 u −v , I1 2 (e − 1) 1 − e = 2 1 − e−v

H (u, v) =

(3.58)

where    q = eu − 1 1 − e−v . Finally, we consider the general case, including both previous ones as particulars. We have 1 H (u, v) = 2

u

u e ds + s

c

v α

e dα c

e−β H (α, β) dβ.

0

The first iterations have the forms 1 H0 (u, v) = 2

x es ds, c

1 H1 (u, v) = 2

u

α1 α2

e dα2 c

v α1

e dα1 c

e−β dβ.

0

We obtain the statement proved by induction: 1 Hn (u, v) = zn (u)yn (v), 2  n v −s e ds n  1 − e−v 0 = , yn (v) = n! n!

110

Transmutations, Singular and Fractional Differential Equations

u zn (u) =

αn+1

eαn+1 dαn+1 c

u 

α2 eα1 dα1 =

eαn dαn . . . 0

0

n+1 es

ds

c

(n + 1)!

.

(3.59)

Therefore

⎞ ⎛ u   √  1 1⎝ es ds ⎠ √ I1 2 q H (u, v) = 2 q c 

   1 eu − ec u c −v = , I1 2 (e − e ) 1 − e 2 1 − e−v

(3.60)

where    q = eu − ec 1 − e−v . Special cases of formula (3.60) are (3.58) for c = 0 and (3.55) for c = −∞. Now we consider the problem of solving Eq. (3.48) (and, therefore, the construction of the transformation operator) for the case of the potential of the opposite sign c = d,

q(x) = −ex ,

c < ∞.

The equation takes the form 1 H (u, v) = − 2

u

u e ds − s

c

v α

e dα c

e−β H (α, β) dβ.

0

The first iteration and general formulas are as follows: 1 H0 (u, v) = − 2

u es ds, c

u Hn+1 (u, v) = −

v α

e dα c

e−β H (α, β) dβ.

0

The assumption proved by induction is 1 Hn (u, v) = (−1)n+1 zn (u)yn (v). 2 The kernel is

∞  1  u (−q)n H (u, v) = − e − ec , 2 n!(n + 1)! n=0

(3.61)

Basic definitions and propositions

111

where u

v es ds ·

q= c

   e−s ds = eu − ec 1 − e−v .

0

For q ∈ C we have (see (1.13)) ∞  n=0

 √  (−q)n 1 = √ J1 2 q , n!(n + 1)! q

(3.62)

where J1 (·) is the Bessel function of the first kind. From (1.13) we have

Jν (z) =

 2 n

∞  z ν 

2

n=0

z 4

n!(ν + n + 1)

,

2 √ and putting ν = 1, q = z4 , z = 2 q we get (3.62). Formula (3.62) is also immediately obtained from (3.53) after replacing q by −q. Therefore, for the kernel in this case, we have the formula



1 H (u, v) = − 2



   eu − ec u − ec ) 1 − e−v . J 2 (e 1 1 − e−v

Let us solve Eq. (3.48) in the case q(x) = e−x ,

c = d,

c = −∞.

We have 1 H (u, v) = 2

u e c

−s

u ds +

e

−α

c

v eβ H (α, β) dβ,

dα 0

1 Hn (u, v) = yn (u)zn (v), 2  −c n+1 e − e−u yn (u) = , (n + 1)! (ev − 1)n zn (v) = , n!   ∞    √  qn 1  −c 1 e−c − e−u −u · H (u, v) = e − e = I1 2 q , √ 2 n!(n + 1)! 2 q n=0

(3.63)

112

Transmutations, Singular and Fractional Differential Equations

  where q = e−c − e−u (ev − 1) or 1 H (u, v) = 2



 

  e−c − e−u −c −u v I1 2 e − e (e − 1) . (ev − 1)

(3.64)

Similarly, in the case q(x) = −e−x , c = d = −∞, solution to (3.48) has the form 1 H (u, v) = 2



 

  e−c − e−u −c − e−u (ev − 1) . e J 2 1 (ev − 1)

(3.65)

Applying the well-known formulas t

t1 dt1

a

a

x

x1 dx1

a

tn−1 dt2 . . . f (t1 )f (t2 ) . . . f (tn ) dtn =

t 

n f (s) ds

a

n!

,

(3.66)

a xn−1

f (xn ) dxn =

dx2 . . . a

x

a

f (t) a

(x − t)n−1 dt, (n − 1)!

(3.67)

we obtain (3.59), (3.57), and (3.52) without using mathematical induction. Usually formulas (3.66) and (3.67) are associated with the name of Dirichlet.

3.3.3 Cases when potential q(x) is constant Now we turn to the consideration of the important case when the potential is constant, q(x) = λ2 = const,

λ ∈ C.

Solution to Eq. (3.48) in this case are kernels of transmutations intertwining D 2 ± λ2 and D 2 . These operators are particular cases of more general operators intertwining the Bessel operator with a spectral parameter and the second derivative S : Bν ± λ2 → D 2 ,

Bν =

d2 2ν + 1 d + . 2 x dx dx

Such operators appear in papers of A. Erdélyi [122–126], I. N. Vekua [581], and J. S. Lowndes [337–339]. Therefore, it is natural to call them Erdélyi–Vekua–Lowndes (EVL). Here we got the simplest EVL operators. So, consider Eq. (3.48) for q(x) = λ2 , c = d ∈ R, i.e., c is a finite number: 1 H (u, v) = λ2 (u − c) + λ2 2

u

v dα

c

H (α, β) dβ. 0

Basic definitions and propositions

113

Iterations are determined by the formulas 1 H0 (u, v) = λ2 (u − c), 2 u v Hn+1 (u, v) = λ2 dα Hn (α, β) dβ. c

0

From the first iterations we get 1  2 2 (u − c)2 λ v, 2 2! 1  2 3 (u − c)3 v 2 H2 (u, v) = λ , 2 3! 2! H1 (u, v) =

and  n+1 1 λ2 (u − c)n+1 v n . H (u, v) = 2 n!(n + 1)!

(3.68)

Summing up the Neumann series using formula (3.53) we obtain H (u, v) =

   1 λ2 I1 2 λ2 (u − c)v . (u − c)  2 λ2 (u − c)v

(3.69)

In particular, for real λ > 0 formula (3.69) for the kernel can be transformed to   λ u−c   (3.70) I1 2λ (u − c)v . H (u, v) = 2 v It has the simplest form when c = 0:  λ u  √  I1 2λ uv . H (u, v) = 2 v

(3.71)

Similar results hold for a negative constant potential. We have q(x) = −λ2 ,

λ ∈ C.

If λ ∈ C, c ∈ R, then

2   u−c   2 λ 1 J1 2 λ (u − c)v . H (u, v) = − √ 2 v λ2 If λ ∈ R+ (λ > 0), c ∈ R, then

  1 u−c   H (u, v) = − λ J1 2λ (u − c)v . 2 v

(3.72)

(3.73)

114

Transmutations, Singular and Fractional Differential Equations

Finally, if λ > 0, c = 0, then  λ u  √  J1 2λ uv . H (u, v) = − 2 v

(3.74)

The formulas for the kernels for q(x) = −λ2 can be obtained from the formulas for q(x) = λ2 by the formal replacement λ → iλ. Note that for λ = 0, all the kernels considered vanish, and transmutations of the form (3.6)–(3.15) are identical. Now we consider the general case when Eq. (3.48) is taken with q(x) = λ2 for different c, d, 1 H (u, v) = λ2 (u − c) + λ2 2

u

v dα

d

H (α, β) dβ. 0

From the first iterations, we easily obtain 1  2 n+1 v n Hn (u, v) = λ yn (u), 2 n!

(3.75)

where yn (u) are defined by y0 (u) = u − c, u y1 (u) = (x1 − c) dx1 , d

u yn+1 (u) =

yn (s) ds. d

Therefore, the formula u yn (u) =

x1 dx1

d

xn−1

(xn − c) dxn

dx2 . . . d

d

is valid. To rewrite this expression we use (3.67). So u yn (u) =

(t − c) d

u yn (u) =

(u − t)n−1 dt, (n − 1)!

n ≥ 1,



(u − t)n  (t − c) (−1) dt n!

d

 u (u − t)n u (u − t)n = (−1)(t − c) − (−1) dt  n! n! d d

Basic definitions and propositions

115

u (u − d)n (u − d)n+1 (u − t)n+1  (u − d)n = (d − c) = − + (d − c) .  n! (n + 1)! d (n + 1)! n! Obviously, the last formula retains meaning even for n = 0. Substituting it into (3.75), we get

1  2 n+1 v n (u − d)n+1 (u − d)n λ + (d − c) Hn (u, v) = 2 n! (n + 1)! n!   2  n n 2     λ (u − d)v λ (u − d)v 1 2 1 2 . λ (u − d) + λ (d − c) = 2 n!(n + 1)! 2 (n!)2 Using formula (3.53) and the relation for I0 (·) of the form (see [2])  2 n z ∞ ∞    √  4 qn I0 (z) = , = I0 2 q , (n!)2 (n!)2 n=0

n=0

we obtain the Neumann series for the sum  √   √  1 1 1 H (u, v) = λ2 (u − d) √ I1 2 q + (d − c) λ2 I0 2 q . 2 q 2

(3.76)

Here q = λ2 (u − d)v. For λ > 0 expression (3.76) allows    λ2   λ u−d   I1 2λ (u − d)v + (d − c) I0 2λ (u − d)v , H (u, v) = 2 v 2 (3.77) which for d = c gives (3.70). Another interesting feature is the choice d = 0:  λ u  √  cλ2  √  I1 2λ uv − I0 2λ uv , H (u, v) = 2 v 2

(3.78)

where the parameter c can take any values. Finally, for the potential q(x) = −λ2 , taking into account the relations for the Bessel function of the imaginary argument, we obtain the following formulas: 1 λ2 1 √ √ H (u, v) = − λ2 (u − d) √ J1 (2 q ) − (d − c) J0 (2 q ), 2 q 2

(3.79)

under conditions λ ∈ C, q = λ2 (u − d)v;    λ u−d   λ2   H (u, v) = − J1 2λ (u − d)v − (d − c) J0 2λ (u − d)v 2 v 2 (3.80)

116

Transmutations, Singular and Fractional Differential Equations

under the condition λ > 0, and  λ u  √  cλ2  √  H (u, v) = − J1 2λ uv + J0 2λ uv 2 v 2

(3.81)

for λ > 0, d = 0. Formulas (3.79)–(3.81) give kernels of transmutations intertwining D 2 − λ2 → D 2 . Now we consider some other relations valid for any linear EVL operators satisfying the identity Sα (D 2 + α)f = D 2 Sα f,

α ∈ C,

(3.82)

for arbitrary functions f . These are formal equalities. Now we consider some other relations valid for any linear EVL operators satisfying Sα D 2 = (D 2 − α)Sα f,

(3.83)

(Sα )2 (D 2 + α) = Sα Sα (D 2 + α) = (D 2 − α)Sα Sα ,

(3.84)

(Sα )2 D 2 = (D 2 − 2α)(Sα )2 ,

(3.85)

(Sα )n D 2 = (D 2 − nα)(Sα )n ,

n ∈ N,

(3.86)

(S−α )2 D 2 = (D 2 + 2α)(S−α )2 , (S−α )n D 2 = (D 2 + nα)(S−α )n ,

(3.87) n ∈ N,

(3.88)

(Sα Sβ )D 4 = (D 2 − α − β)2 (Sα Sβ ),

(3.89)

Sα D 4 = (D 2 − α)Sα ,

(3.90)

Sα (D 2 )n = (D 2 − α)n Sα ,

n ∈ N,

(3.91)

    Sα1 Sα2 · · · Sαk D 2 = D 2 − α1 − α2 − · · · − αk Sα1 Sα2 Sαk , k ∈ N, (3.92)





!

k

Sαn n=1



D2

m

 = D2 −

k  n=1

!m 

k

αn

! Sαn ,

k, n, m ∈ N.

(3.93)

n=1

They turn into identities for functions from the class and each such a class should be defined for each case. The relations (3.83)–(3.93) show that each EVL operator satisfying the property (3.82) generates many different families of other operators intertwining differential expressions of the second and higher orders. In conclusion, we note that each of the operators intertwining differential expressions (3.83)–(3.93) leads to correspondence formulas between the solutions of some

Basic definitions and propositions

117

differential equations (including those with partial derivatives). Fixing the class leads to the fact that some correspondence between the boundary conditions is added to this correspondence.

3.3.4 Estimates of kernels and point formulas for estimating the error for calculating transmutation operators In this subsection, we consider potentials satisfying various uniform and integral inequalities. Let the continuous potential q satisfy in the domain of definition the inequality |q(α − β)| ≤ R(α)T (β).

(3.94)

Here continuous functions R(α) and T (β) are nonnegative and locally integrable. Let us consider (3.48) for c = d. We have 1 H (u, v) = 2

u

u q(s) ds +

c

v q(α − β)H (α, β) dβ.

dα c

(3.95)

0

We estimate successively terms in the Neumann series    u  u      1   1    |H0 (u, v)| ≤  |q(s)| ds  ≤ T (0)  R(s) ds  . 2 2    c

c

The module in front of the integral is needed, because we do not know whether the limits of integration are in the natural order. For example, this will certainly not be the case when c = +∞. We have  u    α     v     1   |H1 (u, v)| ≤  dα |q(α − β)| T (0)  R(s) ds  dβ  2     c c 0  α  v   u         1 ≤ T (0)  R(α) R(s) ds |dα|  T (β) dβ  . 2     c

c

0

We fixed u, v, c. Number α is between u and c. So

1 |H1 | ≤ ± T (0) 2

u 

v T (β) dβ 0

2 R(α) dα

c

2!

.

(3.96)

118

Transmutations, Singular and Fractional Differential Equations

Signs ± are chosen so that the common sign of the expression (3.96) will be a plus. Similarly n 

n+1 c v  T (β) dβ R(α) dα 1 0 u |Hn | ≤ ± T (0) · . 2 n! (n + 1)! Using (3.53) we can estimate a kernel as ⎞ ⎛ (u,c)  + 1 1 ⎟ ⎜ R(α) dα ⎠ $ |H (u, v)| ≤ T (0) ⎝ !  ! % 2 (u,c) % (v,o)  +  + (u,c) & T (β)dβ R(α) dα (v,0)−

(u,c)−

⎛ $ ⎞⎛ ⎞⎞ %⎛ (u,c) % (v,o) + +   ⎜ % ⎟⎜ ⎟⎟ %⎜ × I1 ⎜ T (β) dβ ⎠ ⎝ R(α) dα ⎠ ⎟ ⎠, ⎝2&⎝ (v,0)−

(3.97)

(u,c)−

where I1 (·) is a modified Bessel function of the first kind (1.16). Let (a, b)− = min(a, b),

(a, b)+ = max(a, b).

Consequently, under the condition (3.95), there exists a transmutation operator with a kernel for which inequality (3.97) holds. Next, a standard piece of statements is needed: 2

∂ H (a) From the integral Eq. (3.95), existence of the continuous function ∂u∂v follows. Estimates for this function can be obtained from (3.95), (3.94), and (3.97). ∂H (b) Differentiating (3.95), we obtain the existence of ∂H ∂v and ∂u , and estimates for them of the form (3.97). 2 2 (c) Existence of ∂∂uH2 and ∂∂vH2 and other higher derivatives are obtained if additional smoothness is required from q(x). (d) Solving (3.95) for locally integrable q, we approximate it by continuous potentials:

qn (x) ∈ C, u v −−−→ 0. dα {qn (α − β) − q(α − β)} dβ− n→∞ 0

0

We consider particular cases of formula (3.97).   (1) We have S = S2 , q(x) = ex , eα−β  ≤ eα e−β , R(α) = eα , T (β) = e−β , T (0) = 1, c = −∞, (u, c)− = −∞, (u, c)+ = u,

Basic definitions and propositions

119

v > 0, (v, 0)− = 0, (v, 0)+ = v. So we have an estimate (3.97) for S2 . (2) We have S = S4 , q(x) = ex , R, R(α) = eα , T (β) = e−β , c = −∞, u > 0, v > 0. So we have an estimate (3.97) for S4 . We also can easily verify that the estimate (3.97) holds for S15 , S17 . (3) Let us consider the case when c = +∞, v < 0 (a = ∞) ⇒ β < 0, and |q(x)| ≤

c , xv

v > 1.

Then |q(α − β)| ≤

c . (|α| + |β|)v

Therefore, we can accept R(α) =

1 , αv

T (β) = C.

Therefore, for such a singular potential there exists a transmutation operator of the form ∞ (Sf ) = f (x) −

K(x, t)f (t) dt x

with kernel K(x, t) = H (u, v) satisfying the estimate 

1 c 1 1−v |H (u, v)| ≤ C  I1 2 |v|u 1−v 2 1−v c|v| u =

1 2



(1−v)



1−v 1−v 1 c u 2 c 2 2 I |v| u 2 . 1 c − v |v| 12 c−v

(3.98)

If v = 0 we can obtain an estimate with c = λ2 for constant potential. Remark 8. If q is a monotonous function, |q(α − β)| ≤ |q(α)|, then (3.94) is valid. Also condition (3.94) holds if |q(α − β)| ≤ C|q(α)|. Now consider a slightly weaker integral condition  v    v          |q(α − β)| dβ  ≤ R(α)  T (β) dβ  (3.99)         0

0

where conditions on the functions R(α) and T (β) are the same.

120

Transmutations, Singular and Fractional Differential Equations

Let

  u      W (u) = sup  q(s) ds  , ∀ξ  

(u, s)− ≤ (u, c)+ .

c

We estimate the terms of the Neumann series 1 |H0 (u, v)| ≤ W (u), 2 u v 1 |H1 (u, v)| ≤ ± dα |q(α − β)| W (α) dβ 2 c 0 ⎞⎛ v ⎞ ⎛ u   ≤ ±W (u) ⎝ R(α) dα ⎠ ⎝ T (β) dβ ⎠ . c

0

We used the inequality W (u) ≥ W (α) since α is between u and c. Therefore ∞  qn , |H (u, v)| ≤ ±W (u) (n!)2 n=0

where

  u   v    q =  R(α) dα · T (β) dβ  .   c

0

Using formula (1.16), which takes the form I0 (z) =

∞  (z2 /4)k k=0

(k!)2

,

q=

z2 , 4

√ z=2 q,

 √   qk I0 2 q = (k!)2

(3.100)

as a result, we obtain that there exists a kernel satisfying the inequality ⎞ ⎛ $ % (0,v) (c,u) + %  +  ⎟ ⎜ % % T (β) dβ R(α) dα ⎟ |H (u, v)| ≤ W (u)I1 ⎜ ⎠. ⎝2& (0,v)−

(3.101)

(c,u)−

We consider estimates for a power potential. One of these estimates in the case of an operator of the Levin type will be given later. Unfortunately, in this case it is not possible to precisely construct the potential. Therefore, various estimates are interesting.

Basic definitions and propositions

121

So, let the inequality |q(x)| ≤ A|x|v ,

v > −1,

(3.102)

hold. Consider the simplest equation when c = 0. We estimate the members of the Neumann series, assuming that u, v > 0. We have 1 uν+1 1 (ν + 1) ν+1 |H0 (u, v)| ≤ A = A u , 2 ν + 1 2 (ν + 2) 1 uν+2 |H1 (u, v)| ≤ A(u + v) v 2 (ν + 1)(ν + 2) 1 1 (ν + 1)uν+1  A(u + v)ν · uv , = A 2 (ν + 3)  n 1 (ν + 1) |Hn (u, v)| ≤ A · uν+1 Auv(u + v)ν . 2 (ν + n + 2)(n + 1) Taking into account formula (7) from [455], p. 708, of the form 

 √  qn √ = ( q )−(ν+1) Iv+1 2 q , n!(ν + n + 2)

(3.103)

we obtain    − (v+1) ν√ A 2 · Iν+1 2(u + v) 2 Auv (ν + 1)uν+1 Auv(u + v)ν 2 1−ν (ν+1)   ν(ν−1) 1√ A 2 (ν + 1)  u  2 = (u + v)− 2 Iν+1 2(u + v) 2 Auv . 2 v (3.104)

|H (u, v)| ≤

Of course, this is a rough estimate. However, we note that at a constant potential (v = 0, A = λ2 ), it turns into exact equality. Consider special estimates valid for the negative potential q(x) ≤ 0.

(3.105)

In the equation 1 H (u, v) = 2

u

u q(s) ds +

c

v q(α − β)H (α, β) dβ

dα c

0

we put u > c, v > 0. We define the signs of the members of the Neumann series 1 H0 (u, v) = 2

u q(s) ds ≤ 0, c

122

Transmutations, Singular and Fractional Differential Equations

u H1 (u, v) =

v q(α − β)H0 (α, β) dβ ≥ 0,

dα c

0

and so on. Obviously the signs of the kernels will alternate. Suppose that in the region of variation of the variables u, v, the inequalities   u   v    dα q(α − β) dβ  < A < 1 (3.106)     c

0

hold. Then: (a) the integral equation under consideration has a unique solution, and this solution is bounded; (b) the estimates 1 |H (u, v)| ≤ · 2(1 − A) 1 |H (u, v)| ≤ 2

u |q(s)| ds, c

u |q(s)| ds ≤ const c

are valid; (c) kernel H (u, v) is negative and −

1 2

u |q(s)| ds ≤ H (u, v) ≤ − c

(1 − A) 2

u |q(s)| ds. c

So, the Neumann series is alternating, all members of the series are sign-definite. From the iterative formula  u    v    |Hn+1 (u, v)| =  dα q(α − β)Hn (α, β) dβ    0

0

it follows that all functions Hn (u, v) increase in u and v. Then from the same formula we get  u    v    |Hn+1 (u, v)| ≤ |Hn (u, v)|  dα q(α − β) dβ  ≤ A |Hn (u, v)| ,   0

0

|Hn (u, v)| ≤ An |H0 (u, v)| ,

∀n,

0 < A < 1. Therefore, the inequality

Basic definitions and propositions

123

holds. Consequently, the series converges at least with the speed of geometric progression, and the kernel satisfies the estimate |Hn (u, v)| ≤ An |H0 (u, v)| ,

∀n.

(3.107)

It follows from Fubini’s theorem that the integral for H0 converges,   u      q(s) ds  < ∞,     c

since the integral u q(α − β) dα < ∞ c

is finite. We can put β = 0 in the last integral. Estimate (3.107) can be clarified. Indeed, in the condition of the theorem all the requirements of the Leibnitz theorem are satisfied. Therefore H0 (u, v) ≤ H (u, v) ≤ H0 (u, v) + H1 (u, v). It is equivalent to    u  u         1 1  −  q(s) ds  ≤ H (u, v) ≤ −  q(s) ds  2 2   c

1 + 2

c

u

v |q(α − β)|

dα c

u

1 |q(t)| dt dβ ≤ 2

c

0

u |q(t)| dt (A − 1). c

Therefore, the kernel H in this case is negative and the estimate 1 − 2

u

(1 − A) |q(s) ds| ≤ H (u, v) ≤ − 2

c

u |q(s)| ds c

is correct. The lower A, the higher the accuracy. It follows that 1 |H (u, v)| ≤ 2

u |q(s)| ds. c

This estimate is always more accurate than (3.107).

(3.108)

124

Transmutations, Singular and Fractional Differential Equations

In addition, we obtain estimates of derivatives ∂H 1 = q(u) + ∂u 2

v q(u − β)H (u, β) dβ, 0

1 ∂H ≥ q(u) + ∂u 2 =

(1 − A) 2

v

⎡ ⎤

(1−A) u 1 q(u − β) ⎣ − |q(s)| ds ⎦ dβ 2 c

0 u

v |q(s)| ds ·

c

1 |q(u − β)| dβ − |q(u)|. 2

0

Another derivative is estimated similarly. Remark 9. Most of the previous estimates are also true in the case of the potential q = q(x, λ), in particular, with the condition of boundedness. Let us consider now the following problem having great practical value. Suppose that the kernel of the transmutation operator obtained as a solution of the integral Eq. (3.48) in the form of a Neumann series is approximated by partial sums of this Neumann series in the calculations. As a result, the operator S is replaced by the operator Sn . It is required to estimate the error introduced by such a replacement of S by Sn in the transmutation identity Sn f = Sn (D 2 + q(x))f − D 2 Sn f,

(3.109)

Sf = S(D 2 + q(x))f − D 2 Sf = 0.

(3.110)

Note that a priori we cannot expect that the residual (3.109), which we designated as Sn , will be small in standard norms. The difference (Sn − S)f is really small, but (3.109) contains differentiation operators that are unbounded in standard spaces like Ck . It is all the more surprising that we not only prove the smallness of the residual (3.109), but also derive a simple exact formula for it in terms of the same operators Sn . This formula allows one to evaluate the residual (3.109) in any norms. To highlight the main idea, we first consider the case of a model transmutation operator q(x) = λ2 . Then the main result will be proved, its consequences will be obtained, and possible generalizations will be outlined. Let us consider a transmutation operator intertwining D 2 + λ2 and D 2 with kernel (3.71). Integrated kernels in variables x, t have the forms (see (3.68) with c = 0)

1 x +t K0 (x, t) = , 2 2 1 K1 (x, t) = (x + t)2 (x − t), 32

Basic definitions and propositions

125

and generally Kn (x, t) =

1 22n+2

1 (x + t)n+1 (x − t)n . n!(n + 1)!

Without loss of generality we put λ = 1. We introduce the definition of a “defect” of the operator A. By definition, we accept       (Af )(x) = A D 2 + 1 f (x) − D 2 Af (x). We calculate the defect for operators x (Tn f ) (x) =

Kn (x, t)f (t) dt. 0

We start with the obvious formula for the defect of the unit operator (If )(x) = (If )(x).

(3.111)

Simple calculations using integration by parts lead to formulas 1 1 f (0) − xf  (0), c0 c0 1 1 (T1 f ) (x) = (T1 f ) (x) − (T0 f ) (x) + x 2 f (0) − x 3 f  (0), c1 c1 (T0 f ) (x) = (T0 f ) (x) − f (x) +

(3.112)

where cn =

1 22n+2 n!(n + 1)!

,

1 c0 = , 4

c1 =

1 . 32

A similar formula holds for a defect of an operator Tn : (Tn f ) (x) = (Tn f ) (x) − (Tn−1 f ) (x) +

x 2n x 2n+1  f (0) − f (0). cn cn

(3.113)

Now we sum up formulas (3.111)–(3.113). As a result, for the operator Sn defined by the formula Sn = I +

n 

k,

k=0

we obtain the following formula for a defect: (Sn f ) (x) = (Tn f ) (x) + f (0)

n  x 2k k=0

ck

− f  (0)

n  x 2k+1 k=0

ck

.

(3.114)

126

Transmutations, Singular and Fractional Differential Equations

This is the desired expression. It allows one to evaluate the error from replacing operator S by Sn in any norm. Actually, + n + n + + + x 2k + + x 2k+1 + + + + + Sn f  ≤ Tn f  + |f (0)| · + (3.115) + + |f  (0)| · + +. + + ck + ck + k=0

k=0

In particular, on functions satisfying the boundary condition f (0) = f  (0) = 0 we get Sn f  = Tn f 

(3.116)

with an exact equal sign. Since, for example, in the uniform norm, Tn f is a common term of the Neumann series, which is majorized by rapidly converging series, the value of the defect (3.116) decreases just as rapidly with increasing n. From the same formula (3.116) it can be seen that on the functions f (0) = f  (0) = 0, lim Sn = S = 0.

n→∞

On arbitrary functions, from (3.114) we obtain lim Sn =

n→∞

1 1 I1 (x)f (0) − I1 (x)f  (0), 2x 2

(3.117)

where I1 (·) is a modified Bessel function of the first kind (1.16). If the space under consideration is not only normalized, but also forms a Banach algebra in multiplication (such as, for example, the space C of continuous functions), then formula (3.115) can be simplified as follows: Sn f  ≤ Tn f  +

1 1 I1 (x)|f (0)| + I1 (x)|f  (0)|. 2x 2

(3.118)

We also note the importance of the general formula for the error (3.115). In the practical calculation of operator S on a computer by replacing it with Sn , the quantities f (0) and f  (0) are required. For example, they are necessary for the approximate calculation of the integral Tn f . Even if we assume in the calculations that f (0) = f  (0) = 0, then in a computer these quantities will be replaced by machine zero. As a result of such a replacement, an error occurs, which will then accumulate in the calculations. This error is controlled by the corresponding terms in formula (3.115).

3.4

Transmutations for singular Bessel operator

One of the most important transmutation operators for the singular Bessel operator (9.1) are the Poisson operator, generalized translation, and weighted spherical mean. All them are generated by the singular differential Bessel operator. These classes of transmutations may be used for deriving explicit formulas for solutions of partial

Basic definitions and propositions

127

differential equations with Bessel operators via unperturbed equation solutions. An example is the B-elliptic equation of the form n 

(Bγ )xk u(x1 , . . . , xn ) = f,

(3.119)

k=1

and similar B-hyperbolic and B-parabolic equations. This idea works by the Sonine– Poisson–Delsarte transmutations (cf. [51–53,56,242]). New results follow automatically for new classes of transmutations.

3.4.1 One-dimensional Poisson operator In this section, we consider the one-dimensional Poisson operator. This operator is the one of the most important transmutation operators connected with the Bessel operator. Definition 22. Let γ > 0. The one-dimensional Poisson operator is defined for integrable function f by the equality   γ +1 x   γ −1  2 2C(γ ) 2 γ   , (3.120) Px f (x) = γ −1 f (t) dt, C(γ ) = √ x2 − t 2 x π  γ2 0

or γ

π

Px f (x) = C(γ )

f (x cos ϕ) sinγ −1 ϕ dϕ.

(3.121)

0 γ

The constant C(γ ) is chosen so that Px [1] = 1. Using the example of the Poisson operator, we show how to find out its intertwining property. Let us consider the equation (Bγ )y u(x, y) = Dx2 u(x, y). Looking for the solution to this equation in the form  u = (r), r = y 2 − (x − ξ )2 , where ξ is some variable, we obtain for (r)  ! x − ξ Dx2 (r) = Dx  (r) −  y 2 − (x − ξ )2  !2 x − ξ =  (r) −  y 2 − (x − ξ )2

(3.122)

128

Transmutations, Singular and Fractional Differential Equations

⎛  ⎜ +  (r) ⎝− =  (r)

(Bγ )y (r) =

2

 (x−ξ ) y 2 −(x−ξ )2 − ξ )2

y 2 − (x − ξ )2 + y 2 − (x

⎞ ⎟ ⎠

(x − ξ )2 y2  −  (r) , r2 r3

1 1 y γ +1 Dy y γ Dy (r) = γ Dy  (r)  γ y y y 2 − (x − ξ )2 y2 y 2 − (x − ξ )2

=  (r)

 (γ + 1)y γ y 2 − (x − ξ )2 − 1 +  (r) γ y y 2 − (x − ξ )2 y2 (γ + 1)r 2 − y 2 =  (r) 2 +  (r) , r r3

(Bγ )y (r) − Dx2 (r) =  (r) +

γ +2  y 2 y −(x−ξ )2

γ +1   (r) = 0. r

One solution is u1 = (r) = r −γ . Taking into account the recurrent formula (1.109) the function u2 = y 1−γ r γ −2 is also a solution to (3.122). It is easy to see that x+y 

u1 =

(ξ ) dξ = rγ

x−y

1−γ

(ξ )r

(ξ ) (y 2

x−y x+y 

u2 = y

x+y 

γ

− (x − ξ )2 ) 2

dξ,

x+y 

γ −2

dξ = y

x−y

γ

(ξ )(y 2 − (x − ξ )2 ) 2 −1 dξ,

1−γ x−y

where and  are arbitrary functions with suitable properties. Putting ξ = x + y(2t − 1), summing u1 and u2 , we get a general solution to (3.122), 1 u(x, y) =

(x + y(2t − 1)) 1− γ2

0

(t (1 − t))

1 dt + y

1−γ

(x + y(2t − 1)) γ

0

(t (1 − t)) 2

dt.

Basic definitions and propositions

129

This solution is valid for 0 < γ < 1. Here we change 2γ −1 to and 21−γ  to . Adding to (3.122) the initial conditions u(x, 0) = f (x),

uy (x, 0) = 0,

we get (γ ) u(x, y) = 2  γ   2

1

f (x + y(2t − 1)) γ

0

(t (1 − t))1− 2

dt.

Now we can introduce the new variable z = y(2t − 1) and write (γ )   u(x, y) = γ (2y) −1  2 γ2

y

f (x + z) γ

(t (1 − t))1− 2

−y

dt

⎞ ⎛ y  0 (γ ) f (x + z) f (x + z) ⎠  ⎝ = γ dt + γ dt (2y)γ −1  2 γ2 (t (1 − t))1− 2 (t (1 − t))1− 2 −y

0

=

(γ )   (2y)γ −1  2 γ2

y

f (x + z) + f (x − z) γ

0

(t (1 − t))1− 2

dz.

Using the Legendre duplication formula (1.7) we obtain (γ )

   2 γ2 2γ −1

   γ +1 2   = C(γ ) =√ π  γ2

and C(γ ) u(x, y) = γ −1 y

y

f (x + z) + f (x − z) γ

0

(t (1 − t))1− 2

dz.

We note here that we came to the Poisson operator acting on γ

u(x, y) = Py

f (x+y)+f (x−y) : 2

f (x + y) + f (x − y) . 2

Therefore we obtain that the Poisson operator intertwines the solution of the Cauchy problem for the wave equation Dy2 v(x, y) = Dx2 v(x, y),

v(x, 0) = f (x),

vy (x, 0) = 0

130

Transmutations, Singular and Fractional Differential Equations

and the solution of the Cauchy problem for the hyperbolic equation with the Bessel operator (Bγ )y u(x, y) = Dx2 u(x, y),

u(x, 0) = f (x),

uy (x, 0) = 0.

γ

So we have u(x, y) = Py v(x, y) for γ > 0. It is easy to see that if f (x) ∈ C 2 , then u(x, y) ∈ C 2 too. We can formulate this result in the form of a statement. 2 operator (3.120) acts as a transmutation operator by the Statement 3. For f ∈ Cev formula γ

γ

Px D 2 f = Bγ Px f,

D2 =

d2 , dx 2

Bγ =

d2 γ d + . 2 x dx dx

(3.123)

Theorem 33. The left inverse operator for (3.120) for γ > 0 for any summable function H (x) is defined by γ (Px )−1 H (x) =

√

n x γ 2 πx d    H (z)(x 2 − z2 )n− 2 −1 zγ dz,  2xdx  γ +1  n − γ2 2 0 (3.124)

where n =

γ  2

+ 1.

Proof. Let us find an operator (Px )−1 , such that (Px )−1 H (x)=H (x), where γ H (x)=Px G(x). γ Change of variables by the formulas t 2 = w, x 2 = ξ drive the operator Px to the fractional Riemann–Liouville integral: γ

α (Ia+ ϕ)(x) =

1 (α)

x a

γ

ϕ(t) dt, (x − t)1−α

x > a.

So we have γ Px G(x) =

2C(γ ) x γ −1

x   γ −1 2 G(t) dt x2 − t 2 0

=

C(γ ) ξ

=



γ −1 2

γ  2

ξ

ξ

√ γ G( w) (ξ − w) 2 −1 dz √ w

0

C(γ )

γ −1 2

γ 2

(I0+ )ξ



√

 G( ξ ) = H ( ξ ). √ ξ

Then γ 2

(I0+ )ξ



γ −1 √

 G( ξ ) ξ 2 = γ  H ( ξ ). √ ξ  2 C(γ )

Basic definitions and propositions

131

Using formula (2.14) we obtain √  γ −1   γ  ξ 2 (D0+ G( ξ ) =  γ  )ξ ξ 2 H ( ξ )  2 C(γ ) √ n ξ √ γ −1 γ 1 ξ d  t 2 H ( t)(ξ − t)n− 2 −1 dt, = γ  γ dξ  2 C(γ )  n − 2 0 ,γ n= + 1. 2 Returning to x by the formula ξ = x 2 and putting t = z2 , we get √

n x γ 2 πx d   G(x) =  H (z)(x 2 − z2 )n− 2 −1 zγ dz, γ +1 γ  2xdx  2  n− 2 0 ,γ n= + 1. 2 The proof is complete. For the Bessel function of the first kind Jν the integral representation using the Poisson integral with ν > − 12 (see formula (1) in [591], p. 58) π

xν   Jν (x) = √ π2ν  ν + 12

eix cos ϕ sin2ν ϕdϕ 0

is valid. So we can write for ν =

γ −1 2

  π  γ +1 2 γ   j γ −1 (x) = √ eix cos ϕ sinγ −1 ϕdϕ = Px eix . 2 π  γ2

(3.125)

0

For the Bessel function of the second kind Iν the integral representation using the Poisson integral with ν > − 12 (see formula (9) in [591], p. 94) xν   Iν (x) = √ π2ν  ν + 12

π

is valid. So we can write for ν =

e±x cos ϕ sin2ν ϕdϕ

0 γ −1 2

  π  γ +1 2   i γ −1 (x) = √ e±x cos ϕ sinγ −1 ϕdϕ = Pγ e±x . 2 π  γ2 0

(3.126)

132

Transmutations, Singular and Fractional Differential Equations

Now we consider how the Poisson operator intertwines solutions to equations with the Bessel operator and with the second derivative. The abstract Euler–Poisson–Darboux equation has the form Au = (Bγ )t u,

u = u(x, t; γ ),

where A is a linear operator acting only by variable x = (x1 , ..., xn ). Since the Poisson operator (3.120) intertwines the second derivative and the Bessel γ γ operator, i.e., Pt Dt2 = (Bγ )t Pt , it is possible to use it for finding a solution to the abstract Euler–Poisson–Darboux equation (1.108) using known a solution to the equation Aw=wtt , w=w(x, t), x=(x1 , ..., xn ), t ∈ R. Theorem 34. Let γ > 0. We have a twice continuously differentiable for t > 0 solution u=u(x, t; γ ) to the equation Au = (Bγ )t u,

u = u(x, t; γ ),

x = (x1 , ..., xn ),

t > 0,

(3.127)

associated with the twice continuously differentiable solution to the equation Aw = wtt ,

w = w(x, t),

x = (x1 , ..., xn ),

t ∈ R,

(3.128)

by the formula γ

u(x, t; γ ) = Pt w(x, t),

(3.129)

γ

where Pt is the Poisson operator (3.120) acting by the variable t. Proof. Let us show that function u defined by formula (3.129) satisfies Eq. (3.127). We have   1 2 γ +1 γ 2 γ  u= √ w(x, αt)[1 − α 2 ] 2 −1 dα. π 2 0

Let ξ = αt. Integrating by parts we obtain γ γ 1 u = wξ (x, αt), dv = α[1 − α 2 ] 2 −1 dα, du = twξ ξ (x, αt)dα, v = − [1 − α 2 ] 2 γ

and   γ +1 1 2 γ 2 ∂u γ  α wξ (x, αt)[1 − α 2 ] 2 −1 dα = √ ∂t π 2 0   γ +1 1 2 2 γ t   = √ wξ ξ (x, αt)[1 − α 2 ] 2 dα π γ2 γ 0

Basic definitions and propositions

  2 γ +1 2   = √ π γ2   2 γ +1 2 γ  = √ π 2 For

∂2u ∂t 2

133

1

t γ

γ

wξ ξ (x, αt)[1 − α 2 ] 2 dα 0

1

t γ

γ

Aw(x, αt)[1 − α 2 ] 2 dα. 0

we get

  γ +1 1 γ ∂ 2 u 2 2 γ  = √ α 2 wξ ξ (x, αt)[1 − α 2 ] 2 −1 dα 2 ∂t π 2 0   γ +1 1 2 2 γ γ  = √ Aw(x, αt)α 2 [1 − α 2 ] 2 −1 dα. π 2 0

So (Bγ )t u =

∂ 2 u γ ∂u + = t ∂t ∂t 2

⎡  ⎤ 1 1 2 γ +1 γ γ 2   ⎣ Aw(x, αt)α 2 [1 − α 2 ] 2 −1 dα + Aw(x, αt)[1 − α 2 ] 2 dα ⎦ = √ π γ2 0



0



1 2 γ +1 γ 2 γ   Aw(x, αt)[1 − α 2 ] 2 −1 dα = APt w(x, t) = Au. √ π γ2 0

Hence the function u satisfies Eq. (3.129). The proof is complete. Corollary 3. For 0 < γ < 1 the function

γ f (x + t) + f (x − t) 1−γ 2−γ g(x + t) + g(x − t) u(t, x; γ ) = Pt Pt +t 2 2 satisfies the Cauchy problem uxx = (Bγ )t u, u(0, x; γ ) = f (x),

t ut (t, x; γ )|t=0 = g(x). γ

(3.130) (3.131)

Proof. Let us consider the wave equation ∂ 2u ∂ 2u = 2 ∂x 2 ∂t

(3.132)

134

Transmutations, Singular and Fractional Differential Equations

and its general solution F (x + t) + G(x − t),

(3.133)

where F and G are arbitrary functions. Applying Theorem 34 we obtain that one of the solutions to ∂ 2u = (Bγ )t u, ∂x 2

u = u(x, t; γ )

(3.134)

is a function u1 = 2C(γ )

t

1 t γ −1

γ

[F (x + z) + G(x − z)](t 2 − z2 ) 2 −1 dz.

0

We transform the resulting general solution as follows: C(γ ) u1 = γ −1 t

t

F (x + z) + F (x − z) + G(x + z) + G(x − z) γ

(t 2 − z2 )1− 2

−t

dz.

We introduce a new variable p by the formula z = t (2p − 1), and we obtain

u1 = 2

γ −1

1 C(γ )

(x + t (2p − 1)) γ

0

(p(1 − p))1− 2

dp,

where

(x + z)= [F (x + z)+F (x − z)+G(x + z)+G(x − z)] . From Lemma 7 it follows that if the function u(x, t; γ ) is the solution to (3.134), then the function t 1−γ u(x, t; 2 − γ ) will also be a solution to (3.134). Therefore, the second solution to (3.134) is a function 1 u2 = 2

1−γ

C(2 − γ )t

1−γ

(x + t (2p − 1)) γ

0

(p(1 − p)) 2

dp,

where  is an arbitrary function that, generally speaking, does not coincide with . Composing the values of u1 and u2 we obtain that the general solution to (3.134) has

Basic definitions and propositions

135

the form u = 2γ −1 C(γ )

1

(x + t (2p − 1)) γ

0

(p(1 − p))1− 2 1

+2

1−γ

C(2 − γ )t

1−γ

dp

(x + t (2p − 1)) γ

(p(1 − p)) 2

0

dp.

(3.135)

From the conditions u(0, x; γ ) = f (x) and t γ ut (t, x; γ )|t=0 = g(x) it is easy to find

and . We have for 0 < γ < 1

u(x, 0; γ ) = 2

γ −1

1 C(γ ) (x)

γ

(p(1 − p)) 2 −1 dp = f (x),

0

1

γ

(p(1 − p)) 2 −1 dp =

 γ 2

 2 . (γ )

0

Using the Legendre duplication formula (1.7), we obtain

2

γ −1

1 C(γ )

γ

(p(1 − p)) 2 −1 dp

0

= 2γ −1

       2 2γ −1  γ +1  γ +1  γ2  γ2 2 2   = 1, = √ √ π (γ ) π  γ2 (γ )

and therefore

(x) = f (x).   Let us now find t γ ∂u ∂t |t=0 . We have

 1 γ ∂u  1−γ = (1 − γ )2 C(2 − γ )(x) (p(1 − p))− 2 dp = g(x), t ∂t t=0 γ

0

1 0



γ

(p(1 − p))− 2 dp =

γ 2 2

 1− . (2 − γ )

136

Transmutations, Singular and Fractional Differential Equations

Using the Legendre duplication formula (1.7), we obtain 1 (1 − γ )2

1−γ

C(2 − γ )

γ

(p(1 − p))− 2 dp

0





 2  1 − γ2   = (1 − γ )2 √ π  1 − γ2 (2 − γ )     21−γ  3−γ  1 − γ2 2 = (1 − γ ) = 1 − γ, √ (2 − γ ) π 

1−γ

3−γ 2

and then (x) =

g(x) . 1−γ

We obtain that the solution to (3.130)–(3.131) for 0 < k < 1 is

u=2

γ −1

1 C(γ )

f (x + t (2p − 1)) γ

(p(1 − p))1− 2

0

21−γ C(2 − γ ) 1−γ t + 1−γ

1

dp

g(x + t (2p − 1)) γ

0

(p(1 − p)) 2

dp.

Putting in (3.136) t (2p − 1) = z, we get C(γ ) u = γ −1 t

t

γ

f (x + z)(t 2 − z2 ) 2 −1 dz

−t

C(2 − γ ) + 1−γ =

2C(γ ) t γ −1

t

t

γ

g(x + z)(t 2 − z2 )− 2 dp

−t

γ f (x + z) + f (x − z) 2 (t − z2 ) 2 −1 dz 2

0

t γ g(x + z) + g(x − z) 2 2C(2 − γ ) (t − z2 )− 2 dp + 1−γ 2 0

γ f (x + t) + f (x − t) 2−γ g(x + t) + g(x − t) = Pt + t 1−γ Pt . 2 2

(3.136)

Basic definitions and propositions

137

3.4.2 Multi-dimensional Poisson operator In this section, the multi-dimensional Poisson operator and calculate two  we consider  integrals jγ (rθ, ξ )θ γ dS and iγ (rθ, ξ )θ γ dS. S1+ (n)

S1+ (n)

γ

Definition 23. The multi-dimensional Poisson operator Px acts on the integrable function f by the formula γ Px f (x)=C(γ )

π

π

0



n

C(γ ) = π − 2

γi +1 2  γi   2

sinγi −1 αi dαi ,

(3.137)

i=1

0

where n

n

f (x1 cos α1 , ..., xn cos αn )

...





i=1 γ

such that Px [1] = 1. From (3.123) we get the statement. 2 operator (3.137) acts as a transmutation operator by the Statement 4. For f ∈ Cev formula γ

γ

Px f = γ Px f, where  is the Laplace operator and γ =

n .

Bγi .

i=1

From formulas (3.125) and (3.126) it is easy to obtain representations for (1.30) and (1.31) of the form jγ (x, ξ ) = Pξ [e−ix,ξ  ]

(3.138)

iγ (x, ξ ) = Pξ [e±x,ξ  ],

(3.139)

γ

and γ

where x, ξ  =

n .

xi ξi .

i=1

The part of a sphere of radius r with center at the origin belonging to Rn+ we will denote by Sr+ (n): Sr+ (n)={x ∈ Rn+ : |x|=r}. We obtain the formulas expressing the weight integrals on the part of the sphere S1+ (n) of functions (3.138) and (3.139).

138

Transmutations, Singular and Fractional Differential Equations



Statement 5. The integral

jγ (rθ, ξ )θ γ dS is calculated by the formula

S1+ (n)

n /

 jγ (rθ, ξ )θ γ dS = S1+ (n)

 

i=1

2n−1 



γi +1 2

  j n+|γ | −1 (r|ξ |).

n+|γ | 2

(3.140)

2

Proof. Using formula (3.138) we can write     γ jγ (rθ, ξ )θ γ dS = Pξ e−irθ,ξ  θ γ dS. S1+ (n)

S1+ (n)

Applying formula (3.143) to the last integral we obtain 

n /

 jγ (rθ, ξ )θ dS = √

i=1

γ

S1+ (n)



π2n−1 

γi +1 2





1 

|γ |+n−1 2 −1

e−irp|ξ | (1 − p 2 )

n+|γ |−3 2

dp.

Replacing p by −p we get 1 e

−irp|ξ |

(1 − p ) 2

n+|γ |−3 2

1 dp =

−1

eirp|ξ | (1 − p 2 )

n+|γ |−3 2

dp.

−1

The last integral is found by formula (2.3.5.3) from [455] of the form √

a eitp (a 2 − p 2 )β−1 dp =

1

π (2a)β− 2 (β) 1

t β− 2

−a

(3.141)

Jβ− 1 (at). 2

Therefore  jγ (rθ, ξ )θ γ dS S1+ (n) n / i=1

 

γi +1 2



√

π2

n+|γ | 2 −1

 

n+|γ |−1 2

  n+|γ | π2n−1  n+|γ2 |−1 (r|ξ |) 2 −1   n /  γi 2+1 i=1   j n+|γ | −1 (r|ξ |), = | 2 2n−1  n+|γ 2 =√

which gives (3.140). The proof is complete.

 J n+|γ | −1 (r|ξ |) 2

Basic definitions and propositions

139



Statement 6. The integral

S1+ (n)

iγ (rθ, ξ )θ γ dS is calculated by the formula 

n /

 iγ (rθ, ξ )θ γ dS = S1+ (n)



i=1

2n−1 



γi +1 2



n+|γ | 2

 i n+|γ | −1 (r|ξ |).

(3.142)

2

Proof. Using (3.139) we obtain 



  γ Pξ −erθ,ξ  θ γ dS.

iγ (rθ, ξ )θ γ dS = S1+ (n)

S1+ (n)

Applying formula (3.143) we get n /

 iγ (rθ, ξ )θ γ dS = √ S1+ (n)

 

i=1

π2n−1 

γi +1 2





1 

|γ |+n−1 2 −1

e−rp|ξ | (1 − p 2 )

n+|γ |−3 2

dp.

The last integral is found by formula (2.3.5.1) from [455] of the form a e

−tp

√ (a − p ) 2

2 β−1

dp =

1

π (2a)β− 2 (β) 1

t β− 2

−a

Iβ− 1 (at). 2

So n /

 iγ (rθ, ξ )θ γ dS = S1+ (n)

 

i=1

2n−1 



γi +1 2



n+|γ | 2

 i n+|γ | −1 (r|ξ |), 2

which gives (3.142). The proof is complete. Now we present a known result received by I. A. Kipriyanov and L. A. Ivanov in [247] (see also [242,251] for the particular case). Since this article is difficult to get and it is not translated into English, we present here a formula with proof. The function of the scalar product f (ξ, x) is usually called the function of the “plane wave” type, since it gives a constant value on the plane ξ, x = p. We will γ call function Pξ f (ξ, x) the function of the “weight plane wave” type because in the corresponding integral expressions it will become a function of an ordinary plane wave.

140

Transmutations, Singular and Fractional Differential Equations

Theorem 35. Let f (s) be the integrable function on (−1, 1) of one variable. Then the following formula is valid: 

n / γ Pξ f (σ, ξ )σ γ dSσ

=√

S1+ (n)

 

i=1

π2n−1 

γi +1 2





1 

n+|γ |−1 2 −1

f (|ξ |p)(1 − p 2 )

n+|γ |−3 2

dp.

(3.143) Proof. Let us consider the integral  γ J= Pξ f (ξ, x)x γ dx, BR+ (n)

where BR+ (n) = {x : |x| < R; x1 . . . xn > 0}, ξ ∈ Rn+ . On this integral J we introduce the new coordinates:  x1 = x1 cos α1 ,

 x2 = x1 sin α1 ,

 x3 = x2 cos α2 ,

 x4 = x2 sin α2 , . . . ,

 x2n−1 = xn cos αn ,

 x2n = xn sin αn .

In these coordinates 

n

f ( ξ  , x )

J = C(γ )

γ −1

 x 2ii

d x,

i=1

+ (2n) B R

where  x = ( x1 , . . . , x2n ) ∈ R2n ,  x2i > 0, i = 1, n,  ξ  = (ξ1 , 0, ξ2 , 0, . . . , ξn , 0)∈R2n , +  x : | x | < R; x2i > 0, i = 1, n}. BR (2n) = { Now we consider under the integral J the function of the “plane wave” type. Seξ  , x  = p|ξ |, we obtain lecting integration by  x1 = p and noting that  

R J = C(γ ) −R

n

f (p|ξ |) dp

γ −1

 x2ii

+  B

R 2 −p 2

(2n−1)

d x2 . . . d x2n ,

i=1

+  where B (2n − 1) is a part of a ball centered at the origin with the radius R 2 −p 2  R 2 − p 2 in R2n−1 with  x2i > 0, i = 1, n. Now we introduce a spherical coordinate transformation in the inner integral:

 x1 = ρϑ1 ,

 x2 = ρϑ2 ,

...,

 xn−1 = ρϑn−1

Basic definitions and propositions

141

with Jacobian I = ρ 2n−2 . In terms of the new coordinates we obtain R J = C(γ )



n

f (p|ξ |) dp

ρ

+  B

−R

R 2 −p 2

f (p|ξ |) dp

= C(γ ) −R

ρ

dρ dS

i=1

(2n−1)

 2 2 R  −p

R

γ −1

ρ γi −1 ϑ2ii

2n−2

n+|γ |−2



n

γ −1

ϑ2ii

dρ  S1+ (2n−1)

0

dS,

i=1

where  S1+ (2n−1) is a part of a unit sphere centered at the origin in R2n−1 with  x2i > 0, i = 1, n. Applying formula (1.107) we get 

 n−1

n

γ −1

ϑ2ii

dS =

i=1  S1+ (2n−1)

 / n 1 2

2n−1 



 γi  2

 i=1

n+|γ |−1 2

 =

π

n−1 2

n /



 γi 

2 i=1   2n−1  n+|γ2 |−1

= | S1+ (2n − 1)||γ |−1 and 

2 2 R  −p

ρ

n+|γ |−1

n+|γ |−2

0

(R 2 − p 2 ) 2 dρ = n + |γ | − 1

.

So we obtain  J=

γ

Pξ f (ξ, x)x γ dx

BR+ (n)

C(γ ) = | S + (2n − 1)||γ |−1 n + |γ | − 1 1

R f (|ξ |p)(R 2 − p 2 )

n+|γ |−1 2

dp.

−R

Now on the left side of this equality turning to the spherical coordinates x1 = ρϑ1 . . . xn = ρϑn ,

142

Transmutations, Singular and Fractional Differential Equations

we get R J=

ρ

n+|γ |−1

 dρ

γ

Pξ f (ϑ, ξ )ϑ γ dS

S1+ (n)

0

C(γ ) | S + (2n − 1)||γ |−1 = n + |γ | − 1 1

R f (|ξ |p)(R 2 − p 2 )

n+|γ |−1 2

dp.

−R

Differentiating the last equality by R we write  γ R n+|γ |−1 Pξ f (ϑ, ξ )ϑ γ dS S1+ (n)

=

n + |γ | − 1 C(γ ) | S1+ (2n − 1)||γ |−1 2 n + |γ | − 1 2 R n+|γ |−3 × f (|ξ |p)R(R 2 − p 2 ) 2 dp −R

= C(γ )| S1+ (2n − 1)||γ |−1

R f (|ξ |p)R(R 2 − p 2 )

n+|γ |−3 2

dp.

−R

Putting R = 1, we get 

γ Pξ f (ϑ, ξ )ϑ γ

dS = C(γ )| S1+ (2n − 1)||γ |−1

S1+ (n)

1 f (|ξ |p)(1 − p 2 )

n+|γ |−3 2

dp,

−1

where 

n /

C(γ )| S1+ (2n − 1)||γ |−1 = √



i=1

π2n−1 

γi +1 2





n+|γ |−1 2

.

That gives (3.143). Example 5. Let us consider the case when f (t) = |t|k , k > −1. We have 

n / γ

Pξ |ϑ, ξ |k ϑ γ dS = √ S1+ (n)

 

i=1

π2n−1 

γi +1 2





1 

n+|γ |−1 2 −1

|ξ |k |p|k (1 − p 2 )

n+|γ |−3 2

dp

Basic definitions and propositions

143 n /

 

i=1

γi +1 2



 

n+|γ |−1 2

    k+1 2  |ξ |k

   π2n−1  n+|γ2 |−1  n+|γ2 |+k  /   n  γi 2+1  k+1 2 i=1   |ξ |k . = √ n+|γ |+k n−1 π2  2 =√

3.4.3 Generalized translation In this section we consider the transmutation operator called the generalized translation. Definition 24. Let f = f (x), x ∈ R, γ > 0. The generalized translation is defined by the equality

(

γ

y y Tx f )(x) = γ Tx f =C(γ )

π

 f ( x 2 + y 2 − 2xy cos ϕ) sinγ −1 ϕ dϕ, (3.144)

0

where C(γ ) = 0

y

   γ +1 2 .  √ π γ2 y

Tx = Tx f (x) =

y

For γ = 0 the generalized translation γ Tx is f (x + y) − f (x − y) . 2 y

The generalized translation γ Tx was introduced in the paper [83] and then studied in detail in [317], see also [320]. In particular, in [317] it was shown that u(x, y) = y (γ Tx f )(x) is a unique solution to the Cauchy problem (Bγ )x u(x, y) = (Bγ )y u(x, y), u(x, 0) = f (x),

(3.145)

  ∂ = 0. u(x, y) ∂y y=0

y

2 is a transmutation operator with the followThe operator γ Tx of function f ∈ Cev ing intertwining property: γ

y

y

Tx (Bγ )x f (x) = (Bγ )y γ Tx f (x).

(3.146)

In addition, it satisfies the conditions γ

(

y Tx f )(x)|y=0

= f (x),

  ∂ γ y = 0. ( Tx f )(x) ∂y y=0

(3.147)

144

Transmutations, Singular and Fractional Differential Equations y

Replacing a variable ϕ → π − ϕ it is easy to see that γ Tx can be written in the form   π   γ +1 2 γ y γ y   f ( x 2 + y 2 + 2xy cos ϕ) sinγ −1 ϕdϕ. ( Tx f )(x) = Tx f (x)= √ π  γ2 0

(3.148) Let us present the elementary properties of the generalized translation from [317]. Elementary properties of the generalized translation 1. Linearity and uniformity: γ

y

y

Tx [af (x) + bg(x)] = a γ Tx f (x) + bγ T y g(x),

a, b ∈ R.

y

2. Nonnegativity: γ Tx f (x) ≥ 0 if f (x) ≥ 0. y 3. γ Tx [1] = 1. γ 4. Tx0 f (x) = f (x). y 5. γ Tx f (x) = γ Tyx f (y). y 6. If f (x) ≡ 0 for x ≥ a, then γ Tx f (x) ≡ 0 for |x − y| ≥ a. 7. If a sequence of continuous functions fn (x) converges uniformly in each finite y interval to f (x), then the sequence of functions of the two variables γ Tx fn (x) y converges uniformly in each finite area to γ Tx f (x). y 8. Operator γ Tx is bounded: y

y

|γ Tx f (x)| ≤ γ Tx |f (x)| ≤ sup |f (x)|.

(3.149)

x≥0 y

9. Commutativity of operators γ Tx : γ

y

y

Tx γ Txz f (x) = γ Txz γ Tx f (x).

10. Associativity of operators γ

(3.150)

γ T y: x

y

y

Tyz γ Tx f (x) = γ Txz γ Tx f (x).

(3.151) y

11. The function j γ −1 is an eigenfunction of the generalized translation γ Tx : 2

γ

y

Tx j γ −1 (xξ ) = j γ −1 (xξ ) j γ −1 (yξ ). 2

2

(3.152)

2

We now give some more properties of generalized translation. y

Statement 7. For the generalized translation operator γ Tx the representation  ! 1  4xy y γ Tx f (x) = 2γ −1 C(γ ) f (x + y) 1 − z (x + y)2 0

×z is valid.

γ 2 −1

γ

(1 − z) 2 −1 dz

(3.153)

Basic definitions and propositions

145

Proof. We transform the generalized translation operator as follows. First putting ϕ = 2α in (3.148) we obtain γ

y Tx f (x) = 2C(γ )

π/2  f ( x 2 + y 2 + 2xy cos 2α) sinγ −1 (2α)dα 0

π/2  = 2 C(γ ) f x 2 + y 2 + 2xy(cos2 α − sin2 α) γ

× sin

0 γ −1

α cosγ −1 αdα

π/2  = 2 C(γ ) f x 2 + y 2 + 2xy(1 − 2 sin2 α) γ

0

× sin

γ −1

α(1 − sin2 α)

γ −1 2

dα.

Now let sin α = t. Then α=0 for t=0, α=π/2 for t=1, dα= (1−tdt2 )1/2 , and γ

y Tx f (x) = 2γ C(γ )

1

 γ f ( x 2 + y 2 + 2xy(1 − 2t 2 ))t γ −1 (1 − t 2 ) 2 −1 dt

0

= {t = z} 2

=2

γ −1

1 C(γ )

 γ γ f ( x 2 + y 2 + 2xy(1 − 2z))z 2 −1 (1 − z) 2 −1 dz

0

=2

γ −1

1 C(γ )

 γ γ f ( (x + y)2 − 4xyz)z 2 −1 (1 − z) 2 −1 dz

0

=2

γ −1



1 C(γ )

f 0



! γ γ 4xy (x + y) 1 − z z 2 −1 (1 − z) 2 −1 dz. 2 (x + y)

The proof is complete. y

Statement 8. For the generalized translation operator γ Tx the representation γ

(

y Tx f )(x)=

2γ C(γ ) (4xy)γ −1

x+y 

γ

zf (z)[(z2 − (x − y)2 )((x + y)2 − z2 )] 2 −1 dz

|x−y|

(3.154) is valid.

146

Transmutations, Singular and Fractional Differential Equations

Proof. Changing the variable ϕ to 2α in (3.144), we obtain

γ

y Tx f (x) = 2C(γ )

π/2  f ( x 2 + y 2 − 2xy cos 2α) sinγ −1 (2α)dα 0

π/2  = 2 C(γ ) f x 2 + y 2 − 2xy(cos2 α − sin2 α) γ

× sin

0 γ −1

α cosγ −1 αdα

π/2  = 2 C(γ ) f x 2 + y 2 − 2xy(1 − 2 sin2 α) γ

0

× sinγ −1 α(1 − sin2 α)

γ −1 2

dα.

Now putting sin α=t we get for α=0, t=0, for α=π/2, t=1, dα= (1−tdt2 )1/2 , and

γ

y Tx f (x) = 2γ C(γ )

1

 γ f ( x 2 + y 2 − 2xy(1 − 2t 2 ))t γ −1 (1 − t 2 ) 2 −1 dt.

0

Introducing the variable z by the equality t=

z2 − (x − y)2 4xy



1/2 dt =

,

x 2 + y 2 − 2xy(1 − 2t 2 ) = z, we obtain

zdz , (4xy)1/2 (z2 − (x − y)2 )1/2

z = |x − y| when t = 0, z = x + y when t = 1, and γ

y Tx f (x) =

2γ C(γ ) (4xy)γ −1

x+y 

γ

zf (z)[(z2 − (x − y)2 )((x + y)2 − z2 )] 2 −1 dz.

|x−y|

The proof is complete. y

Statement 9. The generalized translation γ Tx can be written in the form

γ

y Tx f (x) =

21−γ  

2

x+y 

γ +1 2 |x−y|

∞ z

2−γ

j γ −1 (λx)j γ −1 (λy)j γ −1 (λz)λγ dλ.

f (z)dz

2

2

2

0

(3.155)

Basic definitions and propositions

147

Proof. The next formula is valid: γ

[(z2 − (x − y)2 )((x + y)2 − z2 )] 2 −1 (xy)γ −1   ∞ √ π γ2 1−γ   j γ −1 (λx)j γ −1 (λy)j γ −1 (λz)λγ dλ. =z 2 2 2 2 3 γ +1 2 0

(3.156)

This formula follows from formula (2.12.42.14) from [456], p. 204, of the form ∞ λ1−ν Jν (xλ)Jν (yλ)Jν (zλ)dλ 0

=√

1 21−3ν   [(z2 − (x − y)2 )((x + y)2 − z2 )]ν− 2 1 π(xyz)ν  ν + 2

=√

2ν−1 Δ2ν−1  , π(xyz)ν  ν + 12

where |x − y| < z < x + y, x, y, z > 0, Re ν > − 12 , and Δ is the area of a triangle whose sides are equal to x, y, and z. Using (3.154) and (3.156) we get (3.155). Statement 9 is given in [220]. Statement 10. If f (x) is a continuous function such that ∞ |f (x)|x γ dx < ∞ 0

and g(x) is continuous and bounded for all x ≥ 0, then ∞ γ

∞

y Tx f (x)g(y)y γ dy

=

0

y

f (y)γ Tx g(x)y γ dy. 0

Proof. Applying to

∞ γ

y

Tx f (x)g(y)y γ dy representation (3.154), we obtain

0

∞ γ

y

Tx f (x)g(y)y γ dy

0

∞ = (4x)

1−γ γ

2 C(γ )

yg(y)dy 0

(3.157)

148

Transmutations, Singular and Fractional Differential Equations x+y 

×

γ

zf (z)[(z2 − (x − y)2 )((x + y)2 − z2 )] 2 −1 dz

|x−y| x+y x  2 C(γ ) yg(y)dy zf (z)

= (4x)

1−γ γ

x−y

0

γ

× [(z − (x − y) )((x + y) − z2 )] 2 −1 dz 2

2

∞ +

2

γ zf (z)[(z2 − (x − y)2 )((x + y)2 − z2 )] 2 −1 dz .

x+y 

yg(y)dy x

y−x

Converting an expression (z2 − (x − y)2 )((x + y)2 − z2 ) and changing the order of integration we get ∞ γ

y

Tx f (x)g(y)y γ dy

0

x 2 C(γ ) zf (z)dz

= (4x)

1−γ γ

0

x+z γ yg(y)[((z + x)2 − y 2 )(y 2 − (z − x)2 )] 2 −1 dy × x−z ∞

x+z 2 2 2 2 γ2 −1 zf (z)dz yg(y)[((z + x) − y )(y − (z − x) )] dy

+ x

z−x

∞ =

f (z)γ Txz g(y)zγ dz. 0

The commutation is proved.

Statement 10 was proved in [317] in another way. Statement 11. The Hankel transform from generalized translation of the function f ∈Sev (R+ ) has a form y

Fγ [γ Tx f (x)](ξ ) = j γ −1 (yξ ) Fγ [f ](ξ ). 2

(3.158)

Basic definitions and propositions

149

Proof. Using the property of self-adjointness of the generalized translation and (3.152), we obtain Fγ [

γ

y Tx f (x)](ξ ) =

∞

y

j γ −1 (xξ ) γ Tx f (x)x γ dx 2

0

∞ =

γ

y

Tx j γ −1 (xξ ) f (x)x γ dx 2

0

∞ = j γ −1 (y)

j γ −1 (xξ ) f (x)x γ dx

2

2

0

= j γ −1 (yξ ) Fγ [f ](ξ ). 2

Statement 11 was given in [610]. We obtain the formulas for the action of a generalized translation on some elementary and special functions. y (1) For x > 0 the formula representing a generalized translation γ Tx of power function x α is ⎧   γ 4xy ⎪ x = y, ⎨|x − y|α 2 F1 − α2 , 2 , γ ; − (x−y)2     γ y α γ +α γ +1 Tx x = (3.159) α+γ −1 2  2  2 ⎪ ⎩x α  √  x = y α π γ + 2

or

⎧   2 α F − α , 1−α−γ , γ +1 ; y ⎪ x > y, x ⎪ 2 1 2 ⎪ x ⎪ 2  2 2 ⎨ α+γ −1  γ +α  γ +1 2 γ y α 2 2 Tx x = x α  √  x = y, ⎪ π γ + α2 ⎪   ⎪ 2 ⎪ 1−α−γ γ +1 α x α ⎩y 2 F1 − , , 2 ; y2 x < y, 2 2

(3.160)

where 2 F1 is a Gaussian hypergeometric function (1.33). y

Proof. Let first x = y. Using formula (3.153) let us find γ Tx of x α . We have  

α 1 2γ −1  γ +1 2 γ γ 2 4xy γ y α  γ  |x − y|α Tx x = √ z (1 − z) 2 −1 z 2 −1 dz. 1+ (x − y)2 π 2 0

4xy The last integral is a Gaussian hypergeometric function (1.33) for z = − (x−y) 2,

a = − α2 , b = γ2 , c = 2b = γ (c > b > 0), thus    

2γ −1  γ +1  γ2 2 4xy α γ γ y α |x − y|α 2 F1 − , , γ ; − Tx x = . √ 2 2 (x − y)2 π(γ )

150

Transmutations, Singular and Fractional Differential Equations

Using the doubling formula for the gamma function (1.7), we obtain (3.159). For the proof of (3.160) we use the definition of the absolute value in (3.159) and get ⎧   ⎨ (x − y)α 2 F1 − α , γ , γ ; − 4xy 2 x > y, 2 2 γ y α   (x−y) (3.161) Tx x = γ 4xy α ⎩ 2 F1 − , , γ ; − x < y. 2 2 (x−y)2 In [295] the following formula is given:



1 2 4z 1 2a F F , b + ; z a, b, 2b; = (1 + z) a, a − b + , 2 1 2 1 2 2 (1 + z)2 using which we get (3.160) for x = y. For x = y we have   1 2γ −1  γ +1 γ γ +α 2 γ y α α γ  Tx x = (2x) (1 − z) 2 −1 z 2 −1 dz √ π 2 0     γ +α γ −1 2  2  γ +1 2   = (2x)α . √ α π γ + 2 That completes the proof. (2) The generalized translation γ Tx of e−x , x > 0, is   1−γ 2 2 γ y −x 2 Tx e =  γ +1 (xy) 2 e−x −y I γ −1 (2xy) . 2 2

y

(3.162)

2

Proof. Using formula (3.154) we obtain γ

2 y Tx e−x

2γ C(γ ) = (4xy)γ −1

x+y 

γ

ze−z [(z2 − (x − y)2 )((x + y)2 − z2 )] 2 −1 dz. 2

|x−y|

We find the integral x+y 

I=

γ

ze−z [(z2 − (x − y)2 )((x + y)2 − z2 )] 2 −1 dz = {z2 = t} 2

|x−y|

1 = 2

2 (x+y) 

γ

e−t [(t − (x − y)2 )((x + y)2 − t)] 2 −1 dt = {t − (x − y)2 = w}

(x−y)2

1 2 = e−(x−y) 2

4xy γ e−w [w(4xy − w)] 2 −1 dw. 0

Basic definitions and propositions

151

Applying formula (2.3.6.2) from [455] of the form a

x α−1 (a − x)α−1 e−px dx =

√

π(α)

α−1/2 a e−ap/2 Iα−1/2 (ap/2), p

0

(3.163) Re α > 0, we get γ  γ −1 √ 2 2 e−x −y (xy) 2 I γ −1 (2xy) . I = 2γ −2 π 2 2 Then γ

Tx e−x y

2

   γ +1 2   = √ γ −1 (4xy) π  γ2 2γ

x+y 

×

γ

ze−z [(z2 − (x − y)2 )((x + y)2 − z2 )] 2 −1 dz 2

|x−y|

  γ   γ +1 γ −1 2 2γ 2 2 γ −2 √   2 π e−x −y (xy) 2 I γ −1 (2xy) . = √ γ γ −1 2 2 (4xy) π 2 After simplification we get (3.162). (3) The generalized translation γ Tx of x 2 e−x , x > 0, is   γ +1  2 γ y 2 −x 2 Tx x e = 2 +y 2 x e γ +1 2xy

 γ 1−γ e xy 2 2 × I γ −1 (2xy) . + 1; γ + 1; −4xy + (xy) 1 F1 2 (γ + 1) 2 (3.164) 2

y

Proof. Using formula (3.154), let us find γ Tx x 2 e−x : y

γ

2 y Tx e−x

2γ C(γ ) = (4xy)γ −1

x+y 

2

γ

z3 e−z [(z2 − (x − y)2 )((x + y)2 − z2 )] 2 −1 dz. 2

|x−y|

We find the integral x+y 

I= |x−y|

γ

z3 e−z [(z2 − (x − y)2 )((x + y)2 − z2 )] 2 −1 dz = {z2 = t} 2

152

Transmutations, Singular and Fractional Differential Equations 2 (x+y) 

1 = 2

γ

te−t [(t − (x − y)2 )((x + y)2 − t)] 2 −1 dt = {t − (x − y)2 = w}

(x−y)2

4xy γ 1 −(x−y)2 1 2 = e (w + (x − y)2 )e−w [w(4xy − w)] 2 −1 dw = e−(x−y) 2 2 0 ⎛ 4xy γ γ ⎜ × ⎝ e−w w 2 (4xy − w) 2 −1 dw + (x − y)2 0

⎞

4xy

γ ⎟ e−w [w(4xy − w)] 2 −1 dw ⎠ .

× 0

Applying formulas (3.163) and (2.3.6.1) from [455] of the form a

x α−1 (a − x)β−1 e−px dx = B(α, β)a α+β−1 1 F1 (α; α + β; −ap),

0

Re α,

Re β > 0,

we get γ γ   γ 1 2 I = e−(x−y) B , + 1 (4xy)γ 1 F1 + 1; γ + 1; −4xy 2 2 2γ 2 γ −1 γ −2 √ 2 −x 2 −y 2 +2 π (xy) 2 I γ −1 (2xy) . (x − y) e 2 2 Then γ

Tx x 2 e−x y

2

  x+y   γ +1 γ 2 2γ 2   = ze−z [(z2 − (x − y)2 )((x + y)2 − z2 )] 2 −1 dz √ γ γ −1 (4xy) π 2  =



γ +1 2    γ2

|x−y|

 2γ √ (4xy)γ −1 π   γ 1 −(x−y)2  γ γ × B e , + 1 (4xy)γ 1 F1 + 1; γ + 1; −4xy 2 2 2 2

γ  γ −1 √ 2 2 + 2γ −2 π (x − y)2 e−x −y (xy) 2 I γ −1 (2xy) 2 2

Basic definitions and propositions

2γ +1 

=



γ +1 2

153

 e−(x−y) xy 1 F1 2

γ

+ 1; γ + 1; −4xy



(γ + 1) 2   1−γ 2 −y 2 γ +1 −x +  2 (xy) 2 e I γ −1 (2xy) . 2

Transforming we get (3.164). y

(4) The generalized translation γ Tx of j γ −1 (x) is 2

γ

y Tx j γ −1 (x) = j γ −1 (x) j γ −1 (y). 2

2

(3.165)

2

Proof. Using the presentation (1.19) of j γ −1 (x), formulas (3.154) and (2.12.6.1) from 2 [456] of the form b

1

1

x 1−ν (b2 − x 2 )ν− 2 (x 2 − a 2 )ν− 2 Jν (cx)dx a





√ 1 cb + ca cb − ca (b2 − a 2 )ν c−ν Jν Jν , = 2ν−1 π ν + 2 2 2 1 0 < a < b, ν>− , |Re c| < π, 2

we get γ

y Tx j γ −1 (x) = 2

=

2γ C(γ ) (4xy)γ −1

x+y 

γ

zj γ −1 (z)[(z2 − (x − y)2 )((x + y)2 − z2 )] 2 −1 dz 2

|x−y|



γ +1 2γ C(γ ) γ −1 2  2 2 (4xy)γ −1 x+y  γ −1 γ z1− 2 J γ −1 (z)[(z2 − (x − y)2 )((x + y)2 − z2 )] 2 −1 dz × 2

|x−y|

 

γ   γ +1 γ −1 γ −1 γ −1 √ 2 2γ γ +1 2  2 −1 √   = 2 π 2 (4xy) 2 γ γ −1 2 2 (4xy) π 2 × J γ −1 (x)J γ −1 (y) 2

2

= j γ −1 (x)j γ −1 (y), 2

2

which completes the proof. (5) The generalized translation of i γ −1 (x) is 2

γ y Tx i γ −1 (x) = i γ −1 (x)i γ −1 (y). 2

2

2

(3.166)

154

Transmutations, Singular and Fractional Differential Equations

Proof. Using the presentation (1.20) of i γ −1 (x) and formulas (3.154) and (2.15.3.13) 2 from [455] of the form b

1

1

x 1−ν (b2 − x 2 )ν− 2 (x 2 − a 2 )ν− 2 Iν (cx)dx a





1 cb + ca cb − ca 2 2 ν −ν π ν + (b − a ) c Iν Iν , =2 2 2 2 1 a, b > 0, ν>− , 2 ν−1 √

we get γ

y Tx i γ −1 (x) = 2

2γ C(γ ) (4xy)γ −1

x+y 

γ

zi γ −1 (z)[(z2 − (x − y)2 )((x + y)2 − z2 )] 2 −1 dz 2

|x−y|



x+y  γ −1 2γ C(γ ) γ −1 γ +1 2  = 2 z1− 2 I γ −1 (z) γ −1 2 2 (4xy) |x−y|

γ

× [(z − (x − y) )((x + y)2 − z2 )] 2 −1 dz   γ +1

γ  γ  γ −1 γ −1 γ −1 √ 2 2 γ +1 2  2 −1 √   = 2 π 2 (4xy) 2 γ γ −1 2 2 (4xy) π 2 2

2

× I γ −1 (x)I γ −1 (y) 2

2

= i γ −1 (x)i γ −1 (y), 2

2

which completes the proof. y

The generalized translation γ Tx responds to the definition of the generalized convolution as it allows one to correctly generalize distributions theory to the case when instead of the second derivative the Bessel operator is used. Definition 25. The generalized convolution (one-dimensional), generated by the geny eralized translation γ Tx , is (see [242,610]) ∞ (f ∗ g)γ (x) =

y

f (y)γ Tx g(x)y γ dy.

(3.167)

0

Statement 12. Let f, g ∈ Sev (R+ ). The Hankel transform applied to the generalized convolution (3.167) is Fγ [(f ∗ g)γ (x)](ξ ) = Fγ [f (x)](ξ )Fγ [g(x)](ξ ).

(3.168)

Basic definitions and propositions

155

Proof. Using properties of generalized translation we have ∞ Fγ [(f ∗ g)γ (x)](ξ ) = (f ∗ g)γ (x) j γ −1 (xξ )x γ dx 2

0 ∞

∞ =

j γ −1 (xξ )x γ dx 0

y

f (y)γ Tx g(x) y γ dy

2

0

∞ =

∞ γ

f (y)y γ dy

y

Tx g(x) j γ −1 (xξ )x γ dx 2

0 ∞

=

0 ∞

f (y)y γ dy

y

g(x) γ Tx j γ −1 (xξ )x γ dx 2

0 ∞

=

0

∞ g(x) j γ −1 (xξ )x γ dx = Fγ [f ](ξ )Fγ [g](ξ ).

γ

f (y) j γ −1 (xξ )y dy 2

2

0

0

Statement 12 is given in [610]. If in the definitions and statements of classical harmonic analysis we replace the ordinary shift with a generalized translation, then we get weighted harmonic analysis (see, for example, [242,348,442–444,610]) due to the power weight x γ under the sign of integrals. Definition 26. The multi-dimensional generalized translation is defined by the equality y

y

y

y

(γ Tx f )(x) = γ Tx f (x) = (γ1 Tx11 ...γn Txnn f )(x),

(3.169) y

where each one-dimensional generalized translation γi Txii acts for i=1, ..., n according to    γi 2+1 y   (γi Txii f )(x) = √ π γ2i π  × f (x1 , ..., xi−1 , xi2 + τi2 − 2xi yi cos ϕi , xi+1 , ..., xn ) sinγi −1 ϕi dϕi . 0

Obviously, the properties of a one-dimensional generalized translation are transferred to the multi-dimensional one. So, for example, the following equalities are true γ

y

Tx jγ (x; ξ ) = jγ (x; ξ )jγ (y; ξ ),

(3.170)

156

Transmutations, Singular and Fractional Differential Equations γ

y

Tx iγ (x; ξ ) = iγ (x; ξ )iγ (y; ξ ),

where jγ (x; ξ ) =

n /

j γi −1 (xi ξi ), iγ (x; ξ ) =

i=1

2

(3.171) n / i=1

i γi −1 (xi ξi ), γ1 > 0, ..., γn > 0, func2

tion jν is given by (1.19), and iν is given by (1.20). From Statement 10 if follows that if f (x) is continuous on Rn+ such that ∞ |f (x)|x γ dx < ∞ 0

and g(x) is continuous and bounded on Rn+ , then   y γ y Tx f (x)g(y)y γ dy = f (y)γ Tx g(x)y γ dy. Rn+

(3.172)

Rn+

Definition 27. Generalized convolution generated by a multi-dimensional generaly ized translation γ Tx is given by  y (f ∗ g)γ (x) = (f ∗ g)γ = f (y)(γ Tx g)(x)y γ dy. (3.173) Rn+

We will use also the mixed generalized convolution product defined by the formula  y f ∗ gγ = f (τ, y)(γ Tx g)(t − τ, x)y γ dτ dy, (3.174) Rn+1 + y

where γ Tx is the multi-dimensional generalized translation y

y

y

(γ Tx f )(t, x)=(γ1 Tx11 ...γn Txnn f )(t, x).

(3.175) y

Each one-dimensional generalized translation γi Txii is defined for i=1, ..., n by formula (3.144). From (12) it follows that for f, g ∈ Sev (Rn+ ) the Hankel multi-dimensional transform applied to the generalized convolution (3.173) is Fγ [(f ∗ g)γ (x)](ξ ) = Fγ [f (x)](ξ )Fγ [g(x)](ξ ).

(3.176)

For the generalized convolution (3.173), Young’s inequality is known, which we present for convenience with the proof. Statement 13. Let p, q, r ∈ [1, ∞] and 1 1 1 + =1+ . p q r

(3.177)

Basic definitions and propositions γ

157

γ

If f ∈ Lp , g ∈ Lq , 1 ≤ p, q, r ≤ ∞, and q1 = p1 + 1r − 1, then a generalized convolution (f ∗g)γ is bounded almost everywhere and the Hausdorff–Young inequality is valid, ||(f ∗ g)γ ||r,γ ≤ ||f ||p,γ ||g||q,γ . If

1 p

+

(3.178)

= 1, then

1 q

||(f ∗ g)γ ||∞,γ ≤ ||f ||p,γ |||g||q,γ .

(3.179)

Proof. Let 1 1 1 + = 1. + r p1 p2

(3.180)

We apply to the expression |(f ∗ g)γ (x)| the Hölder inequality for three functions (see [438], p. 64):  y |(f ∗ g)γ (x)| = f (y)(γ Tx g)(x)y γ dy Rn+



y

y

|f (y)|1−a |(γ Tx g)(x)|1−b |f (y)|a |(γ Tx g)(x)|b y γ dy

≤ Rn+

⎛

⎜ ≤⎝

⎞1/r ⎛



⎟ y |f (y)|(1−a)r |(γ Tx g)(x)|(1−b)r y γ dy ⎠

Rn+

⎛

⎜ ×⎝



⎞1/p1 ⎟ |f (y)|ap1 y γ dy ⎠

Rn+

⎞1/p2 ⎟ y |(γ Tx g)(x)|bp2 y γ dy ⎠

⎜ ⎝



.

Rn+

Consider the integral

 Rn+

y

y

|(γ Tx g)(x)|bp2 y γ dy. By producing in (γ Tx g)(x) the change

of variables z2i−1 = yi cos αi , z2i = yi sin αi , 0 ≤ αi ≤ π, i = 1, ..., n, 2n and putting  R2n + = {z = (z1 , ..., z2n ) ∈ R : z2i > 0, i = 1, .., n}, we obtain  y |(γ Tx g)(x)|bp2 y γ dy Rn+



=  R2n +

  2 )|bp2 |g( (z1 − y1 )2 + z22 , ..., (z2n−1 − yn )2 + z2n

n

γ −1

z2ii i=1

dz

158

Transmutations, Singular and Fractional Differential Equations

 = {(z2i−1 − yn )→z2i−1 } =

n

γ −1

z2ii

dz

i=1

 R2n +

 =

  2 2 )|bp2 |g( z12 + z22 , ..., z2n−1 + z2n

|g(x)|bp2 x γ dx.

Rn+

Therefore ⎛ ⎜ |(f ∗ g)γ (x)| ≤ ||f ||aap1 ||g||bbp2 ⎝



⎞1/r ⎟ y |f (y)|(1−a)r |(γ Tx g)(x)|(1−b)r y γ dy ⎠

.

Rn+

We raise the last inequality to the power of r, multiply both parts by x γ , and integrate over Rn+ : ||(f ∗ g)γ ||rr,γ

 

rb ≤ ||f ||ra ap1 ||g||bp2

y

|f (y)|(1−a)r |(γ Tx g)(x)|(1−b)r y γ dy x γ dx Rn+ Rn+



rb = ||f ||ra ap1 ||g||bp2



|f (y)|(1−a)r y γ dy Rn+

y

|(γ Tx g)(x)|(1−b)r x γ dx

Rn+

(1−a)r (1−b)r rb = ||f ||ra ap1 ||g||bp2 ||f ||(1−a)r ||g||(1−b)r .

Choosing a and b so that (1 − a)r = ap1 and (1 − b)r = bp2 , i.e., a = r , we can write b = r+p 2 ra+ap1

||(f ∗ g)γ ||rr,γ ≤ ||f ||ap1

rb+bp2

||g||bp2

r r+p1

and

= ||f ||rap1 ||g||rbp2

or putting ap1 = p and bp2 = q ||(f ∗ g)γ ||r,γ ≤ ||f ||p ||g||q . It remains to show that with this choice of p1 and p2 , (3.180) is valid: 1 1 a 1 b 1 1 p 1  q 1 + = + + = + + 1− + 1− = r p1 p2 r p q r p r q r 1 1 1 1 1 1 1 1 + − + − = + − = 1. r p r q r p q r Inequality (3.179) is obtained from (3.178) by tending to the limit with r → ∞ using (1.47) (p and q should be such that 1/p + 1/q = 1).

Basic definitions and propositions

159

3.4.4 Weighted spherical mean In this subsection, we give the definition and properties of the spherical weighted mean, the main function of which is its effect on the operator γ . Of great interest among various researchers is a generalization of the classical spherical mean of  1 u(x + βr)dS. (3.181) M(x, r, u) = |Sn (1)| Sn (1)

So, in paper [597] a spherical mean in space with negative curvature was considered, and in [160] and [119] a generalization of the spherical mean generated by the Charles Dunkl transformation operator was studied. Here we consider the spherical weighted mean, which is the transformation operator intertwining the multi-dimensional operan . tor (γ )x = (Bγi )xi , x ∈ Rn+ and the one-dimensional Bessel operator (Bn+|γ |−1 )t , i=1

t > 0. Such spherical mean is closely related to the B-ultrahyperbolic equation of the form n 

(Bν )xj u =

j =1

n  (Bν )yj u,

u = u(x1 , ..., xn , y1 , ..., yn ).

(3.182)

j =1

When constructing a weighted spherical mean, instead of the usual shift, a multidimensional generalized translation (3.169) is used. Definition 28. The weighted spherical mean of function f (x), x ∈ Rn+ , for n ≥ 2 is  1 γ γ γ tθ Tx f (x)θ γ dS, (3.183) Mt [f (x)] = (Mt )x [f (x)] = + |S1 (n)|γ S1+ (n)

where θ γ =

n / i=1

θi i , S1+ (n)={θ :|θ|=1, θ ∈Rn+ } is a part of a sphere in Rn+ , and |S1+ (n)|γ γ

γ

is given by (1.107). For n = 1 let Mt [f (x)] = γ Txt f (x). γ

Theorem 36. The weighted spherical mean Mt [f (x)] is the transmutation operator 2 : intertwining (γ )x and (Bn+|γ |−1 )t for f ∈ Cev γ

γ

(Bn+|γ |−1 )t Mt [f (x)] = Mt [(γ )x f (x)].

(3.184)

2 (1.104) gives Proof. First of all we note that for the function f ∈ Cev

|S1+ (n)|γ

t λ

n+|γ |−1

γ Mλ [f (x)]dλ =

0

t 0

λn+|γ |−1 dλ



= Br+ (n)

 (T λy f )(x) y γ dSy

S1+ (n)

(T z f )(x)zγ dz.

(3.185)

160

Transmutations, Singular and Fractional Differential Equations

Let us apply the operator γ to both sides of the relation (3.185) with respect to x. Then we obtain |S1+ (n)|γ

t λ

n+|γ |−1

 n 

(γ )x (γ Txz f )(x)zγ dz

Bt+ (n)

0

=



γ (γ )x Mλ [f (x)]dλ =

(Bγi )zi (γ Txz f )(x)zγ dz.

i=1 + Bt (n)

Formula (3.146) gives (Bγi )zi γi Txzii f (x) = (Bγi )xi γi Txzii f (x) and therefore (γ )x (γ Txz f )(x) = (γ )z (γ Txz f )(x). Then |S1+ (n)|γ

t λ

n+|γ |−1



γ (γ )x Mλ [f (x)]dλ =



 (γ )z (γ Txz f )(x) zγ dz.

Bt+ (n)

0

(3.186) By applying formula (1.101) to the right side of relation (3.186), we obtain  n 

(Bγi )zi (T z f )(x)zγ dz =

i=1 + Bt (n)

n  

∂ (T z f )(x) cos(ν , ei ) zγ dSz , ∂zi

i=1 + St (n)

where ei is the direction of the axis Ozi , i = 1, ..., n. Now, by using the fact that the direction of the outward normal to the boundary of a ball with as center the origin coincides with the direction of the position vector of the point on the ball, we obtain the relation  n 

(Bγi )zi (T f )(x)z dz = t z

γ

i=1 + Bt (n)

= |S1+ (n)|γ t n+|γ |−1

n+|γ |−1



S1+ (n)

∂ tθ (T f )(x) θ γ dSθ ∂t

∂ γ M [f (x)]. ∂t t

Returning to (3.186), we obtain n   i=1 0

t

λn+|γ |−1 Bγi Mλ [f (x)]dλ = t n+|γ |−1 γ

∂ γ M [f (x)]. ∂t t

(3.187)

Basic definitions and propositions

161

By differentiating relation (3.187) with respect to t, we obtain n 

t n+|γ |−1 Bγi Mt [f (x)] = (n + |γ | − 1)t n+|γ |−2 γ

i=1

+ t n+|γ |−1

∂ γ M [f (x)] ∂t t

∂2 γ M [f (x)] ∂t 2 t

or n 

γ

Bγi Mt [f (x)] =

i=1

n + |γ | − 1 ∂ γ ∂2 γ Mt [f (x)] + 2 Mt [f (x)], t ∂t ∂t

and so γ

(γ )x Mt [f (x)] =

n + |γ | − 1 ∂ γ ∂2 γ Mt [f (x)] + 2 Mt [f (x)]. t ∂t ∂t

(3.188)

γ

Now let us consider (γ )x Mt [f (x)]. Using the commutativity of Bγi and Txtθi i (see [242]) we obtain  1 γ γ tθ (γ )x Tx f (x)θ γ dSθ (γ )x Mt [f (x)] = + |S1 (n)|γ =

1 |S1+ (n)|γ

S1+ (n)

 γ

γ

Txtθ [(γ )x f (x)]θ γ dSθ = Mt [(γ )x f (x)],

S1+ (n)

which with (3.188) gives (3.184). A similar proof can be found in [506]. We note the simplest properties of the weighted spherical mean. 1. Linearity and uniformity: γ

γ

γ

Mt [af (x) + bg(x)] = aMt [f (x)] + bMt [g(x)],

a, b ∈ R.

γ

2. Positivity: If f (x) ≥ 0, then Mt [f (x)] ≥ 0. γ 3. Mt [1] = 1. 4. For t = 0 equalities γ Mt [f (x)]|t=0

= f (x),

  ∂ γ =0 M [f (x)] ∂t t t=0

are valid. 2 , then M γ ∈ C 2 (Rn ) by x and 5. If f (x) ∈ Cev t + ev γ

γ

(γ )x Mt [f (x)] = Mt [γ f (x)].

(3.189)

162

Transmutations, Singular and Fractional Differential Equations

6. From (3.140) and (3.142), γ

(Mt )x [ jγ (x, ξ )] = jγ (x, ξ ) j n+|γ | −1 (t|ξ |),

(3.190)

2

γ

(Mt )x [ iγ (x, ξ )] = iγ (x, ξ ) i n+|γ | −1 (t|ξ |) 2

follow.

(3.191)

Weighted generalized functions generated by quadratic forms

4

In this chapter we consider certain types of weighted generalized functions associated with nondegenerate indefinite quadratic forms. Such functions and their derivatives are used for constructing fundamental solutions of iterated ultrahyperbolic equations with a Bessel operator and for constructing negative real powers of hyperbolic and ultrahyperbolic operators with a Bessel operator.

4.1 The weighted generalized function associated with a positive quadratic form and concentrated on a part of a cone We consider the weighted generalized functions δγ (P ) concentrated on a part of a cone and give formulas for its derivatives in this section. More precisely, a weighted generalized function is the function whose action on the test function is equal to the limit of the integral of a delta-like sequence of functions approximating a delta function on a part of a cone in Rn+ with a weight x γ .

4.1.1 B-ultrahyperbolic operator Here we discuss the B-ultrahyperbolic operator and a method of the weighted generalized function which is proposed to study this operator and obtain its fundamental solution for appropriate test functions. I. M. Gelfand and G. E. Shilov in [177] proposed the idea of finding fundamental solutions of second order differential operators by studying coefficients of Laurent series of the weighted generalized function generated by the corresponding quadratic form of this operator. This method is convenient because if we have information about the residues of the generalized function, we can obtain a solution to the equation containing the iterated operator. Depending on the relation between the iteration order and the dimension of the space we obtain a fundamental solution or a solution of a homogeneous equation. The weighted generalized function generated by the indefinite quadratic form 2 2 −...−xp+q P = x12 +...+xp2 −xp+1 Transmutations, Singular and Fractional Differential Equations With Applications to Mathematical Physics https://doi.org/10.1016/B978-0-12-819781-3.00010-0 Copyright © 2020 Elsevier Inc. All rights reserved.

164

Transmutations, Singular and Fractional Differential Equations

is used for the construction of a fundamental solution to the ultrahyperbolic operators with Bessel operator, i.e., γ = γ  ,γ  =

γp ∂ ∂2 γ1 ∂ ∂2 + +...+ + 2 2 x1 ∂x1 ∂xp xp ∂xp ∂x1 −

γp+1 ∂ γp+q ∂ ∂2 ∂2 − −...− + , 2 2 x ∂x xp+q ∂xp+q ∂xp+1 ∂xp+q p+1 p+1

(4.1)

where γi >0, i = 1, ..., p+q, p, q∈N. Operator γ = γ  ,γ  is a B-ultrahyperbolic operator. Let p, q ∈ N, n = p+q, x = (x1 , ..., xn ) = (x  , x  ) ∈ Rn+ , x  = (x1 , ..., xp ), x  = (xp+1 , ..., xp+q ). Weighted generalized functions used for solution to the general Euler–Poisson–Darboux equation and the construction of fractional powers of γ are P = |x  |2 − |x  |2 ,

|x  |2 = x12 + ... + xp2 ,

2 2 |x  |2 = xp+1 + ... + xp+q (4.2)

and P(x) =

n 

gk xk2 ,

gk ∈ C,

k = 1, ..., n.

(4.3)

k=1

Next, we define and study the following weighted generalized functions associated with quadratic forms (4.2) and (4.3) for λ ∈ C (see [505]): • δγ (P ); • Pγλ,+ , Pγλ,− ; • Pγλ , P = P1 + iP2 , where P1 is an indefinite quadratic form with real coefficients and P2 is a positive defined quadratic form; • (P + i0)λγ , (P − i0)λγ ; • (w 2 − |x|2 )λγ ; • (c2 + P + i0)λγ , (c2 + P − i0)λγ ; • Pγλ f (P, λ), where f (z, λ) is an entire function. In Section 1.56 the Hankel transforms of Pγλ , (P ± i0)λγ , Pγλ,± , (w 2 −|x|2 )λ+,γ , and (c2 + P ± i0)λγ will be found.

4.1.2 Weighted generalized function associated with a positive quadratic form Here we consider the weighted generalized function r λ , r = |x|, λ ∈ C. This function is studied in [242] in the case when in the weighted functional (rγλ , ϕ)γ weight was taken only by one variable.

Weighted generalized functions generated by quadratic forms

165

The weighted generalized function rγλ is defined by  (rγλ , ψ)γ =

r λ ψ(x)x γ dx, ψ ∈ Sev .

Rn+

This function is an analytic function of λ for Re λ > −(n + |γ |). For Re λ ≤ −(n + |γ |) we may define the weighted generalized function rγλ by analytic continuation. For Re λ > −(n + |γ |), rγλ can be differentiated by parameter λ, i.e., ∂ λ (r , ϕ)γ = ∂λ



r λ ln r ϕ(x)(x  )γ dx.

R+ N

Let us move on to spherical coordinates x = r, r = |x|, in (rγλ , ϕ)γ , writing it in the form ∞ (rγλ , ϕ)γ

=

r

λ+n+|γ |−1

 ϕ(r)γ dS dr

S1+ (n)

0

= |S1+ (n)|γ

∞

r λ+n+|γ |−1 Mϕγ (r) dr,

0

where |S1+ (n)|γ is given by (1.107) and γ

Mϕγ (r) = M0 [ϕ(r)] =

1 + |S1 (n)|γ

 ϕ(r)θ γ dS, S1+ (n)

is the weighted spherical mean (3.183) at 0. γ

Theorem 37. For Mϕ (r), ϕ ∈ Sev , the representation 1 1 (Mϕγ ) (0)r 2 + ... + (M γ )(2k) (0)r 2p + ... 2! (2p)! ϕ ∞  (γ )p ϕ(0)r 2p + = |S1 (n)|γ 2p p!(n + |γ |)(n + |γ | + 2)...(n + |γ | + 2p − 2)

Mϕγ (r) = ϕ(0) +

p=0

(4.4) is valid. γ

Proof. In the beginning, we note that the function Mϕ (r) is infinitely differentiable by r for r > 0 and decreases at r → ∞ faster than any degree 1r , which follows from the similar properties of the function ϕ(x).

166

Transmutations, Singular and Fractional Differential Equations γ

It is easy to show that the function Mφ (r) is infinitely differentiable at r = 0, which is enough to expand the function φ(x) using the Taylor formula and make sure that all γ its odd derivatives are equal to zero when r = 0. So the function Mφ (r) ∈ Sev . The regular nonweighted generalized function  μ x+

=

x ≤ 0, x>0

0 xμ

acts on ψ ∈ Sev by the formula μ (x+ , ψ) =

∞ x μ ψ(x) dx. 0

The expression (rγλ , ϕ)γ = |S1+ (n)|γ

∞

r λ+n+|γ |−1 Mϕγ (r) dr

0

can be considered as the result of applying the nonweighted generalized function γ μ |S1+ (n)|γ x+ , where μ = λ + n + |γ | − 1 to the test function Mϕ (x): (r λ , ϕ(x))γ = (|S1+ (n)|γ x+ , Mϕγ (x)). μ

(4.5)

Therefore, we can use the results of the study of this generalized function given in the book [177]. μ The generalized function x+ is an analytical function for Re μ > −1 or Re λ > −n − |γ |. Its analytical extension to the set Re μ > −n − 1, with excluded points μ = −1, −2, −3, ..., is determined by the equality

μ (x+ , ψ) =

∞

1 x ψ(x) dx = μ

0

... −

 x ψ(x) − ψ(0) − xψ  (0) − ... μ

0

x n−1 (N − 1)!

 ∞ n  (n−1) (0) dx + x μ ψ(x) dx + ψ 1

k=1

ψ (k−1) (0) . (k − 1)!(μ + k)

(4.6)

Note that in the strip of the complex plane −n − 1 < Re μ < −n for 1 ≤ k ≤ n the equality 1 − = μ+k

∞ x μ+k−1 dx 1

(4.7)

Weighted generalized functions generated by quadratic forms

167

holds. By virtue of equality (4.7), in the strip −n − 1 < Re μ < −n formula (4.6) can be converted to a simpler one: μ (x+ , ψ) =

∞ 0

  x n−1 x μ ψ(x) − ψ(0) − xψ  (0) − ... − ψ (n−1) (0) dx. (4.8) (n − 1)!

The right side of expression (4.6) gives a regularized value of the integral to the μ left in this expression. The generalized function x+ is defined for all μ = −1, −2, ... and as a function of μ has first order poles at points μ = −1, −2, ... (λ = −(n + |γ |), −(n + |γ | + 1), −(n + |γ | + 2), ...). We calculate its residue at μ = −k: μ

res [(x+ , ψ)] =

μ=−k

(−1)k−1 (k−1) (x), ψ(x)). (δ (k − 1)! μ

Therefore, the generalized function x+ as a function of μ has at μ = −k a simple pole (−1)k−1

with the residue (k−1)! δ (k−1) (x), k = 1, 2, .... γ μ The residue of (|S1+ (n)|γ x+ , Mϕ (x)) at μ = −k (λ = −(N + |γ | + k − 1)) is res [(|S1+ (n)|γ x+ , Mϕγ (x))] = |S1+ (n)|γ μ

μ=−k

γ

(Mϕ )(k−1) (0) . (k − 1)! γ

But since all the odd derivatives of the function Mϕ (r) vanish at r = 0, only a series of poles corresponding to odd values μ = −1, −3, ..., −2p − 1, ..., p = 0, 1, 2, ..., (λ = −(n + |γ |), −(n + |γ | + 2), ..., −(n + |γ | + 2p), ...) remains, and we can write res

μ=−2p−1

[(|S1+ (n)|γ x+ , Mϕγ (x))] = |S1+ (n)|γ μ

= |S1+ (n)|γ

γ

(δ (2p) (x), Mϕ (x)) (2p)! γ

(Mϕ )(2p) (0) . (2p)!

Now let us return to weighted generalized functions. Considering formula (4.5) we note that the residue of (rγλ , ϕ)γ as a function of λ at λ = −(n + |γ | + 2p), p = 0, 1, 2, ..., is res

λ=−(n+|γ |+2p)

[(rγλ , ϕ)γ ] = |S1+ (n)|γ

γ

(Mϕ )(2p) (0) . (2p)!

(4.9)

In particular, when p = 0 at λ = −(n + |γ |), the weighted generalized function rγλ has a simple pole with residue res

λ=−(n+|γ |)

[(rγλ , ϕ)γ ] = |S1+ (n)|γ Mϕγ (0).

Using the first of condition (3.189) we obtain res

λ=−(n+|γ |)

[(rγλ , ϕ)γ ] = |S1+ (n)|γ ϕ(0) = |S1+ (n)|γ (δ(x), ϕ(x)).

(4.10)

168

Transmutations, Singular and Fractional Differential Equations

It means that rγλ as a function of λ at λ = −(n + |γ |) has a simple pole with residue |S1+ (n)|γ δ(x). γ Following P. Pizetti [440] (see also [177]) let us express the value (Mϕ )(2p) (0) directly through the function ϕ. To do this, we construct another expression for the residue of the weighted generalized function rγλ . Prove that for Re λ > −n − |λ|, γ (rγλ+2 ) = (λ + 2)(λ + N + |γ |)rγλ .

(4.11)

For Re λ > 0 formula (4.11) is proved by direct calculation of the left side. For other values of λ it is valid due to analytic continuation. Iterating formula (4.11), we get for any integer p the equality p λ+2p

rγλ =

γ rγ . (λ + 2)...(λ + 2p)(λ + n + |γ |)...(λ + n + |γ | + 2p − 2)

(4.12)

Now the residue of rγλ at λ = −(n + |γ | + 2p) is calculated as a residue of the right side of equality (4.12) at this λ. But for λ = −(n + |γ | + 2p) the denominator of expression (4.12) does not vanish; hence it suffices to find the residue of the numerator. p p As for any test function ϕ ∈ Sev , if the equality (γ r λ+2p , ϕ)γ = (r λ+2p , B ϕ)γ λ+2p p is true, then it is needed to find a residue of (rγ , γ ϕ)γ at λ = −(n + |γ | + 2p). Such residue was calculated in (4.10) and it equals res

τ =−(n+|γ |)

[(r τ , pγ ϕ)γ ] = |S1+ (n)|γ pγ ϕ(0).

Then the residue of rγλ as a function of λ at λ = −(n + |γ | + 2p) is res

λ=−(n+|γ |+2p)

[(rγλ , ϕ(x))γ ]

|S1+ (n)|γ (γ δ(x), ϕ(x)) , 2p p!(n + |γ |)(n + |γ | + 2)...(n + |γ | + 2p − 2) p

=

(4.13)

or |S1+ (n)|γ γ δ(x) . (4.14) 2p p!(n + |γ |)(n + |γ | + 2)...(n + |γ | + 2p − 2) p

res

λ=−(n+|γ |+2p)

[rγλ ] =

Comparing expression (4.13) with the previously obtained formula (4.9), we obtain an expression for the derivative with respect to r at zero of order 2p, p = 0, 1, 2..., for the weighted spherical mean: p

(Mϕγ )(2p) (0) =

(2p)!γ ϕ(0) . p 2 p!(n + |γ |)(n + |γ | + 2)...(n + |γ | + 2p − 2) γ

This makes it possible to write the decomposition of the function Mϕ (r) in the Taylor series, which gives the statement. The proof is complete.

Weighted generalized functions generated by quadratic forms

169

Formula (4.4) makes it possible to write a regularization of the weighted generalized function rγλ for Re λ>−(n+|γ |+2p), λ = −(n+|γ |), −(n+|γ |+2), ..., −(n+|γ |+2p−2), in the form

(r

λ

, ϕ)γ = |S1+ (n)|γ

∞

r λ+n+|γ |−1 Mϕγ (r) dr = |S1+ (n)|γ (r λ+n+|γ |−1 , Mϕγ (r))

0

= |S1+ (n)|γ

1

r λ+n+|γ |−1 [Mϕγ (r) − ϕ(0) − ... −

1 (M γ (0))(2p) r 2p ] dr (2p)! ϕ

0

+ |S1+ (n)|γ

∞

r λ+n+|γ |−1 Mϕγ (r) dr + |S1+ (n)|γ

2p  k=0

1

γ

(Mϕ (0))(2k) . (2k)!(λ + n + |γ | + 2k) (4.15)

Define the weighted functional (rγλ lnm r, ϕ)γ by the formula  (rγλ lnm

r, ϕ)γ =

rγλ lnm rϕ(x)x γ dx.

(4.16)

Rn+

We regularize (rγλ lnm r, ϕ)γ at Re λ > −(n + |γ | + 2p), λ = −(n + |γ |), −(n + |γ | + 2), ..., −(n + |γ | + 2p − 2), using m-times differentiation by λ of formula (4.15): (rγλ lnm

r, ϕ)γ = |S1+ (n)|γ

1

r λ+n+|γ |−1 lnm r[Mϕγ (r) − ϕ(0) − ...

0

1 (M γ (0))(2p) r 2p ] dr − (2p)! ϕ ∞ + + |S1 (n)|γ r λ+n+|γ |−1 lnm rMϕγ (r) dr 1

+ |S1+ (n)|γ

2p  k=0

γ

(−1)m m!(Mϕ (0))(2k) . (2k)!(λ + n + |γ | + 2k)m+1

(4.17)

4.1.3 Weighted generalized function δγ (P ) In this subsection we will study the singular generalized function δγ defined by the equality (see Section 1.2.3) (δγ , ϕ)γ = ϕ(0),

ϕ(x) ∈ Sev .

170

Transmutations, Singular and Fractional Differential Equations

Let p, q∈N, n = p+q, and 2 2 − ... − xp+q , P = |x  |2 − |x  |2 = x12 + ... + xp2 − xp+1   where x = (x1 , ..., xn ) = (x  , x  )∈R+ n , x = (x1 , ..., xp ), x = (xp+1 , ..., xp+q ).

Definition 29. Let ϕ∈Sev vanish at the origin. For such ϕ we define the generalized function δγ (P ) concentrated on the part of the cone P = 0 belonging to R+ n by the formula  (δγ (P ), ϕ)γ = δγ (|x  |2 − |x  |2 )ϕ(x)x γ dx. (4.18) Rn+

If the function ϕ∈Sev does not vanish at the origin, then (δγ (P ), ϕ)γ is defined by regularizing the integral. Lemma 12. Let ϕ∈Sev vanish at the origin. For (δγ (P ), ϕ(x))γ when p > 1, q > 1, the representation 1 (δγ (P ), ϕ(x))γ = 2

∞ 0





{|ω |=1}+

{|ω |=1}+

ϕ(tω)t n+|γ |−3 ωγ dS  dS  dt,

(4.19)

where {|ω | = 1}+ = {ω ∈ R+ : |ω | = 1}, {|ω | = 1}+ = {ω ∈ R+ : |ω | = 1}, ω = (ω , ω ) is valid. When p = q = 1, P = x 2 − y 2 , p

(δγ (x 2 − y 2 ), ϕ(x, y))γ =

q

1 2

∞

ϕ(y, y)y |γ |−1 dy.

0

When p = 1, q = n − 1 > 1, P = x12 − |x  |2 , γ  = (γ2 , ..., γn ), 1 (δγ (P ), ϕ(x))γ = 2

∞ 



ϕ(y, yσ )y n+|γ |−3 dy σ γ dS.

0 S + (n−1) 1

When q = 1, p = n − 1 > 1, P = |x  |2 − xn2 , γ  = (γ1 , ..., γn−1 ), (δγ (P ), ϕ(x))γ =

1 2

∞ 



ϕ(xσ, x)x n+|γ |−3 dx σ γ dS.

0 S + (n−1) 1

Proof. Let first p > 1, q > 1. Turning to bipolar coordinates x1 = rω1 , ..., xp = rωp , xp+1 = sωp+1 , ..., xp+q = sωp+q ,

(4.20)

Weighted generalized functions generated by quadratic forms

where r =



171



2 2 , and putting ω = (ω , ..., ω ) ∈ + ... + xp+q xp+1 1 p  q = (ωp+1 , ..., ωn ) ∈ R+ , |ω | = ω12 + ... + ωp2 = 1, |ω | =

x12 + ... + xp2 , s =

R+ , ω  2 2 ωp+1 + ... + ωp+q = 1, we obtain p

(δγ (P ), ϕ(x))γ = ∞∞





δ(r 2 − s 2 )ϕ(rω , sω )r p+|γ

 |−1

s q+|γ

 |−1

ωγ dS  dS  drds,

0 0 {|ω |=1}+ {|ω |=1}+

where {|ω | = 1}+ = {ω ∈ R+ : |ω | = 1}, p

{|ω | = 1}+ = {ω ∈ R+ : |ω | = 1}, q

dS  is the surface element of {|ω | = 1}+ and dS  is the surface element of {|ω | = 1}+ . Replacing variables by formulas r 2 = u, s 2 = v, we obtain dr = 1 1 − 12 du, ds = 12 v − 2 dv, and 2u 1 (δγ (P ), ϕ(x))γ = 4 p+|γ  | 2 −1

×u ∞ 1 = 4



∞



v

∞∞





√ √ δ(u − v)ϕ( uω , vω )

0 0 {|ω |=1}+ {|ω |=1}+

q+|γ  | −1 2

ωγ dS  dS  dudv



p+|γ  | q+|γ  | √ √ ϕ( vω , vω )v 2 −1 v 2 −1 ωγ dS  dS  dv

0 {|ω |=1}+ {|ω |=1}+

1 = 4



n+|γ | √ ϕ( vω)v 2 −2 ωγ dS  dS  dv.

0 {|ω |=1}+ {|ω |=1}+

Going back to the variable s by the formula v = s 2 we get (4.19). For p = q = 1 the quadratic form P is x 2 − y 2 and (δγ (x 2 − y 2 ), ϕ(x, y))γ ∞∞ = δ(x 2 − y 2 )ϕ(x, y)x γ1 y γ2 dxdy = {x 2 = u, y 2 = v} 0 0 ∞∞

=

1 4

0 0

γ1 −1 γ2 −1 √ √ δ(u − v)ϕ( u, v)u 2 v 2 dudv

172

Transmutations, Singular and Fractional Differential Equations

1 = 4

∞

|γ | √ √ 1 ϕ( v, v)v 2 −1 dv = {v = y 2 } = 2

0

∞

ϕ(y, y)y |γ |−1 dy.

0

Let us consider now p = 1, q = n − 1 > 1. We have (δγ (P ), ϕ(x))γ = (δ(x12 − |x  |2 ), ϕ(x))γ =



δ(x12 − |x  |2 )ϕ(x)x γ dx.

Rn+

Passing to spherical coordinates x  = ρσ , σ ∈ Rn−1 + , we obtain (δγ (P ), ϕ(x))γ = (δ(x12 − |x  |2 ), ϕ(x))γ ∞∞  γ = δ(x12 − ρ 2 )ρ n+|γ |−2 x1 1 dx1 dρ





ϕ(x1 , ρσ )σ γ dS

S1+ (n−1)

0 0

= {x12 = u, ρ 2 = v} ∞∞  γ1 −1 n+|γ  |−1 √ √ 1  δ(u − v)ϕ( u, vσ )u 2 v 2 −1 dudvσ γ dS = 4 0 0 S + (n−1) 1

1 = 4 1 = 2

∞ 

n+|γ | √ √  ϕ( v, vσ )v 2 −2 dvσ γ dS = {v = y 2 }

0 S + (n−1) 1

∞ 



ϕ(y, yσ )y n+|γ |−3 dyσ γ dS.

0 S + (n−1) 1

The case q = 1, p = n − 1 > 1 is considered similarly. Lemma 13. The derivative of order k of function δγ (P ) for p > 1, q > 1 has two (k) (k) representations denoted δγ ,1 (P ) and δγ ,2 (P ) of the form (k) (δγ ,1 (P ), ϕ(x))γ

∞ = 0

(k) (δγ ,2 (P ), ϕ(x))γ

1 ∂ 2s ∂s

k

∞

= (−1)

k 0

ψ(r, s)s

1 ∂ 2r ∂r

q+|γ  |−2

k ψ(r, s)r



r p+|γ

 |−1

dr, (4.21)

s 2 =r 2

p+|γ  |−2



s q+|γ

 |−1

ds,

r 2 =s 2

(4.22)

Weighted generalized functions generated by quadratic forms

where ψ(r, s) =

1 2





{|ω |=1}+

{|ω |=1}+

173

ϕ(rω , sω )ωγ dS  dS  ,

ϕ ∈ Sev .

(4.23)

Integrals (4.21) and (4.22) converge and coincide for k< n+|γ2 |−2 . If k≥ n+|γ2 |−2 , then these integrals need to be understood in the sense of regularized values. Proof. Let us find the derivative of the order k of δγ (P ). After tending to bipolar coordinates (4.20) we obtain P = r 2 − s 2 and ∞∞ (δγ(k) (P ), ϕ(x))γ







= 0 0 {|ω |=1}+ {|ω |=1}+

× ϕ(rω , sω )r p+|γ

 |−1

∂k 2 2 δ(r − s ) ∂P k

s q+|γ

 |−1

ωγ dS  dS  drds. 1

(4.24) 1

Replacing variables by formulas r 2 = u, s 2 = v, dr = 12 u− 2 du, ds = 12 v − 2 dv, we ∂ ∂ obtain P = u − v, ∂P = ∂u , and (δγ(k) (P ), ϕ(x))γ 1 = 4

∞∞







0 0 {|ω |=1}+ {|ω |=1}+

∂k δ(u − v) ∂uk

p+|γ  | q+|γ  | √ √ × ϕ( uω , vω )u 2 −1 v 2 −1 ωγ dS  dS  dudv k

∞∞   √  √  p+|γ  | −1 ∂ k1 2 δ(u − v) ϕ( uω , vω )u = (−1) 4 ∂uk

0 0 {|ω |=1}+ {|ω |=1}+

×v

q+|γ  | −1 2

= (−1)k ×v

1 4

ωγ dS  dS  dudv ∞   0 {|ω |=1}+ {|ω |=1}+

q+|γ  | −1 2

∂ k √  √  p+|γ  | −1 2 ϕ( uω , vω )u ∂uk u=v

ωγ dS  dS  dv.

Remembering that u = r 2 , v = s 2 we can write (δγ(k) (P ), ϕ(x))γ = (−1)

k1

2

× s q+|γ

∞





0 {|ω |=1}+ {|ω |=1}+  |−1

ωγ dS  dS  ds.



1 ∂ 2r ∂r

k





ϕ(rω , sω )r

p+|γ  |−2



r 2 =s 2

174

Transmutations, Singular and Fractional Differential Equations

Entering the designation (4.23), we obtain the following formula for δγ(k) (P ): ∞ (δγ(k) (P ), ϕ(x))γ

= (−1)

1 ∂ 2r ∂r

k

k ψ(r, s)r

p+|γ  |−2



0

s q+|γ

 |−1

ds.

r 2 =s 2

(4.25) Returning to formula (4.24) and putting r 2 = −u, s 2 = −v, u < 0, v < 0, dr = 1 ∂ ∂ ds = − 12 (−v)− 2 dv (u < 0, v < 0), we get P = v − u, ∂P = ∂v , and

1 − 12 (−u)− 2 du,

(δγ(k) (P ), ϕ(x))γ 1 = 4

−∞ −∞







0 {|ω |=1}+ {|ω |=1}+

0

∂k δ(v − u) ∂v k

√ √ p+|γ  | q+|γ  | × ϕ( −uω , −vω )(−u) 2 −1 (−v) 2 −1 ωγ dS  dS  dudv = (−1)k

1 4

−∞ −∞

 δ(v − u)

0 {|ω |=1}+ {|ω |=1}+

0





∂k ∂v k

 √ √ q+|γ  | p+|γ  | p q   −1 2 × ϕ( −uω , −vω )(−v) (−u) 2 −1 ωγ dS1 dS1 dudv = (−1)

k1

4

× (−u)

−∞







0 {|ω |=1}+ {|ω |=1}+ p+|γ  | 2 −1

 √ q+|γ  | ∂k √   −1 2 ϕ( −uω , −vω )(−v) k ∂v v=u

ωγ dS  dS  du.

Remembering that −u = r 2 , −v = s 2 , we return to the variables r and s: (δγ(k) (P ), ϕ(x))γ 1 = 2

∞







0 {|ω |=1}+ {|ω |=1}+

× r p+|γ

 |−1

1 ∂ 2s ∂s

k





ϕ(rω , sω )s

q+|γ  |−2



s 2 =r 2

ωγ dS  dS  dr.

Using the designation (4.23), we write ∞ (δγ(k) (P ), ϕ(x))γ

= 0

1 ∂ 2s ∂s

k ψ(r, s)s

q+|γ  |−2



s 2 =r 2

r p+|γ

 |−1

dr. (4.26)

Weighted generalized functions generated by quadratic forms

175

(k) Further for functions (4.25) and (4.26) we will use notations δγ(k) ,1 (P ) and δγ ,2 (P ),

so (k) (δγ ,1 (P ), ϕ(x))γ

∞ =

1 ∂ 2s ∂s

k ψ(r, s)s

q+|γ  |−2



∞ = (−1)

k

 |−1

dr,

s 2 =r 2

0

(k) (δγ ,2 (P ), ϕ(x))γ

r p+|γ

1 ∂ 2r ∂r

k ψ(r, s)r

p+|γ  |−2



0

s q+|γ

 |−1

ds.

r 2 =s 2

(k) and δγ ,2 (P ) converge and coincide for k < n+|γ2 |−2 for any The integrals ϕ ∈ Sev . If, on the other hand, k ≥ n+|γ2 |−2 , these integrals must be understood in the (k) sense of their regularizations. Specifically, let us make in δγ(k) ,1 (P ) and δγ ,2 (P ) the formal change of variables r 2 = u, s 2 = v. Then we may write (k) δγ ,1 (P )

(k) (δγ ,1 (P ), ϕ(x))γ

1 = 4

∞ 0

(δγ(k) ,2 (P ), ϕ(x))γ

(−1)k = 4

 q+|γ  | p+|γ  | √ √ ∂k −1 2 2 −1 du, ψ( u, v)v u ∂v k v=u ∞ 0

 p+|γ  | q+|γ  | √ √ ∂k 2 −1 ψ( u, v)u v 2 −1 dv. k ∂u u=v

√ √ The function ψ( u, v) ∈ Sev for u and v. Then   k q+|γ  | q+|γ  | √ √ ∂ −1 2 ψ( u, v)v = u 2 −1−k 1 (u), k ∂v v=u 

 p+|γ  | p+|γ  | √ √ ∂k −1 2 2 −1−k 2 (v), ψ( u, v)u = v ∂uk u=v

where 1 (u), 2 (v) ∈ Sev , thus (k) (δγ ,1 (P ), ϕ(x))γ

1 = 4

∞ u

n+|γ |−k −1 2

1 1 (u)du = (uλ+ , 1 ), 4

0

(δγ(k) ,2 (P ), ϕ(x))γ

(−1)k = 4

∞ v 0

 uλ uλ+ = 0

u > 0, u ≤ 0.

n+|γ |−k −1 2

2 (v)dv =

(−1)k λ (v+ , 2 ), 4

176

Transmutations, Singular and Fractional Differential Equations

The regularization of this function is the generalized function uλ+ , which for λ = −1, −2, ... is obtained by analytic continuation uλ+ from Re λ>0. For λ = −1, −2, ..., (n+|γ | = −1, −2, ...) this analytic generalized function has simple poles and the generalized function u−m + , m ∈ N, is defined as the constant term in the Laurent expansion for uλ+ about λ = −m (see [177]). Remark 7. Note that when k = 0 formulas (4.21) and (4.22) are equivalent to formula (4.19). We will use integrals (4.21) and (4.22) at k∈N ∪ {0}. We have been studying the case when p > 1 and q > 1. The cases in which either p or q is equal to unity are special cases, since in this case the transition to bipolar coordinates loses its meaning. Let us start from the case p = q = 1. Lemma 14. For p = q = 1 and ϕ ∈ Sev the derivative of order k of the weighted (k) (k) generalized function δγ (P ) has two representations, denoted by δγ ,1 (P ) and δγ ,2 (P ), of the form (k) (δγ ,1 (x 2

1 − y ), ϕ(x, y))γ = 2

∞

2

1 ∂ 2y ∂y

k ϕ(x, y)y

γ2 −1



x γ1 dx (4.27)

y=x

0

and (k) (δγ ,2 (x 2

− y ), ϕ(x, y))γ = (−1) 2

k1

∞

2

1 ∂ 2x ∂x

k ϕ(x, y)x

γ1 −1



0

y γ2 dy.

x=y

(4.28) Proof. The quadratic form P for p = q = 1 is P = x 2 − y 2 . Let us find the derivative of order k of δγ (x 2 − y 2 ): ∞∞ (δγ(k) (P ), ϕ(x, y))γ

=

∂k 2 2 δ(x − y ) ϕ(x, y)x γ1 y γ2 dxdy. ∂P k

(4.29)

0 0

Now let us choose the new variables u and v by formulas x 2 = u, y 2 = v. We obtain 1 1 ∂ ∂ dx = 12 u− 2 du, dy = 12 v − 2 dv, P = u − v, ∂P = ∂u , and (δγ(k) (P ), ϕ(x, y))γ

∞∞ k γ1 −1 γ2 −1 √ √ 1 ∂ = δ(u − v) ϕ( u, v)u 2 v 2 dudv k 4 ∂u 0 0

= (−1)

k1

∞∞

4 0 0



γ1 −1 γ2 −1 ∂k √ √ 2 2 dudv δ(u − v) ϕ( u, v)u v ∂uk

Weighted generalized functions generated by quadratic forms

= (−1)

k1

∞

4 0

177

γ1 −1 γ −1 ∂k √ √ v 22 dv. 2 ϕ( u, v)u ∂uk u=v

Remembering that u = x 2 , v = y 2 , we can return to the variables x and y. Denot(k) (k) (k) ing δγ (P ) through δγ ,2 (P ) we get (4.28) δγ (P ). Similarly, making the change of variables r 2 = −u, s 2 = −v, u0}, {P (x) 1. Transforming Pγλ,+ using bipolar coordinates (4.20) we obtain ∞r (Pγλ,+ , ϕ)γ

=

(r 2 − s 2 )λ ψ(r, s)r p+|γ

 |−1

s q+|γ

 |−1

drds,

(4.36)

0 0

where ψ(r, s) =

1 2





{|ω |=1}+

{|ω |=1}+

ϕ(rω , sω )ωγ dS  dS  (see (4.23)). Now let us

pass in (4.36) to variables u = r 2 , v = s 2 : (Pγλ,+ , ϕ)γ

1 = 4

∞u (u − v)λ ψ1 (u, v)u

p+|γ  | 2 −1

s

q+|γ  | −1 2

dudv,

0 0

where ψ1 (u, v) = ψ(r, s) with u = r 2 , v = s 2 . Finally, we write v = ut, which transforms Pγλ,+ to the form ∞ (Pγλ,+ , ϕ)γ

=

uλ+ 0

p+q+|γ | −1 2

(λ, u)du,

(4.37)

180

Transmutations, Singular and Fractional Differential Equations

where 1 (λ, u) = 4

1 (1 − t)λ t

q+|γ  | −1 2

ψ1 (u, tu)dt.

(4.38)

0

Formula (4.37) shows that Pγλ,+ has two sets of poles. The first of these consists of the poles of function (λ, u). Namely, for t = 1 the function (λ, u) has simple poles at λ = −1, −2, ..., −k, ...

(4.39)

with residues   q+|γ  |−2 1 (−1)k−1 ∂ k−1 2 res (λ, u) = ψ1 (u, tu) . t λ=−k 4 (k − 1)! ∂t k−1 t=1

(4.40)

Besides, at regular points of (λ, u) function (4.37) has poles at λ=−

n + |γ | n + |γ | n + |γ | ,− − 1, ..., − − k, ..., 2 2 2

(4.41)

with residues res

| λ=− n+|γ 2 −k

(Pγλ,+ , ϕ)γ =



1 ∂k n + |γ | .  − − k, u k k! ∂u 2 u=0

(4.42)

Sets of poles were obtained for p > 1, q > 1; however, the same sets are obtained when p ≥ 1, q ≥ 1. All following results concerning weighted generalized functions related to indefinite quadratic forms are valid for p ≥ 1, q ≥ 1. So we have three cases. The first case is when λ is a point in the first set (4.39) but does not belong to the second set (4.41). The second case is when λ belongs to the second set (4.41) but λ = −k, k∈N. The third case is when λ is in both sets (4.39) and (4.41) simultaneously. We study these three cases separately, presenting the results in the form of the following three theorems. Residues of Pγλ,± at poles are expressed through the weighted generalized function (4.18). Theorem 38. If p ≥ 1, q ≥ 1, λ = −k, k ∈ N, and n + |γ |∈R\N or n + |γ |∈N and | n + |γ | = 2k − 1, k∈N, and besides, if n+|γ | is even and k< n+|γ 2 , then the weighted λ generalized function Pγ ,+ has simple poles at such λ with residues res Pγλ,+ =

λ=−k

(−1)k−1 (k−1) (P ). δ (k − 1)! γ ,1

(4.43)

Proof. Let us write (λ, u) in the neighborhood of λ = −k in the form (λ, u) =

0 (u) + 1 (λ, u), λ+k

0 (u) = res (λ, u), λ=−k

Weighted generalized functions generated by quadratic forms

181

where 1 (λ, u) is regular at the λ = −k weighted function. We obtain (Pγλ,+ , ϕ)γ =

1 λ+k

∞ uλ+

n+|γ | 2 −1

∞ uλ+

0 (u)du+

0

n+|γ | 2 −1

1 (λ, u)du. (4.44)

0

Integrals in (4.44) are regular at λ for λ = −k. So (Pγλ,+ , ϕ)γ has a simple pole at such a point and using (4.40) we can write res

λ=−k

(Pγλ,+ , ϕ) =

(−1)k−1 4(k−1)!

∞ u

n+|γ | 2 −k−1

0

  q+|γ  | ∂ k−1 −1 2 ψ (u, tu) du. t 1 ∂t k−1 t=1 (4.45)

If in (4.45) we put tu = v, then we obtain res

λ=−k

(Pγλ,+ , ϕ) =

(−1)k−1 4(k−1)!

∞ 0

  q+|γ  | p+|γ  | ∂ k−1 −1 2 2 −1 du, ψ (u, v) u v 1 ∂v k−1 v=u (4.46)

where integral is understood in the sense of its regularization at k ≥ n2 . Note that if we write u = r 2 and v = s 2 in the formula for the (k − 1)-th derivative of function δγ (P ), defined by formula (4.21), we get (k−1)

(δγ ,1 (P ), ϕ)γ =

1 2

∞ 0

 p+|γ  | ∂ k−1 q+|γ  | −1 2 v ψ (u, v) u 2 −1 du, 1 k−1 ∂v v=u

(4.47)

where ψ1 (u, v) =

1 2

 

√ √ ϕ( uω , vω )ωγ dSp dSq .

Sp+ Sq+

Formulas (4.46) and (4.47) give (4.42). For k ≥ n2 integral in (4.47) is to be understood in the sense of its regularization. In the case n + |γ |∈R\N or n + |γ |∈N and n + |γ | = 2k − 1, k∈N, the regularization of integral in (4.47) is defined as its analytical continuation. Now let us consider the case when the singular point λ is in the second set (4.41), | but not in the first (4.39). If λ = − n+|γ 2 −k, k = 0, 1, 2, ..., and n + |γ |∈R\N or n + |γ |∈N and n + |γ | = 2k − 1, k∈N, then the weighted generalized function (λ, u) is | λ regular in the neighborhood of λ = − n+|γ 2 −k. So (Pγ ,+ , ϕ)γ will have a simple pole

| with residue given by (4.42). Thus the residue of Pγλ,+ at λ = − n+|γ 2 −k is a weighted functional concentrated at the origin.

182

Transmutations, Singular and Fractional Differential Equations

Before getting the expression for residue

res

| λ=− n+|γ 2 −k

(Pγλ,+ , ϕ) through derivatives

of function ϕ(x) at the origin we get one useful formula. Consider the B-ultrahyperbolic operator (4.1):   , − Bγp+q γ = γ  ,γ  = Bγ1 + ... + Bγp − Bγp+1

Bγi =

∂2 γi ∂ + . 2 xi ∂xi ∂xi

Applying γ to the (λ + 1)-th power of quadratic form 2 2 −...−xp+q , n = p + q, p > 1, q > 1, P (x) = x12 +...+xp2 −xp+1

we obtain γ P



λ+1

n+|γ | (x) = 4(λ+1) λ+ P λ (x). 2

(4.48)

Theorem 39. Let p ≥ 1, q ≥ 1, n + |γ | is not a natural number or n + |γ |∈N and n + |γ | = 2k − 1, k∈N. Then either p + |γ  | is not natural or p+|γ  |∈N, p+|γ  | = 2m−1, | m∈N, and q + |γ  | is even. In this case Pγλ,+ has simple poles at λ = − n+|γ 2 −k, k∈N∪{0}, with residues   n   γi 2+1 q+|γ  | (−1) 2 i=1   k δγ (x). res Pγλ,+ = n+2k n+|γ | 2 k!  n+|γ | + k γ λ=− 2 −k 2 If p + |γ  | is even, then Pγλ,+ is a regular generalized weighted function at λ =

| − n+|γ 2 −k, k∈N∪{0}.

| Proof. Consider first λ = − n+|γ 2 . Using formula (4.42) we can write



res (Pγλ,+ , ϕ)γ | λ=− n+|γ 2

1 n+|γ | q+|γ  | n+|γ | ψ1 (0, 0) = − (1−t)− 2 t 2 dt ,0 = 2 4 0     q+|γ  | n+|γ | + 1  −  2 2 1   = ψ1 (0, 0) . (4.49) | p+|γ 4  − 2 +1

From the last formula it follows that if p+|γ  | is even, then

res (Pγλ,+ , ϕ) = 0. | λ=− n+|γ 2  + |γ |∈N, p + |γ  | = 2k − 1, k∈N,

Now let p + |γ  | not be a natural number or p and q+|γ  | is even. We have   ψ1 (0, 0) = ψ(0, 0) = ϕ(0) ωγ dSp dSq = ϕ(0)|S1+ (p)|γ  |S1+ (q)|γ  , Sp+ Sq+

(4.50)

Weighted generalized functions generated by quadratic forms

183

where p 

|S1+ (p)|γ  =

 

i=1

2p−1 



γi +1 2



p+|γ  | 2

q 

|S1+ (q)|γ  =

,



γi +1 2



i=1

2q−1 





q+|γ  | 2

.

(4.51)

After a simple calculation, we get

res

| λ=− n+|γ 2

(Pγλ,+ , ϕ)γ =

q+|γ  | 2

(−1) 2n

   γi 2+1 i=1   ϕ(0). |  n+|γ 2 n 

Besides,

res

| λ=− n+|γ 2

Pγλ,+ =

q+|γ  | 2

(−1) 2n

   γi 2+1 i=1   δγ (x). |  n+|γ 2 n 

(4.52)

Using the Green theorem and formula (4.48) we obtain    ϕ(x)[γ P λ+1 (x)] − P λ+1 (x)[γ ϕ(x)] x γ dx = 0, {P (x)>0}+

so (Pγλ,+ , ϕ)γ =

1 (P λ+1 , γ ϕ)γ . 2(λ + 1)(2λ + n + |γ |) γ ,+

(4.53)

Applying formula (4.53) k times gives (Pγλ,+ , ϕ)γ =

k (Pγλ+k ,+ , γ ϕ)γ  .   | n+|γ | 22k (λ + 1)...(λ + k) λ+ n+|γ +k−1 ... λ+ 2 2

Consequently, res

| λ=− n+|γ 2 −k

×

(Pγλ,+ , ϕ)γ =

res

| λ=− n+|γ 2 −k



1

k (Pγλ+k ,+ , γ ϕ)γ





| | 22k (λ+1)...(λ+k) λ+ n+|γ ... λ+ n+|γ 2 2 +k−1



and res

| λ=− n+|γ 2 −k

k (Pγλ+k ,+ , γ ϕ)γ =

res

| λ=− n+|γ 2

(Pγλ,+ , kγ ϕ)γ .

| λ=− n+|γ 2 −k

(4.54)

184

Transmutations, Singular and Fractional Differential Equations

Therefore, if p + |γ  | is even, then residues vanish. If p + |γ  | is not natural or p + |γ  |∈N, p + |γ  | = 2k − 1 and k∈N, then (4.52) gives

res

| λ=− n+|γ 2 −k

(Pγλ,+ , ϕ)γ =



n 

q+|γ  | 2



γi +1 2



(−1) i=1   (kγ δγ (x), ϕ)γ . 2n+2k k!  n+|γ | + k 2

The proof is complete. Theorem 40. Let p ≥ 1, q ≥ 1. If n+|γ | is even, p + |γ  | and q + |γ  | are even, | k∈N∪{0}, then Pγλ,+ has simple poles at λ = − n+|γ 2 −k with residues res

| λ=− n+|γ 2 −k

Pγλ,+



= 

n+|γ | 2





1 +k

q+|γ  |

(−1) 2 + 22k k!

 (−1)

n+|γ | 2 +k−1



n+|γ | 2 +k−1

δγ ,1

(P )



n  γi + 1 k  γ δγ (x) . 2 i=1

If p+|γ  | and q+|γ  | are not natural or p+|γ  |, q+|γ  |∈N, and p+|γ  | = 2m − 1, q+|γ  | = 2k − 1, m, k∈N, then Pγλ,+ has a second order pole at points | λ λ = − n+|γ 2 −k. Coefficients c−2 and c−1 of decomposition Pγ ,+ in the Laurent se(k)

(k)

| ries in the neighborhood of points λ = − n+|γ 2 − k are (0) c−1

=

1





  (−1)

n+|γ | 2 +k−1



n+|γ | 2 +k−1

δγ ,1

n+|γ |

(−1) 2 (P ) + 22k k!

−1

|  n+|γ 2 +k 







n  γi +1 p+|γ  | p+|γ  | n+|γ | ×  sin π ψ −ψ kγ δγ (x) , 2 2 2 2 i=1

(k)

c−2 = (−1) where ψ(x) =

n+|γ | 2 +1

sin π(p+|γ 2

n 

 |)

 

γi +1 2



i=1   n+|γ |+k 2

2n+2k k!π

kγ δγ (x),

  (x) (x) .

| λ Proof. Let n+|γ | be even and λ = − n+|γ 2 −k, k∈N∪{0}. We first write (Pγ ,+ , ϕ)γ in the form

(Pγλ,+ , ϕ)γ

1 = λ+k

∞

| λ+ n+|γ 2 −1

u 0

∞ uλ+

0 (u)du+ 0

n+|γ | 2 −1

1 (λ, u)du, (4.55)

Weighted generalized functions generated by quadratic forms

where 0 (u) =

res

| λ=− n+|γ 2 −k

185

| (λ, u) and 1 (λ, u) is regular at λ = − n+|γ 2 −k. Each

| λ integral in (4.55) can have a simple pole at λ = − n+|γ 2 −k, so (Pγ ,+ , ϕ)γ can have

| a second order pole at λ = − n+|γ 2 −k. In the neighborhood of such a point we may λ expand Pγ ,+ in the Laurent series (k)

Pγλ,+ =  λ+

(k)

c−2 n+|γ | 2

+k

c−1

2 +

(k)

λ+

n+|γ | 2

+k

+ ... .

(k)

Let us find coefficients c−1 and c−2 . We have (k) (c−2 , ϕ)γ

∞ =

uλ+

res

| λ=− n+|γ 2 −k

n+|γ | 2 −1

0 (u)du =

1 (k)  (0). k! 0

0

(0)

If k = 0, then c−2 = 0 (0). In accordance with (4.38) we get 1 0 (0) = ψ1 (0, 0) res 4 λ=− n+|γ | 2

= ψ1 (0, 0)

1 (1 − t)λ t 0



q+|γ  | 2

q+|γ  |−2 2

dt



(λ + 1) . q+|γ  | | λ=− n+|γ + 1 2 4 λ + 2 



res

Taking into account that ψ1 (0, 0) = ϕ(0)|S1+ (p)|γ  |S1+ (q)|γ  , where |S1+ (p)|γ  and |S1+ (q)|γ  are given by (4.51), we can write (0)

(c−2 , ϕ)γ = (−1)

n+|γ | 2 +1

 B

p+|γ  | q+|γ  | 2 , 2

 sin

4π

π(p + |γ  |) + |S1 (p)|γ  |S1+ (q)|γ  ϕ(0). 2

+ |γ  |

Then p is even (in this case q + |γ  | is also even) and we have c−2 = 0. Thus |  (Pγλ,+ , ϕ)γ has just a simple pole at λ = − n+|γ 2 . If, on the other hand, p + |γ | is not   natural or p + |γ |∈N and p + |γ | = 2k − 1, k∈N, then

(0)

c−2 = (−1)

(k)

n+|γ | 2 +1

sin π(p+|γ 2

 |)

2n π

n 

i=1

 

n+|γ | 2

γi +1 2



 δγ (x).

Using the same way as in the proof of Theorem 39, if p + |γ  | and q + |γ  | are even, |   then Pγλ,+ has a simple pole at λ = − n+|γ 2 − k. If p + |γ | and q + |γ | are not natural

186

Transmutations, Singular and Fractional Differential Equations

or p+|γ  |, q+|γ  |∈N and p+|γ  | = 2m−1, q+|γ  | = 2k−1, m, k∈N, then   n  |)  sin π(p+|γ  γi 2+1 2 n+|γ | i=1 (k)   kγ δγ (x). c−2 = (−1) 2 +1 n+|γ |+k n+2k 2 k!π 2 (k)

Let us find now c−1 . We have ∞

(k)

(c−1 , ϕ) =

u−k−1 0 (u)du +

0

∞ ×

| λ+ n+|γ 2 −1

u

res

| λ=− n+|γ 2 −k

n + |γ | 1 − − k, u du. 2

0

Since 0 (u) = res (λ, u), using formulas (4.40) and (4.47) we obtain λ=−k

∞

(−1) 

u−k−1 0 (u)du =

n+|γ | 2



0

n+|γ | ( +k−1)  δγ ,12 (P ), ϕ .

n+|γ | 2 +k−1

+k−1

γ

Therefore ∞ uλ+

res

| λ=− n+|γ 2 −k

=

1 k!

0

∂ k 1

n + |γ | 1 − − k, u du 2  − k, u = (αγ(k) , ϕ)γ

n+|γ | 2 −1

 | − n+|γ 2 ∂uk

u=0

and (−1) 

(k)

c−1 =



n+|γ | 2 +k−1

n+|γ | 2

( n+|γ | +k−1)

+k−1

 δγ ,12

(P ) + αγ(k) .

For k = 0 we get

n + |γ | = 1 − ,0 . 2   | To find 1 − n+|γ 2 , 0 consider (λ, 0). Using (4.49)–(4.51), we can write (αγ(0) , ϕ)γ

  n   γi 2+1 (λ + 1)  .  i=1  (λ, 0) = ϕ(0) p+|γ  | q+|γ  | n 2  + 1  λ + 2 2

Weighted generalized functions generated by quadratic forms

The formula (1 − x)(x) = 

(λ, 0) =

sin π λ +

π sin πx

q+|γ  | 2

187

gives



sin πλ

    n  |  −λ − q+|γ  γi 2+1 2  i=1  ϕ(0). p+|γ  | (−λ)  2

If p+|γ  | and q+|γ  | are even, then    | sin π λ+ q+|γ 2

lim

| λ→− n+|γ 2

sin πλ

= (−1)

q+|γ  | 2

.

| and Since (λ, 0) is regular at λ = − n+|γ 2





n + |γ | n + |γ | ,0 =  − , 1 − 2 2 we have   γi +1  2 q+|γ  | i=1  ϕ(0).  (αγ(0) , ϕ)γ = (−1) 2 |  n+|γ 2 n 

If p+|γ  | and q+|γ  | are not natural or p+|γ  |, q+|γ  |∈N and p+|γ  | = 2m−1, | q+|γ  | = 2k−1, m, k∈N, then (λ, 0) has a pole at λ = − n+|γ 2 . In this case



n  n+|γ | n + |γ | γi + 1 , 0 = (−1) 2 −1 (αγ(0) , ϕ)γ = 1 −  2 2 i=1        | | | sin p+|γ − ψ n+|γ ψ p+|γ 2 π 2 2   × ϕ(0), |  n+|γ 2 where ψ(x) = (0) c−1

  (x) (x) .



= 

From this we obtain 

1 n+|γ | 2

 (−1)

 n+|γ | 2 −1

where θ = (−1)

q+|γ  | 2

n  γi + 1  2 i=1



n+|γ | 2 −1

δγ ,1

 (P ) + θ δγ (x) ,

188

Transmutations, Singular and Fractional Differential Equations

if p+|γ  | and q+|γ  | are even. If p+|γ  | and q+|γ  | are not natural or p+|γ  |, q+|γ  |∈N and p+|γ  | = 2m−1, q+|γ  | = 2k−1, m, k∈N, then



n  n+|γ | γi +1 p+|γ  |  sin π θ = (−1) 2 −1 2 2 i=1



 n + |γ | p + |γ | −ψ . × ψ 2 2 (k)

Finally, in order to obtain c−1 for any k ≥ 1, we use formula (4.54) again. This completes the proof. One can likewise obtain results for the weighted generalized function Pγλ,− of the form  λ (−P (x))λ ϕ(x)x γ dx, ϕ ∈ Sev (Pγ ,− , ϕ)γ = {P (x)0}+

and  (Aλγ ,− , ϕ)γ = {A(x)0}+ = {x∈Rn+ :A(x)>0}, {A(x) 0, λ Aγ ,+ = 0 A ≤ 0,  Aλγ ,− =

A ≥ 0, A < 0,

0 (−A)λ

where A is defined by (4.67). Then 

A ≥ 0, A < 0.

Aλ e±λπi |A|λ

(A ± i0)λγ = lim (A ± iε|x|)λγ = ε→0

Since the analytic continuation is unique, formulas (4.70) and (4.71) can be used for Re λ ≤ 0. For λ = −k, k ∈ N, the weighted generalized function Pγλ does not have poles and (A ± i0)λγ in this case is introduced by formulas (4.70) and (4.71). So we deduce that the weighted generalized functions (A ± i0)λγ are analytic in λ ∈ C every-

| where except at λ = − n+|γ 2 − k, k = 0, 1, 2, ..., where they have simple poles with residues

res

| λ=− n+|γ 2 −k

(A ± i0)λγ = lim

res

ε→0 λ=− n+|γ | −k

(A ± iε|x|)λγ .

2

Since the quadratic form A has in canonical representation p positive and q negative terms, from (4.63) and (4.64) we obtain lim

ε→0

n 

(ε − iak )−

1+γk 2

=

k=1

n 

|ak |−

1+γk 2

(−i)−

p+|γ  | 2

i−

q+|γ  | 2

k=1

=e

πi   4 (p+|γ |−q−|γ |)

n 

|ak |−

1+γk 2

,

k=1

lim

ε→0

n 

−

(ε + iak )

k=1

1+γk 2

=

n 

|ak |−

1+γk 2

i−

p+|γ  | 2

(−i)−

q+|γ  | 2

k=1

=e

πi   4 (−p−|γ |+q+|γ |)

n  k=1

|ak |−

1+γk 2

,

194

Transmutations, Singular and Fractional Differential Equations

where |γ  | = γ1 + ... + γp , |γ  | = γp+1 + ... + γp+q . Therefore, applying formulas (4.65) and (4.66) we get (4.68) and (4.69). 2 −...−x 2 , If in (4.67) all ak = 1, then the quadratic form is P = x12 +...+xp2 −xp+1 n n = p+q, and for this form the weighted generalized functions (P + i0)λγ and (P − i0)λγ for Re λ > 0 are defined by the formulas

(P + i0)λγ = lim (P + iP  )λγ , ε→0

(P − i0)λγ = lim (P − iP  )λγ , ε→0

where the limit can be taken under the integral sign in

 Rn+

P λ ϕx γ dx. For Re λ < 0,

| λ λ λ = −k, λ = − n+|γ 2 − k + 1, k ∈ N, expressions (P + i0)γ and (P − i0)γ are defined by analytical continuation and then the limit ε → 0 is taken. We have

(P + i0)λγ = Pγλ,+ + eπλi Pγλ,− , (P

= Pγλ,+

− i0)λγ

+e

−πλi

(4.72)

Pγλ,− .

(4.73)

From formulas (4.68) and (4.69) it follows that residues of (P + i0)λγ and (P − i0)λγ

| at λ = − n+|γ 2 − k, k = 0, 1, 2, ..., are weighted generalized functions concentrated on the vertex of the P (x) = 0 cone:   π(q+|γ  |) | e−i 2 |S1+ (n)|γ  n+|γ 2 kγ δγ (x) (P + i0)λγ = (4.74) res   n  | n+|γ | λ=− n+|γ −k k 2 4 k!  + k 2 k=1

and

res

| λ=− n+|γ 2 −k

(P − i0)λγ =

ei

  | |S1+ (n)|γ  n+|γ 2 kγ δγ (x).   n  n+|γ | k 4 k!  2 +k

π(q+|γ  |) 2

(4.75)

k=1

By (4.72) and (4.73), formulas λ P+,γ =−

  1 e−πλi (P + i0)λγ − eπλi (P − i0)λγ 2i sin λπ

(4.76)

and λ = P−,γ

follow.

  1 (P + i0)λγ − (P − i0)λγ 2i sin λπ

(4.77)

Weighted generalized functions generated by quadratic forms

4.3

195

Other weighted generalized functions associated with a quadratic form

λ and (P ± i0)λ are used to obtain a fundamental soluThe introduced functions P±,γ γ tion of the B-ultrahyperbolic equation and to construct hyperbolic Riesz B-potentials. However, these functions are not enough to obtain solutions to the Cauchy problem for the general Euler–Poisson–Darboux equation. In this section, we consider the functions that will be required to solve this problem.

4.3.1 Functions (w2 − |x|2 )λ+,γ and (c2 + P ± i0)λγ Here we consider the weighted generalized function associated with the positive definite quadratic form (w 2 − |x|2 )λ+,γ and the weighted generalized function associated with an indefinite quadratic form (c2 + P ± i0)λγ , where c and w do not depend on x ∈ Rn+ . Definition 33. Let x ∈ Rn+ and w does not depend on x. We define the weighted generalized function (w 2 −|x|2 )λ+,γ by the formula  ((w

2

− |x|2 )λ+,γ , ϕ)γ

=

(w 2 − |x|2 )λ ϕ(x)x γ dx,

ϕ ∈ Sev ,

λ ∈ C,

{|x| 0, where c does not depend on x. We define the weighted generalized functions (c2 + P ± iP  )λγ by ((c + P 2

+ iP  )λγ , ϕ(x))γ

 =

(c2 + P + iP  )ϕ(x)x γ dx

Rn+

and ((c + P 2

− iP  )λγ , ϕ(x))γ

 = Rn+

(c2 + P − iP  )ϕ(x)x γ dx.

196

Transmutations, Singular and Fractional Differential Equations

The weighted generalized functions (c2 + P + i0)λγ and (c2 + P − i0)λγ for Re λ > 0 are (c2 + P + i0)λγ = lim (c2 + P + iP  )λγ , ε→0

(c2 + P − i0)λγ = lim (c2 + P − iP  )λγ , ε→0

where the limit ε → 0 can be taken under the integral sign in

 Rn+

P λ ϕx γ dx.

4.3.2 General weighted generalized functions connected with quadratic form In this subsection we introduce the new family of weighted generalized functions 2 −...−x 2 , n = p + q. connected with the quadratic form P (x) = x12 +...+xp2 −xp+1 p+q Functions of this family are generalizations of studied earlier functions. Let P be a quadratic form with real coefficients, let P1 be a positive definite quadratic form, and P = P ± iP1 . Definition 35. Let ϕ ∈ Sev , and f (z, λ) is the entire function of z and λ. The weighted generalized functions Pγλ f (P, λ) are given by  (Pγλ f (P, λ), ϕ(x))γ = Pγλ · f (P, λ)ϕ(x)dx, Rn+

where λ ∈ C, Re λ > −1, and P is a complex quadratic form with positive definite imaginary part. For Re λ > −1 the function Pγλ f (P, λ) is analytic of λ. For other meanings of λ the weighted generalized function Pγλ f (P, λ) is defined as analytical continuation. From the decomposition of f (z, λ) to power series by z it follows that for the quadratic form with real coefficients P limits (P ± i0)λγ f (P ± i0, λ) = lim Pγλ f (P, λ), P1 →0

P = P ± iP1 ,

exist. From formulas (4.72) and (4.73) we get (P + i0)λγ f (P , λ) = Pγλ,+ f (P+ , λ) + eπλi Pγλ,− f (P− , λ),

(4.79)

(P − i0)λγ f (P , λ) = Pγλ,+ f (P+ , λ) + e−πλi Pγλ,− f (P− , λ).

(4.80)

The family of functions given by Definition 35 is quite wide. In particular weighted generalized functions generated by Bessel functions J n+|γ | +λ (P 1/2 ), (1) K n+|γ | +λ (P 1/2 ), H n+|γ | 2

this family.

2

2

(2)

+λ

(P 1/2 ), H n+|γ | 2

+λ

(P 1/2 ), and I n+|γ | +λ (P 1/2 ) are from 2

Weighted generalized functions generated by quadratic forms

4.4

197

Hankel transform of weighted generalized functions generated by the quadratic form

The purpose of this section is to prove the formulas for the multi-dimensional Hankel transform of weighted generalized functions generated by the definite and indefinite quadratic form considered earlier.

4.4.1 Hankel transform of rγλ In this subsection we present a Hankel transform of rγλ following [177,251,242]. In [251,242] an analogue of formula (4.83) was obtained in the case when instead of Hankel transform a mixed Fourier–Bessel transform is used and in the weighted functional (rγλ , ϕ)γ weight was taken only by one variable. Now, we perform the formula expressing Hankel transform of any radial function γ from L1 (Rn+ ). γ

Lemma 16. Let ϕ(s) be a function of one variable and ϕ(|x|)∈L1 (Rn+ ). The Hankel transform of the radial function ϕ(|x|) is the radial function and the following formula is valid: n 

Fγ [ϕ(|x|)](ξ ) =

 

i=1

2n−1 



γi +1 2



n+|γ | 2

∞



ϕ(r) j n+|γ |−2 (|ξ |r) r n+|γ |−1 dr.

(4.81)

2

0

Proof. In the Hankel transform of the radial function replacing jγ (x, ξ ) to Pξ [e−i x,ξ ] by formula (3.138) and going over the spherical coordinates x = rσ we get  Fγ [ϕ(|x|)](ξ ) = ϕ(|x|)jγ (x, ξ ) x γ dx γ

Rn+

∞ =

ϕ(r)r

n+|γ |−1

 dr

Pξ [e−ir σ,ξ ] σ γ dS. γ

S1+ (n)

0

Using formulas (3.143) and (3.141) we can write  γ Pξ [e−ir σ,ξ ] σ γ dS S1+ (n) n 

=√

 

i=1

π2n−1 

γi +1 2





1 

n+|γ |−1 2 −1

e−i|ξ |rp (1 − p 2 )

n+|γ |−3 2

dp

(4.82)

198

Transmutations, Singular and Fractional Differential Equations n 

=√

 

i=1

π2n−1 

γi +1 2





n+|γ |−1 2



√

π2

n+|γ |−2 2



n + |γ | − 1  2



2−n−|γ |

× (|ξ |r) 2 J n+|γ |−2 (|ξ |r) 2   n  γi +1  2 i=1   j n+|γ |−2 (|ξ |r). = | 2 2n−1  n+|γ 2 Returning to (4.82) we get n 

 

i=1

Fγ [ϕ(|x|)](ξ ) =

2n−1 



γi +1 2



n+|γ | 2

∞



ϕ(r) j n+|γ |−2 (|ξ |r) r n+|γ |−1 dr. 2

0

Theorem 42. The Hankel transform of rγλ in the sense of (1.83) is ⎧ −n−|γ |−λ ⎪ ⎨|ξ | λ Fγ [rγ ](ξ ) = Dn,γ (λ) (−γ )λ/2 ξ ϕ)δγ ⎪ ⎩ −n−|γ |−λ ln |ξ | |ξ |

λ = 2k, λ = −(n + |γ | + 2k), λ = 2k, λ = −(n + |γ | + 2k), (4.83)

where δγ = δγ (ξ ) is the weighted delta-function, k = 0, 1, 2, ..., and ⎧     n  γ +1 2|γ |+λ  i 2  n+|γ2 |+λ ⎪ ⎪ ⎪ i=1 ⎪   ⎪ ⎪ ⎨  −λ

λ = 2k, λ = −(n + |γ | + 2k),

2

Dn,γ (λ) = 1 ⎪ ⎪ n+|γ |+λ ⎪   n ⎪ |γ |+λ+1 2 ⎪ γ +1 (−1) ⎪  2   ⎩  i2 n+|γ |+λ λ −

i=1

2

! −2

λ = 2k, λ = −(n + |γ | + 2k).

Proof. Let Re λ > −(n + |γ |). Then rγλ is a locally summable function. Assuming that −(n + |γ |) < Re λ < −(n + |γ |)/2, in this case formula (4.81) is valid and n 

Fγ [rγλ ](ξ ) =

=

 

i=1

2n−1  2 |ξ |



γi +1 2



n+|γ | 2

|γ |−n 2 n+|γ |−2 2

∞



j n+|γ |−2 (|ξ |r) r n+|γ |−1+λ dr 2

0



∞ n  n+|γ | γi + 1  r 2 +λ J n+|γ |−2 (|ξ |r) dr. 2 2 i=1

0

Weighted generalized functions generated by quadratic forms

199

To calculate the integral, using the formula for the Weber type integral, we get   n+|γ | n+|γ |+λ ∞ n+|γ | 2 +λ  2 2 +λ rγ 2 J n+|γ |−2 (|ξ |r) dr = .  n+|γ | 2 2 +λ+1  − λ |ξ | 2 0 Then, we need to show that the Hankel transform (Fγ [rγλ ](ξ ), ϕ)γ according to (1.83) is equal to

1 Gn,γ (λ) , ϕ = (rγλ , Fγ ϕ)γ , (4.84) |ξ |n+|γ |+λ γ 2|γ |+λ

where Gn,γ (λ) =

n 

 

i=1

γi +1 2

    n+|γ2 |+λ  ,

  − λ2

λ = 2k, λ = −(n + |γ | + 2k), k =

0, 1, 2, .... The right side of equality (4.84) is defined and analytic for all λ ∈ C, since Fγ ϕ ∈ Sev . The left part is analytic for all λ ∈ C except for λ = 2k and λ = −(n + |γ | + 2k), k = 0, 1, 2, .... In the case λ = 2k the Hankel transform of rγλ has the form 

 (Fγ [rγ2k ], ϕ)γ =

|x|2k

Rn+

jγ (x, ξ )ϕ(ξ ) ξ γ dξ x γ dx.

Rn+

Note that if |x|2k jγ (x, ξ ) = (−γ )kξ jγ (x, ξ ), then we have (Fγ [rγ2k ], ϕ)γ

  =

(−γ )kξ jγ (x, ξ )ϕ(ξ ) ξ γ dξ x γ dx

Rn+ Rn+



=

Fγ [1](ξ )(−γ )kξ ϕ(ξ )) ξ γ dξ = (δγ , (−γ )kξ ϕ))γ .

Rn+

Consider the case λ = λk = −(n + |γ | + 2k). We can write equality (4.84) for λ in the neighborhood of λk in the following way:

1 , ϕ , (λ − λk )(rγλ , Fγ ϕ)γ = a(λ) |ξ |n+|γ |+λ γ where a(λ) = (λ − λk )Gn,γ (λ). Differentiating this equality by λ we obtain 

d a (λ) + a(λ) ln |ξ | λ ,ϕ . ((λ − λk )rγ , Fγ ϕ)γ = dλ |ξ |n+|γ |+λ γ

(4.85)

200

Transmutations, Singular and Fractional Differential Equations

Then from (4.85) for λ → λk it follows that

ln |ξ | , ϕ . (rγλk , Fγ ϕ)γ = a(λk ) |ξ |N+|γ |+λk γ We calculate the constant a(λk ), λk = −(n + |γ | + 2k): a(λk ) = lim a(λ) = λ→λk



n  γi + 1 (−1)k  ,  2 22k+n−1 k! n+|γ |+2k i=1

2

and since λ = λk = −(n + |γ | + 2k), k = − n+|γ2 |+λ and n+|γ |+λ

n  γi + 1 (−1) 2 2|γ |+λ+1    .  a(λk ) = 2 − n+|γ |+λ !  − λ

i=1

2

2

The proof is complete.

4.4.2 Hankel transforms of functions Pγλ , (P ± i0)λγ , and Pγλ,± In this subsection to find the multi-dimensional Hankel transform of weighted generalized functions related to the indefinite quadratic form, we will use formula (4.83) when λ = −(n + |γ | + 2k), λ = 2k, k = 0, 1, 2, ...: 2|γ |+λ

n 

 

i=1

Fγ [r λ ](ξ ) =

    n+|γ2 |+λ |ξ |−n−|γ |−λ .    − λ2 γi +1 2

Theorem 43. The Hankel transform of Pγλ for λ = k, λ = − 0, 1, 2, ..., is Fγ [Pγλ ](ξ ) =



n+|γ | 2

 +k , k =

   |

− n+|γ | −λ  n+|γ 2 2 +λ 2 2 ξ ξ i=1 1   + ... + n . α1 αn  (−λ) (−iα1 )1+γ1 ... (−iαn )1+γn (4.87)

22λ+|γ | e−

n+|γ | 4 iπ

n 



(4.86)



γi +1 2

Proof. If the weighted generalized function Pγλ is an analytic function of α1 , ..., αn in the region Im αk > 0, k = 1, 2, ..., n, then the Hankel transform of Pγλ is also an analytic function in the same region. Therefore in order to find Fγ [Pγλ ] we need only treat the case in which all αk are imaginary and then analytically continue the found Hankel transform to the whole complex plane. Putting αk = ibk , bk > 0, k =

Weighted generalized functions generated by quadratic forms

201

1, 2, ..., n, we obtain π



Fγ [Pγλ ](ξ ) = e 2 λi

(b1 x12 + ... + bn xn2 )λ j(x, ξ )x γ dx. Rn+

Change of variables by xi = π − Fγ [Pγλ ](ξ ) = e 2 λi b1

√yi , bi 1+γ1 2

i = 1, ..., n, transforms this to the form 

− 1+γn ....bn 2



y r j √ , ξ y γ dy. b 2λ

Rn+

The Fourier transform of the weighted generalized function rγ2λ has already been calculated in Section 4.4.1. Using (4.86) we have    |  n+|γ 2 +λ π i=1  Fγ [Pγλ ](ξ ) = e 2 λi  1+γ 1+γ  (−λ) b1 1 ... bn n

−n−|γ |−2λ ξ12 ξn2 × + ... + b1 bn     n  |  γi 2+1  n+|γ + λ 2 π i=1   = 22λ+|γ | e 2 λi  (−λ) (−iα1 )1+γ1 ... (−iαn )1+γn

 −n−|γ | −λ 2 ξ12 ξn2 × + ... + . −iα1 −iαn 22λ+|γ |

n 





γi +1 2

Taking the factor −i out of the bracket

ξ12 −iα1

+ ... +

ξn2 −iαn

−n−|γ | −λ 2

we get (4.87).

Now the uniqueness of analytic continuation implies that (4.87) remains valid also for positive definite. The square roots  any quadratic  form whose imaginary part is √ 1 1 1+γ 1+γ n 1 (−iα1 ) ... (−iαn ) are calculated by z = |z| 2 e 2 i arg z .   | Theorem 44. The Hankel transforms of (P ± i0)λγ for λ = k, λ = − n+|γ 2 +k , k = 0, 1, 2, ..., are Fγ [(P + i0)λγ ](ξ ) = e− Fγ [(P − i0)λγ ](ξ ) = e

q+|γ  | iπ 2

q+|γ  | iπ 2

βn,γ (λ)(Q − i0)−

βn,γ (λ)(Q + i0)−

where 2 2 − ... − ξp+q , Q = ξ12 + ... + ξp2 − ξp+1

n+|γ | 2 −λ

n+|γ | 2 −λ

,

,

(4.88) (4.89)

202

Transmutations, Singular and Fractional Differential Equations n 

βn,γ (λ) = 22λ+|γ |

i=1

 

γi +1 2

   | + λ  n+|γ 2 .

 (−λ)

Proof. Let in formula (4.87) αk = ak + ibk , k = 1, ..., n. Then    | + λ  n+|γ 2 n+|γ | i=1   Fγ [Pγλ ](ξ ) = 22λ+|γ | e− 4 iπ  (−λ) (b1 − ia1 )1+γ1 ... (bn − iak )1+γn

− n+|γ | −λ 2 ξ12 ξn2 × + ... + . (4.90) a1 + ib1 an + ibn n 





γi +1 2

Putting a1 = 1,...,ap = 1, ap+1 = −1, ...,ap+q = −1 in (4.90) and tending to limits b1 →0,...,bn →0, we obtain Fγ [(P + i0)λγ ](ξ )

   | +λ  n+|γ 2 n+|γ | i=1  = 22λ+|γ | e− 4 iπ   √  (−λ) (−i)1+γ1 ... (−i)1+γp i 1+γp+1 i 1+γn |

− n+|γ 2 2 −λ ξp+1 ξp2 ξ12 ξn2 × + ... + + + ... + 1 + i0 1 + i0 −1 + i0 −1 + i0     n  |  γi 2+1  n+|γ 2 +λ q+|γ  | i=1 2λ+|γ | iπ =2 e 2  (−λ)  2 2 2 (−1 − i0) + ... × ξ1 (1 − i0) + ... + ξp (1 − i0) + ξp+1 n 





γi +1 2

− n+|γ | −λ 2 + ξn2 (−1 − i0)     n  |  γi 2+1  n+|γ +λ 2  q+|γ | n+|γ | i=1 = 22λ+|γ | e− 2 iπ (Q − i0)− 2 −λ .  (−λ) That gives (4.88). Formula (4.89) is obtained similarly. Further for formulas (4.88) and (4.89) we will use a short notation: Fγ [(P ± i0)λγ ] = e−

q+|γ  | iπ 2

βn,γ (λ)(P ∓ i0)−

n+|γ | 2 −λ

.

(4.91)

Weighted generalized functions generated by quadratic forms

Theorem 45. The Hankel transforms of Pγλ,± for λ = k, λ = − 0, 1, 2, ..., are Fγ [P+λ ] =

203



n+|γ | 2 +k

 , k=





n 22λ+|γ |−1  γi + 1 n + |γ |   + λ  (1 + λ) iπ 2 2 i=1

 

 |   |  | n+|γ | −iπ λ+ q+|γ iπ λ+ q+|γ − n+|γ −λ 2 2 × e (Q − i0) 2 −e (Q + i0)− 2 −λ

(4.92) and



n 22λ+|γ |−1  γi + 1 n + |γ |   + λ  (λ + 1) iπ 2 2 i=1

q+|γ  | n+|γ | q+|γ  | n+|γ | × e− 2 iπ (Q − i0)− 2 −λ − e 2 iπ (Q + i0)− 2 −λ ,

Fγ [P−λ ] = −

(4.93)

where 2 2 − ... − ξp+q . Q = ξ12 + ... + ξp2 − ξp+1

Proof. Using formulas (4.76), (4.77), (4.88), (4.89), and (z)(1 − z) =

π , sin πz

after some elementary operations we obtain (4.92) and (4.93).

4.4.3 Hankel transforms of functions (w2 − |x|2 )λ+,γ and (c2 + P ± i0)λγ Here we obtain the formulas of the Hankel transform of some functions from Section 4.3.1. Theorem 46. The following formula holds:  (Fγ )x

(w 2

− |x|2 )λ+,γ

(λ + 1)



  n   γi 2+1 w n+|γ |+2λ  i=1  j n+|γ | +λ (w|x|), (ξ ) = | 2 n 2  n+|γ + λ + 1 2 (4.94)

k−n−|γ |−1

where (w 2 − |x|2 )+,γ 2

is defined by formula (4.78), w > 0.

204

Transmutations, Singular and Fractional Differential Equations

Proof. Let first Re λ>−1. We perform the integration in Fγ (w 2 −|x|2 )λ+,γ by going to spherical coordinates and applying formula (3.140):  2 2 λ (Fγ )x (w − |x| )+,γ = jγ (x, ξ )(w 2 − |x|2 )λ x γ dx = {x = rθ, r = |x|} + Bw (n)

w  2 2 λ n+|γ |−1 dr = (w − r ) r S1+ (n)

0 n 

=

 

i=1

2n−1 

= |x|1−



γi +1 2



n+|γ | 2

n+|γ | 2

2

jγ (rθ, x)θ γ dS

w  (w 2 − r 2 )λ j n+|γ | −1 (r|x|)r n+|γ |−1 dr

|γ |−n 2

2

0



w n  n+|γ | γi + 1  (w 2 − r 2 )λ J n+|γ | −1 (r|x|)r 2 dr. 2 2 i=1

0

Using formula (2.12.4.6) from [456] of the form w r ν+1 (w 2 − r 2 )β−1 Jν (μr)dr =

2β−1 w β+ν (β) Jβ+ν (μw), μβ

(4.95)

0

w > 0,

Re β > 0,

Re ν > −1,

we obtain n+|γ | w n+|γ | 2λ w 2 +λ (λ + 1) 2 2 λ 2 (w − r ) J n+|γ | −1 (r|x|)r dr = J n+|γ | +λ (|x|ω) 2 2 |x|λ+1

0

for Re λ > −1 and

(Fγ )x (w 2 − |x|2 )λ+,γ

  n   γi 2+1 w n+|γ |+2λ (λ + 1) i=1   j n+|γ | +λ (w|x|), = n+|γ | 2 2n  2 +λ+1

which coincides with (4.94). So we get (4.94) for Re λ > −1. For other values of λ such that λ = −1, −2, −3, ..., equality (4.94) remains valid by analytic continuation in λ. (w 2 −|x|2 )λ Residues of (λ+1)+,γ at λ = −m, m ∈ N , have forms (see Section 4.2.1) lim

λ→−m

(w 2 − |x|2 )λ+,γ (λ + 1)

= δγ(m−1) (w 2 − |x|2 ).

Weighted generalized functions generated by quadratic forms

205

Then for λ = −m we get  (Fγ )x

(w 2 − |x|2 )λ+,γ



(λ + 1)

 jγ (x, ξ )δγ(m−1) (w 2 − |x|2 )x γ dx

(ξ ) = Rn+

  n   γi 2+1 w n+|γ |−2m i=1   j n+|γ | −m (w|x|). = n+|γ | 2 n 2  − m + 1 2 The proof is complete.

Theorem 47. The following formulas hold:

2

|γ |−n 2 +λ+1

Fγ (w 2 + P + i0)λγ = ⎡

1

e− 2 qπi w

n+|γ | 2 +λ

√ (−λ) || 1

1 2

(wQγ ,+ ) iπ | ⎢ K n+|γ 2 +λ   ×⎢ + ⎣ 1 n+|γ | 2 2 2 +λ Qγ ,+

n 

 

i=1

γi +1 2



⎤

(1) (wQγ2 ,− ) ⎥ | − n+|γ 2 −λ ⎥   ⎦ 1 n+|γ | 2 2 +λ Qγ ,−

H

(4.96)

and 2

|γ |−n 2 +λ+1

Fγ (w 2 + P − i0)λγ = ⎡ 1 2

(wQγ ,+ ) iπ | ⎢ K n+|γ 2 +λ   ×⎢ − ⎣ n+|γ | 1 2 2 2 +λ Qγ ,+ where Q =

n  i=1

1 2 ai ξi

1

e 2 qπi w

n+|γ | 2 +λ

√ (−λ) ||

n 

 

i=1

1



⎤

(2) (wQγ2 ,− ) ⎥ | − n+|γ 2 −λ ⎥,   ⎦ 1 n+|γ | 2 2 +λ Qγ ,−

H

is a quadratic form dual to P =

n  i=1

(1)

γi +1 2

(4.97)

ai xi2 ,  is the coefficient

(2)

matrix determinant P , Hα and Hα are Hankel functions of the first and second kind (1.14) and (1.15), respectively, and Kα is a modified Bessel function (1.17). Proof. We first consider the Hankel transform of the weighted generalized function | (w 2 + P )λγ , where P = |x|2 is a positive definite quadratic form and Re λ < − n+|γ 2 .

206

Transmutations, Singular and Fractional Differential Equations

Applying (3.140) we obtain  Fγ [(w

2

+ P )λγ ](ξ ) =

jγ (x, ξ )(c2 + |x|2 )λ x γ dx = {x = rθ, r = |x|}

Rn+

∞  2 2 λ n+|γ |−1 = (w + r ) r dr S1+ (n)

0 n 

=

 

i=1

2n−1 



γi +1 2

| 1− n+|γ 2

= |ξ |



n+|γ | 2

2

jγ (rθ, x)θ γ dS

∞  (w 2 + r 2 )λ j n+|γ |−2 (r|ξ |)r n+|γ |−1 dr 2

0

|γ |−n 2



∞ n  n+|γ | γi + 1  (w 2 + r 2 )λ J n+|γ | −1 (r|ξ |)r 2 dr. 2 2 i=1

0

Using formula (2.12.4.28) from [456], ∞

cρ−1 zν−ρ+1 Kν−ρ+1 (cz), 2ρ−1 (ρ)

x ν+1 (x 2 + z2 )−ρ Jν (cx)dx =

0

we get 2

|γ |−n 2 +λ+1

w

n+|γ | 2 +λ

n 

 

i=1

Fγ [(w 2 + P )λγ ](ξ ) =

|ξ |

n+|γ | 2 +λ

γi +1 2

 K n+|γ | +λ (w|ξ |), 2

(−λ)

(4.98) | . For other values of λ the Hankel transform Fγ (w 2 + P )λγ remains where λ < 1−n−|γ 4 valid by analytic continuation in λ. Now let P be any real quadratic form. Let us consider the weighted generalized functions (w 2 + P + i0)λγ and (w 2 + P − i0)λγ . In accordance with the uniqueness of the analytic continuation, (4.98) gives

Fγ [(w 2 + P ± i0)λγ ](ξ ) 2

|γ |−n 2 +λ+1

1

e∓ 2 qπ w

= (Q ∓ i0) where Q =

n  i=1

1 2 ai ξi

1 2



n+|γ | 2 +λ

 n+|γ | 2 +λ

n  i=1

 

γi +1 2

√ (−λ) ||

 1

K n+|γ | +λ (w(Q ∓ i0)γ2 ),

is a quadratic form dual to P =

(4.99)

2

n  i=1

ai xi2 and  is the coefficient

matrix determinant of P . Taking into account the definitions of the modified Bessel

Weighted generalized functions generated by quadratic forms

207

function of the first and second kind (1.16) and (1.17), we obtain 1

K n+|γ | +λ (w(Q ∓ i0)γ2 ) 2

(Q ∓ i0) 

= 2 sin ⎛ ×⎝

1 2





n+|γ | 2 +λ

π n+|γ | 2

1

1

(w(Q ∓ i0)γ2 ) − I n+|γ | +λ (w(Q ∓ i0)γ2 ) | π I− n+|γ 2 −λ 2   =    1 n+|γ | 2 n+|γ | 2 2 +λ sin 2 + λ π (Q ∓ i0)

∞  1   + λ π m=0 m! n+|γ |

w 2m− 2  n+|γ | 22m− 2 −λ  m − n+|γ |

w 2m+ 2  − n+|γ | 22m+ 2 +λ  m +

−λ

| m−λ− n+|γ 2

n+|γ | 2

 (Q ∓ i0)γ

−λ+1

+λ

n+|γ | 2

+λ+1

⎞

⎠.  (Q ∓ i0)m γ

The weighted generalized functions (Q + i0)λγ and (Q − i0)λγ are expressed through Qλγ ,+ and Qλγ ,− by formulas ∓πμi (Q ∓ i0)μ Qγ ,− . γ = Qγ ,+ + e μ

μ

So 1

K n+|γ | +λ (w(Q ∓ i0)γ2 ) 2

(Q ∓ i0) 

= 2 sin ⎛ ×⎝

1 2





n+|γ | 2 +λ

π n+|γ | 2

∞  1   + λ π m=0 m! n+|γ |

w 2m− 2  n+|γ | 22m− 2 −λ  m −

−λ

n+|γ | 2



−λ+1

 

| | | m−λ− n+|γ m−λ− n+|γ ∓πi m−λ− n+|γ 2 2 2 × Qγ ,+ +e Qγ ,− n+|γ |

w 2m+ 2  − n+|γ | 22m+ 2 +λ  m + 1

+λ

n+|γ | 2

+λ+1

⎞   ∓πim m  Qm Qγ ,− ⎠ γ ,+ + e 1

2 2 | (wQγ ,+ ) − I | (wQγ ,+ ) π I−λ− n+|γ λ+ n+|γ 2 2 =    1 n+|γ | 2 | sin n+|γ + λ π (Qγ2 ,+ )λ+ 2 2

208

Transmutations, Singular and Fractional Differential Equations   ± λ+ n+|γ | π

1

1

2 2 J−λ− n+|γ | (wQγ2 ,− ) | (wQγ ,− ) − e iπ Jλ+ n+|γ 2 2 + .     1 | 2 n+|γ | λ+ n+|γ 2 2 i sin − λ + 2 π (Qγ ,− )

Noting that Kα (x) =

π I−α (x) − Iα (x) , 2 sin(απ)

Hα(1) (x) =

J−α (x) − e−απi Jα (x) , i sin(απ)

Hα(2) (x) =

J−α (x) − eαπi Jα (x) , −i sin(απ)

we get 1

K n+|γ | +λ (w(Q + i0)γ2 ) 2

(Q ∓ i0)

1 2





n+|γ | 2 +λ

=

2

1 2

| λ+ n+|γ 2

1

(Q ∓ i0)

1 2



+

(Qγ ,+ )

K n+|γ | +λ (w(Q − i0)γ2 ) 2

1

1

Kλ+ n+|γ | (wQγ2 ,+ )



n+|γ | 2 +λ

(1) 2 | (wQγ ,− ) −λ− n+|γ 2

H

iπ 2

1

(Qγ2 ,− )λ+

Kλ+ n+|γ | (wQγ2 ,+ ) 2

1 2

(Qγ ,+ )λ+

,

1

1

=

n+|γ | 2

n+|γ | 2

(2) 2 | (wQγ ,− ) −λ− n+|γ 2

H

iπ − 2

1

(Qγ2 ,− )λ+

n+|γ | 2

.

Considering (4.99) finally we obtain (4.96) and (4.97).

Corollary 4. If P is a positive definite quadratic form, then

2

|γ |−n 2 +λ+1

Fγ (w 2 + P + i0)λγ =

1

e− 2 qπi w

n+|γ | 2 +λ





n 



i=1

γi +1 2

 1

K n+|γ | +λ (wQγ2 ,+ )

 1 n+|γ | √ 2 2 +λ (−λ) ||Qγ ,+

2

(4.100) and 2 Fγ (w 2 + P − i0)λγ =

|γ |−n 2 +λ+1

1

e 2 qπi w

n+|γ | 2 +λ



n  i=1

 

 1 n+|γ | √ 2 2 +λ (−λ) ||Qγ ,+

γi +1 2

 1

K n+|γ | +λ (wQγ2 ,+ ). 2

(4.101)

Weighted generalized functions generated by quadratic forms

209

If P is a negative definite quadratic form, then Fγ (w 2 + P + i0)λγ iπ2

|γ |−n 2 +λ

=

1

e− 2 qπi w

n+|γ | 2 +λ

n 

 

i=1



γi +1 2

 1

(1) (wQγ2 ,− ) | − n+|γ 2 −λ

 1 n+|γ | √ 2 2 +λ (−λ) ||Qγ ,−

H

(4.102)

and Fγ (w 2 + P − i0)λγ iπ2

|γ |−n 2 +λ

=−

where Q =

n  i=1

1

e 2 qπi w

n 

n+|γ | 2 +λ



 

i=1

1 2 ai ξi



n+|γ | 2 +λ

√ (−λ) ||Qγ ,− 1 2

is dual to the P =

n  i=1

γi +1 2

 1

(2) (wQγ2 ,− ), | − n+|γ 2 −λ

H

(1)

ai xi2 quadratic form, Hα

(4.103)

(2)

and Hα

are

Hankel functions of the first and second kinds (1.14) and (1.15), respectively, and Kα is a modified Bessel function (1.17).

Buschman–Erdélyi integral and transmutation operators

5

The term “Buschman–Erdélyi transmutations” was introduced by S. M. Sitnik and is now widely accepted. Integral equations with these operators were studied in the mid-1950s. S. M. Sitnik was the first to prove the transmutational nature of these operators. The classical Sonine and Poisson operators are special cases of the Buschman– Erdélyi transmutations, and Sonine–Dimovski and Poisson–Dimovski transmutations are their generalizations for hyper-Bessel equations and functions. The Buschman–Erdélyi transmutations have many modifications. S. M. Sitnik introduced a convenient classification of them. Due to this classification we introduce Buschman–Erdélyi transmutations of the first kind; their kernels are expressed in terms of Legendre functions of the first kind. In the limiting case we define Buschman– Erdélyi transmutations of zero order smoothness being important in applications. The kernels of Buschman–Erdélyi transmutations of the second kind are expressed in terms of Legendre functions of the second kind. Some combination of operators of the first kind and the second kind leads to operators of the third kind. For the special choice of parameters they are unitary operators in the standard Lebesgue space. S. M. Sitnik proposed the terms “Sonine–Katrakhov” and “Poisson–Katrakhov” transmutations in honor of V. Katrakhov, who introduced and studied these operators. The study of integral equations and invertibility for the Buschman–Erdélyi operators was started in the 1960s by P. Buschman and A. Erdélyi. These operators were also investigated by Higgins, Ta Li, Love, Habibullah, K. N. Srivastava, Ding Hoang An, Smirnov, Virchenko, Fedotova, Kilbas, Skoromnik, and others. During this period, for this class of operators only problems of solving integral equations, factorization, and invertibility were considered (cf. [494]). The most detailed study of the Buschman–Erdélyi transmutations was started in [533,535] and continued in [230,234,534,535,537] and some other papers. Interesting and important results were proved by N. Virchenko and A. Kilbas and their disciples.

5.1 Buschman–Erdélyi transmutations of the first kind 5.1.1 Sonine–Poisson–Delsarte transmutations Let us first consider the most well-known transmutations for the Bessel operator and the second derivative:   2ν + 1 d2 D, D 2 = 2 , ν ∈ C. T (Bν ) f = D 2 Tf, Bν = D 2 + x dx Transmutations, Singular and Fractional Differential Equations With Applications to Mathematical Physics https://doi.org/10.1016/B978-0-12-819781-3.00011-2 Copyright © 2020 Elsevier Inc. All rights reserved.

(5.1)

212

Transmutations, Singular and Fractional Differential Equations

The Poisson transmutation is defined by (see (3.120) where γ = 2ν + 1)  x ν− 1 1 1 2 2 2 Pν f = − t f (t) dt, Re ν > − . x ν 2ν 2 (ν + 1)2 x 0

(5.2)

The Sonine transmutation is defined by 1

Sν f =

2ν+ 2

d 1 ( 2 − ν) dx



x



x2 − t 2

−ν− 1

0

2

1 t 2ν+1 f (t) dt, Re ν < . 2

(5.3)

The operators (5.2)–(5.3) intertwine by the formulas Sν Bν = D 2 Sν , Pν D 2 = Bν Pν .

(5.4)

The definition may be extended to ν ∈ C. We will use the historically more exact term Sonine–Poisson–Delsarte transmutations [532]. An important generalization for the Sonine–Poisson–Delsarte transmutations are the transmutations for the hyper-Bessel operators and functions. Such functions were first considered by Kummer and Delerue. The detailed study on these operators and hyper-Bessel functions was carried out by Dimovski, and further by Kiryakova. The corresponding transmutations were called Sonine–Dimovski and Poisson–Dimovski transmutations by Kiryakova [252]. In hyper-Bessel operators theory the leading role is for the Obrechkoff integral transform [252]. It is a transform with Meijer’s Gfunction kernel which generalizes the Laplace, Meijer, and many other integral transforms introduced by different authors. Various results on the hyper-Bessel functions, connected equations, and transmutations were many times reviewed. The same is true for the Obrechkoff integral transform. In our opinion, the Obrechkoff transform and the Laplace, Fourier, Mellin, Stankovic transforms are essential basic elements from which many other transforms are constructed with corresponding applications.

5.1.2 Definition and main properties of Buschman–Erdélyi transmutations of the first kind Let us define and study some main properties of the Buschman–Erdélyi transmutations of the first kind. This class of transmutations for some choice of parameters generalizes the Sonine–Poisson–Delsart transmutations, Riemann–Liouville and Erdélyi– Kober fractional integrals, and the Mehler–Fock transform. Definition 36. Define the Buschman–Erdélyi operators of the first kind by ν,μ B0+ f

=

x 

x2 − t 2

− μ 2

Pνμ

x 

f (t)d t,

(5.5)

  t f (t)d t, x

(5.6)

t

0 ν,μ E0+ f

=

x  0

x −t 2

2

− μ 2

Pμ ν

Buschman–Erdélyi integral and transmutation operators

ν,μ B− f

=

∞

t −x 2

2

− μ 2

Pνμ

213

  t f (t)d t, x

(5.7)

x ν,μ E− f

=

∞

t 2 − x2

− μ 2

Pμ ν

x  t

(5.8)

f (t)d t.

x μ

μ

Here Pν (z) is the Legendre function of the first kind (1.42), Pν (z) is this function on the cut −1 ≤ t ≤ 1 (1.44), and f (x) is a locally summable function with some growth conditions at x → 0, x → ∞. The parameters are μ, ν ∈ C, Re μ < 1, Re ν ≥ −1/2. Now we consider some main properties for this class of transmutations, following essentially [533,535], and also [230,234,532,537]. All following functions are defined on the positive semiaxis. So we use notations L2 for the functional space L2 (0, ∞) and L2,k for the power weighted space L2,k (0, ∞) equipped with the norm ∞ |f (x)|2 x 2k+1 dx,

(5.9)

0

where N denotes the set of naturals, N0 positive integers, Z integers, and R real numbers. First, add to Definition 36 the case of parameter μ = 1. It defines a very important class of operators. Definition 37. Define for μ = 1 the Buschman–Erdélyi operators of zero order smoothness by ν,1 B0+ f

ν = 1 S0+ f

d = dx

x Pν

x  t

f (t) dt,

(5.10)

0

x ν,1 f = 1 P−ν f = E0+

Pν

  t df (t) dt, x dt

(5.11)

   df (t) t − dt, x dt

(5.12)

0

∞ ν,1 ν f = 1 S− f= B−

Pν x

  ∞   d x ν,1 ν f = 1 P0+ f= − Pν f (t) dt, E− dx t

(5.13)

x

where Pν (z) = Pν0 (z) is the Legendre function. Theorem 48. The next formulas hold true for factorizations of Buschman–Erdélyi transmutations for suitable functions via Riemann–Liouville fractional integrals and

214

Transmutations, Singular and Fractional Differential Equations

Buschman–Erdélyi operators of zero order smoothness: ν, μ

1−μ

B0+ f = I0+ ν, μ

ν 1 S0+ f , 1−μ

1−μ

ν, μ

B− f = 1 P−ν I− ν, μ

1−μ

ν E0+ f = 1 P0+ I0+ f, E− f = I−

f,

(5.14)

ν 1 S− f.

(5.15)

These formulas allow to separate parameters ν and μ. We will prove soon that operators (5.10)–(5.13) are isomorphisms of L2 (0, ∞) except for some special parameters. So, operators (5.5)–(5.8) roughly speaking are of the same smoothness in L2 as integro-differentiations I 1−μ and they coincide with them for ν = 0. It is also possible to define Buschman–Erdélyi operators for all μ ∈ C. Definition 38. Define the number ρ = 1 − Re μ as smoothness order for Buschman– Erdélyi operators (5.5)–(5.8). So for ρ > 0 (otherwise for Re μ > 1) the Buschman–Erdélyi operators are smoothing and for ρ < 0 (otherwise for Re μ < 1) they decrease smoothness in L2 -spaces. Operators (5.10)–(5.13) for which ρ = 0 due to Definition 5 are of zero smoothness order in accordance with their definition. For some special parameters ν, μ the Buschman–Erdélyi operators of the first kind are reduced to other known operators. For μ = −ν or μ = ν + 2 they reduce to Erdélyi–Kober operators, for ν = 0 they reduce to fractional integro-differentiation 1−μ 1−μ I0+ or I− , for ν = − 12 , μ = 0, or μ = 1 kernels reduce to elliptic integrals, and ν, 0 for μ = 0, x = 1, v = it − 12 the operator B− differs only by a constant from the Mehler–Fock transform. As a pair for the Bessel operator consider a connected one    ν(ν + 1) d ν ν d = − + , (5.16) Lν = D 2 − dx x dx x x2 which for ν ∈ N is an angular momentum operator from quantum physics. Their transmutational relations are established in the next theorem. Theorem 49. For a given pair of transmutations Xν , Yν , Xν Lν = D 2 Xν , Yν D 2 = Lν Yν ,

(5.17)

define the new pair of transmutations by formulas Sν = Xν−1/2 x ν+1/2 , Pν = x −(ν+1/2) Yν−1/2 .

(5.18)

Then for the new pair Sν , Pν the following formulas are valid: Sν Bν = D 2 Sν , Pν D 2 = Bν Pν .

(5.19) ν, μ

Theorem 50. Let Re μ ≤ 1. Then an operator B0+ on proper functions is a Sonine type transmutation and (5.17) is valid.

Buschman–Erdélyi integral and transmutation operators

215 ν, μ

The same result holds true for other Buschman–Erdélyi operators, E− is Sonine ν, μ ν, μ type and E0+ , B− are Poisson type transmutations. From these transmutation connections, we conclude that the Buschman–Erdélyi operators link the corresponding eigenfunctions for the two operators. They lead to formulas for the Bessel functions via exponents and trigonometric functions, and vice versa which generalize the classical Sonine and Poisson formulas.

5.1.3 Factorizations of the first kind Buschman–Erdélyi operators and the Mellin transform Now consider factorizations of the Buschman–Erdélyi operators using standard fractional integrals. First let us list the main forms of fractional integro-differentiations: Riemann–Liouville (2.25) and (2.26), Erdélyi–Kober (2.34) and (2.35), and the fractional integral by function g(x) (2.38) and (2.39). Theorem 51. The following factorization formulas are valid for the Buschman– Erdélyi operators of the first kind via the Riemann–Liouville and Erdélyi–Kober fractional integrals:

ν, μ

ν+1−μ −(ν+1) I 0+; 2, ν+ 12

B0+ = I0+ ν, μ

 x ν+1

 ν+1 2 , x −(ν+μ)

I ν+1 1 I0+ , 0+; 2, − 2 2  ν+1 2 ν, μ −(ν+1) ν−μ+2 I−; 2, ν+1 I− , B− = x  x ν+1 ν, μ −(ν+μ) ν+1 E − = I− I−; 2, 0 . 2 E0+ =

(5.20) (5.21) (5.22) (5.23)

The Sonine–Poisson–Delsarte transmutations also are special cases for this class of operators. Now let us study the properties of the Buschman–Erdélyi operators of zero order smoothness, defined by (5.10)–(5.13). A similar operator was introduced by Katrakhov by multiplying the Sonine operator with a fractional integral; his aim was to work with transmutation obeying good estimates in L2 (0, ∞). We use the Mellin transform presented in Definition 11. The Mellin convolution is defined by (1.63) ∞ (f1 ∗ f2 )(x) = 0

  dy x f1 f2 (y) , y y

(5.24)

216

Transmutations, Singular and Fractional Differential Equations

so the convolution operator with kernel K acts under the Mellin transform as a multiplication on multiplicator ⎡∞ ⎤    x dy M[Af ](s) = M ⎣ K f (y) ⎦ (s) = M[K ∗ f ](s) = mA (s)Mf (s), y y 0

(5.25) mA (s) = M[K](s). We observe that the Mellin transform is a generalized Fourier transform on the semiaxis with Haar measure dy y [162]. It plays an important role for the theory of special functions; for example, the gamma function is a Mellin transform of the exponential. With the Mellin transform the important breakthrough in evaluating integrals was made in the 1970s when, mainly by O. Marichev, the famous Slater theorem was adapted for calculations. The Slater theorem taking the Mellin transform as input gives the function itself as output via hypergeometric functions (see [361]). This theorem turned out to be the milestone of powerful computer methods for calculating integrals for many problems in differential and integral equations. The package Mathematica of Wolfram Research is based on this theorem in calculating integrals. ν defined Theorem 52. The Buschman–Erdélyi operator of zero order smoothness 1 S0+ by (5.10) acts under the Mellin transform as convolution (5.25) with the multiplicator

m(s) =

(−s/2 + ν2 + 1)(−s/2 − ν2 + 1/2) (1/2 − 2s )(1 − 2s )

(5.26)

for Re s < min(2 + Re ν, 1 − Re ν). Its norm is a periodic in ν and equals ν,1 L2 = B0+

√

1

min(1, 1 − sin πν)

.

(5.27)

This operator is bounded in L2 (0, ∞) if ν = 2k + 1/2, k ∈ Z, and unbounded if ν = 2k + 1/2, k ∈ Z. Proof. First let us prove formula (5.26) with a proper multiplicator. Using consequently formulas (7), p. 130, (2), p. 129, and (4), p. 130, from [361], we evaluate ⎡∞ ⎤   (2 − s) x dy x ν,1 M(B0+ )(s) = M⎣ H ( − 1)Pν ( ) {yf (y)} ⎦ (s − 1) (1 − s) y y y 0

   (2 − s) 2 = M (x − 1)0+ Pν0 (x) (s − 1)M f (s), (1 − s) we use notations from [361] for Heaviside and cutting power functions   α if x  0, x 1 if x  0, α 0 H (x) = x+ = = x+ 0 if x < 0, 0 if x < 0.

Buschman–Erdélyi integral and transmutation operators

217

Further using formulas (14)(1), p. 234, and (4), p. 130, from [361], we evaluate

√  ( 12 + ν2 − s)(− ν2 − s) M (x − 1)0+ Pν0 ( x) (s) = , (1 − s)( 12 − s) 

1 ( 1 + ν − s−1 )(− ν2 − s−1 2 ) M (x 2 − 1)0+ Pν0 (x) (s − 1) = · 2 2 s−12 1 s−1 2 (1 − 2 )( 2 − 2 ) =

1 (− 2s + ν2 + 1)(− 2s − ν2 + 12 ) · 2 (− 2s + 32 )(− 2s + 1)

under conditions Re s < min(2 + Re ν, 1 − Re ν). Now we evaluate the formula for ν,1 )(s) = M(B0+

1 (2 − s) s 3 s · · (− + )(− + 1). 2 (1 − s) 2 2 2

Applying to (2 − s) the Legendre duplication formula (1.7) we evaluate 2−s (− 2s + ν,1 )(s) = √ · M(B0+ π

ν 2

+ 1)(− 2s − (1 − s)

ν 2

+ 12 )

.

We apply the Legendre duplication formula once more to (1 − s) and the formula for the multiplicator (5.26) is proved. In the paper [535] it was shown that restrictions may be reduced to 0 < Re s < 1 for proper ν. These restrictions may be weakened because they were derived for the class of all hypergeometric functions but we need just one special case of the Legendre function for which specified restrictions may be easily verified directly. Now we prove formula (5.27) for a norm. From the multiplicator value we just found and Theorem 4.7 from [524] on the line Re s = 1/2, s = iu + 1/2, it follows that   1  (−i u2 − ν2 + 14 )(−i u2 + ν2 + 34 )  ν,1 |M(B0+ )(iu + 1/2)| = √  .  ( 1 − iu) 2π  2

Below the operator√symbol in the multiplicator will be omitted. We use formulas for the modulus |z| = z¯z and the gamma function (z) = (¯z) following from its definition as integral. The last property is true in general for the class of real analytic functions. So we derive ν,1 |M(B0+ )(iu + 1/2)|  1  (−i u2 − =√  2π 

ν 2

+ 14 )(i u2 −

ν 1 u ν 2 + 4 )(−i 2 + 2 ( 12 − iu)( 12 + iu)

+ 34 )(i u2 +

ν 2

 + 34 )  . 

218

Transmutations, Singular and Fractional Differential Equations

In the numerator we combine outer and inner terms and transform three pairs of gamma functions by formula (1.6). As a result we evaluate  cos(πiu) ν,1 |M(B0+ )(iu + 1/2)| = ν 1 2 cos π( 2 + 4 + i u2 ) cos π( ν2 + 14 − i u2 )  ch(πiu) = . ch πu − sin πν We further denote t = ch πu, 1  t < ∞. So we derive once more applying Theorem 4.7 from [524]  t 1 . sup |m(iu + )| = sup 2 u∈R 1t max(Re ν, −1 − Re ν), (5.28)

Re s > 0.

(5.29)

The following formulas are valid for norms of the Buschman–Erdélyi operator of zero order smoothness in L2 : √ ν  = 1 P−ν  = 1/ min(1, 1 − sin πν), 1 S0+ √ ν ν 1 P0+  = 1 S −  = max(1, 1 − sin πν). Similar results are proved in [230] and [535] for power weight spaces. Corollary 5. The norms of operators (5.10)–(5.13) are periodic in ν with period 2 X ν  = X ν+2 , and X ν is any of operators (5.10)–(5.13). ν , P ν are not bounded in general, every Corollary 6. The norms of the operators 1 S0+ 1 − norm is greater than or equal to 1. The norms are equal to 1 if sin πν ≤ 0. The opν , P ν are unbounded in L if and only if sin πν = 1 (or ν = (2k) + 1/2, erators 1 S0+ 1 − 2 k ∈ Z). ν , S ν are all bounded in ν, and every Corollary 7. The norms of the operators 1 P0+ 1 − √ norm is not greater than 2. The norms are equal to 1 if sin πν ≥ 0. The operators √ ν ν 2 is 1 P0+ , 1 S− are bounded in L2 for all ν. The norm maximum which equals to achieved if and only if sin πν = −1 (ν = −1/2 + (2k), k ∈ Z).

The most important property of the Buschman–Erdélyi operators of zero order smoothness is the unitarity for integer ν. It is just the case if we interpret for these parameters the operator Lν as angular momentum operator in quantum mechanics. Theorem 54. The operators (5.10)–(5.13) are unitary in L2 if and only if the paramν , P ν ) and ( S ν , P ν ) eter ν is an integer. In this case the pairs of operators (1 S0+ 1 − 1 − 1 0+ are mutually inverse. To formulate an interesting special case, let us suppose that operators (5.10)–(5.13) act on functions permitting outer or inner differentiation in integrals. It is enough to suppose that xf (x) → 0 for x → 0. Then for ν = 1 1 1 P0+ f

1 = (I − H1 )f, 1 S− f = (I − H2 )f,

(5.30)

where H1 , H2 are the famous Hardy operators, 1 H1 f = x

x

∞ f (y)dy, H2 f =

0

and I is the identic operator.

x

f (y) dy, y

(5.31)

220

Transmutations, Singular and Fractional Differential Equations

Corollary 8. The operators (5.30) are unitary in L2 and mutually inverse. They are transmutations for the pair of differential operators d 2 /dx 2 and d 2 /dx 2 − 2/x 2 . The unitarity of the shifted Hardy operators (5.30) in L2 is a known fact [305]. Now we list some properties of the operators acting as convolutions by formula (5.25) and with some multiplicator under the Mellin transform and being transmutations for the second derivative and angular momentum operator in quantum mechanics. Theorem 55. Let an operator Sν act by formulas (5.25) and (5.17). Then: (a) its multiplicator satisfies a functional equation m(s) = m(s − 2)

(s − 1)(s − 2) ; (s − 1)(s − 2) − ν(ν + 1)

(5.32)

(b) if any function p(s) is periodic with period 2 (p(s) = p(s − 2)), then a function p(s)m(s) is a multiplicator for a new transmutation operator S2ν also acting by the rule (5.17). This theorem confirms the importance of studying transmutations in terms of the Mellin transform and multiplicator functions. Define the Stieltjes transform by (cf. [494]) ∞ (Sf )(x) = 0

f (t) dt. x+t

This operator also acts by formula (5.25) with multiplicator p(s) = π/ sin(πs), it is bounded in L2 . Obviously p(s) = p(s − 2). So from Theorem 55 a convolution of the Stieltjes transform follows with bounded transmutations (5.10)–(5.13), and also transmutations of the same class bounded in L2 . In this way many new classes of transmutations were introduced with special functions as kernels.

5.2 Buschman–Erdélyi transmutations of the second and third kind 5.2.1 Second kind Buschman–Erdélyi transmutation operators Now we consider Buschman–Erdélyi transmutations of the second kind.

Buschman–Erdélyi integral and transmutation operators

221

Definition 39. Define a new pair of Buschman–Erdélyi transmutations of the second kind with Legendre functions of the second kind in kernels f= ⎛ x ⎞      ∞ 1 1 2⎝ x x − (x 2 − y 2 )− 2 Q1ν f (y)dy + (y 2 − x 2 )− 2 Q1ν f (y)dy ⎠ , π y y

2S

ν

x

0

(5.33) ν f= ⎛ x ⎞  ∞     1 1 2⎝ y y − (x 2 − y 2 )− 2 Q1ν f (y)dy − (y 2 − x 2 )− 2 Q1ν f (y)dy ⎠ , π x x

2P

x

0

(5.34) μ

μ

where Qν is the Legendre functions of the second kind (1.43) and Qν is the Legendre function of the second kind on the cut (1.45). These operators are analogues of Buschman–Erdélyi transmutations of zero order smoothness. If y → x ± 0, then integrals are defined by principal values. It is proved that they are transmutations of Sonine type for (5.33) and of Poisson type for (5.34). Theorem 56. Operators (5.33) and (5.34) are of the form (5.25) with multiplicators m2 S ν (s) = p(s) m1 S−ν (s), m2 P ν (s) =

1 m P ν (s), p(s) 1 −

(5.35) (5.36)

ν , P ν defined by (5.28) and (5.29), where the with multiplicators of operators 1 S− 1 − period 2 function p(s) equals

p(s) =

sin πν + cos πs . sin πν − sin πs

Theorem 57. The following formulas for norms are valid: √ 2 S ν L2 = max(1, 1 + sin πν), √ 2 P ν L2 = 1/min(1, 1 + sin πν).

(5.37)

(5.38) (5.39)

Corollary 9. Operator 2 S ν is bounded for all ν. Operator 2 P ν is not bounded if and only if sin πν = −1. Theorem 58. Operators 2 S ν and 2 P ν are unitary in L2 if and only if ν ∈ Z. Theorem 59. Let ν = iβ + 1/2, β ∈ R. Then  2 S ν L2 = 1 + ch πβ, 2 P ν L2 = 1.

(5.40)

222

Transmutations, Singular and Fractional Differential Equations

Theorem 60. The following formulas are valid: 0 2S f =

2 π

∞ 0

2S

−1

2 f= π

y f (y) dy, x2 − y2

∞ 0

(5.41)

x f (y) dy. x2 − y2

(5.42)

So in this case the operator 2 S ν reduces to a pair of semiaxis Hilbert transforms [494]. For operators of the second kind we also introduce more general ones with two parameters analogously to Buschman–Erdélyi transmutations of the first kind by formulas ⎛ x    2⎝ ν,μ 2 2 − μ2 −μπi μ x S f = (x + y ) e Q f (y) dy 2 ν π y 0 ⎞   ∞ μ x (5.43) f (y) dy ⎠ , + (y 2 + x 2 )− 2 Qμ ν y x μ

μ

where Qν (z) is the Legendre function of the second kind (1.43), Qν (z) is this function on the cut (1.45), and Re ν < 1. The second operator may be defined as formally conjugate in L2 (0, ∞) to (5.43). Theorem 61. The operator (5.43) on C0∞ (0, ∞) is well defined and acts by M[2 S ν ](s) = m(s) · M[x 1−μ f ](s),  m(s) = 2

μ−1

cos π(μ − s) − cos πν sin π(μ − s) − sin πν





( 2s )( 2s + 12 )) ( 2s +

1−ν−μ )( 2s 2

+1+

ν−μ 2 )

.

5.2.2 Sonine–Katrakhov and Poisson–Katrakhov transmutations Now we construct transmutations which are unitary for all ν. They are defined by formulas πν ν πν ν 2 S f + cos 1 S− f, 2 2 πν πν ν ν PUν f = − sin 2 P f + cos 1 P− f. 2 2

SUν f = − sin

(5.44) (5.45)

For all values ν ∈ R they are linear combinations of Buschman–Erdélyi transmutations of the first and second kinds of zero order smoothness. Also they are in the defined

Buschman–Erdélyi integral and transmutation operators

223

below class of Buschman–Erdélyi transmutations of the third kind. The following integral representations are valid: SUν f

  ∞   πν d x = cos Pν − f (y) dy 2 dx y x ⎛ x    πν ⎝ 2 2 2 − 12 1 x (x − y ) Qν f (y) dy + sin π 2 y 0 ⎞   ∞ 1 x f (y) dy ⎠ , − (y 2 − x 2 )− 2 Q1ν y

(5.46)

x

y  d  f (y) dy x dy 0 ⎛ x  y  1 πν ⎝ 2 − (x 2 − y 2 )− 2 Q1ν f (y) dy − sin π 2 x 0 ⎞ ∞ y  1 f (y) dy ⎠ . − (y 2 − x 2 )− 2 Q1ν x

PUν f = cos

πν 2

x

Pν

(5.47)

x

Theorem 62. Operators (5.44)–(5.45) and (5.46)–(5.47) for all ν ∈ R are unitary, mutually inverse, and conjugate in L2 . They are transmutations acting by (5.16); SUν is a Sonine type transmutation and PUν is a Poisson type one. Transmutations like (5.46)–(5.47) but with kernels into more complicated form with hypergeometric functions were first introduced by Katrakhov in 1980. For this reason S. M. Sitnik termed this class of operators Sonine–Katrakhov and Poisson– Katrakhov. In S. M. Sitnik’s papers these operators were reduced to more simple forms of Buschman–Erdélyi ones. This made it possible to include this class of operators in general composition (or factorization) methods [146,229,234,525,537].

5.2.3 Buschman–Erdélyi transmutations of the third kind with arbitrary weight function Define sine and cosine Fourier transforms with inverses  Fc f =

2 π

∞ 0

f (y) cos(ty) dy, Fc−1 = Fc ,

(5.48)

224

Transmutations, Singular and Fractional Differential Equations

 Fs f =

2 π

∞

f (y) sin(ty) dy, Fs−1 = Fs .

(5.49)

0

We define the Hankel (Fourier–Bessel) transform here a little different and its inverse in (1.56)–(1.57) by 1 Fν f = ν 2 (ν + 1) ∞ =

∞ f (y) jν (ty) y 2ν+1 dy 0

Jν (ty) 2ν+1 1 f (y) y dy = ν ν (ty) t

0

Fν−1 f =

∞ f (y)Jν (ty) y ν+1 dy,

(5.50)

0

1 (y)ν

∞ f (t)Jν (yt) t ν+1 dt.

(5.51)

0

Here Jν (·) is the Bessel function (1.13) and jν (·) is the normalized Bessel function (1.19). Operators (5.48) and (5.49) are unitary self-conjugate in L2 (0, ∞). Operators (5.50) and (5.51) are unitary self-conjugate in the power weighted space L2, ν (0, ∞). Now define on proper functions the first pair of Buschman–Erdélyi transmutations of the third kind   1 (ϕ) −1 Sν, = F F (5.52) ν , c c ϕ(t) Pν,(ϕ)c = Fν−1 (ϕ(t)Fc ) ,

(5.53)

and the second pair by   1 (ϕ) −1 Fν , Sν, s = Fs ϕ(t) Pν,(ϕ)s = Fν−1 (ϕ(t)Fs ) ,

(5.54) (5.55)

with ϕ(t) being an arbitrary weight function. The operators defined on proper functions are transmutations for Bν and D 2 . They may be expressed in the integral form. Theorem 63. Define transmutations for Bν and D 2 by formulas   1 (ϕ) −1 S  s  = F s  Fν , ν, c ϕ(t) c   (ϕ) −1     ϕ(t)F s . P s = Fν ν,

c

c

Buschman–Erdélyi integral and transmutation operators

225

Then for the Sonine type transmutation an integral form is valid, 

 (ϕ) S s  f ν, c

 (x) =

2 π

∞ K(x, y)f (y) dy,

(5.56)

0

where

 ∞ K(x, y) = y ν+1

sin(xt) cos(xt) Jν (yt)dt. ϕ(t) t ν

0

For the Poisson type transmutation an integral form is valid, 

 (ϕ) P s  f ν, c

 (x) =

2 π

∞ G(x, y)f (y) dy,

(5.57)

0

where 1 G(x, y) = ν x



∞ ϕ(t) t

ν+1

sin(yt) J (xt)dt. cos(yt) ν

0

The above introduced unitary transmutations of Sonine–Katrakhov and Poisson– Katrakhov are special cases of this class of operators. For this case we must choose a weight function ϕ(t) as a power function depending on the parameter ν.

5.2.4 Some applications of Buschman–Erdélyi transmutations In this subsection we briefly discuss some applications of Buschman–Erdélyi operators. Section 8.1 contains other applications in expanded form. The above classes of transmutations may be used for deriving explicit formulas for solutions of partial differential equations with Bessel operators via unperturbed equation solutions. An example is the B-elliptic equation of the form n 

Bν,xk u(x1 , . . . , xn ) = f,

(5.58)

k=1

and also similar B-hyperbolic and B-parabolic equations. This idea works by Sonine– Poisson–Delsarte transmutations (cf. [51–53,56,242]). New results follow automatically for new classes of transmutations. Now let us consider the Euler–Poisson–Darboux equation in a half-space Bα, t u(t, x) =

∂ 2 u 2α + 1 ∂u + = x u + F (t, x), t ∂t ∂t 2

226

Transmutations, Singular and Fractional Differential Equations

with t > 0, x ∈ Rn . Let us consider a general plan for finding different initial and boundary conditions at t = 0 with guaranteed existence of solutions. Define any transmutations Xα, t and Yα, t satisfying (5.16). Suppose that functions Xα, t u = v(t, x),

Xα, t F = G(t, x)

exist. Suppose that the unperturbed Cauchy problem ∂2 v = x v + G, v|t=0 = ϕ(x), vt |t=0 = ψ(x) ∂t 2

(5.59)

−1 is correctly solvable in a half-space. Then if Yα, t = Xα, t , then we receive the following initial conditions:

Xα u|t=0 = a(x), (Xα u) |t=0 = b(x).

(5.60)

By this method the choice of different classes of transmutations (Sonine–Poisson– Delsarte, Buschman–Erdélyi of the first, second, and third kinds, Buschman–Erdélyi of the zero order smoothness, unitary transmutations of Sonine–Katrakhov and Poisson–Katrakhov, transmutations with general kernels) will correspond to different kinds of initial conditions [535]. In the monograph of Pskhu [459] this method is applied for solving an equation with fractional derivatives with the usage of the Stankovic transform. Glushak applied Buschman–Erdélyi operators in [185]. This class of operators was thoroughly studied by Levitan [316,327]. It has many applications to partial differential equations, including Bessel operators [317]. Generalized translations are used for moving singular points from the origin to any location. They are explicitly expressed via transmutations [317]. Due to this fact new classes of transmutations lead to new classes of generalized translations. In recent years Dunkl operators were thoroughly studied. These are difference– differentiation operators consisting of combinations of classical derivatives and finite differences. In higher dimensions Dunkl operators are defined by symmetry and reflection groups. For this class there are many results on transmutations which are of Sonine–Poisson–Delsarte and Buschman–Erdélyi types (cf. [560] and references therein). It has been known for many years that a problem of describing polynomial solutions for the B-elliptic equation does not need the new theory. The answer is in the transmutation theory. The simple fact that Sonine–Poisson–Delsarte transmutations transform power functions into other power functions means that they also transform explicitly so-called B-harmonic polynomials into classical harmonic polynomials and vice versa. The same is true for generalized B-harmonics because they are restrictions of B-harmonic polynomials onto the unit sphere. This approach is thoroughly applied by Rubin [485]. Usage of Buschman–Erdélyi operators refreshes this theory with new possibilities.

Buschman–Erdélyi integral and transmutation operators

227

Now let us construct integral operators of Buschman–Erdélyi with more general ν . It has the form functions as kernels. Consider an operator 1 S0+ ν 1 S0+

d = dx

x

  x K f (y) dy, y

(5.61)

0

with kernel K expressed by K(z) = Pν (z). Simple properties of special functions lead ν is a special case of (5.61) with the Gegenbauer function kernel to the fact that 1 S0+ K(z) =

(α + 1) (2β) 2

p− 12

1

(α + 2β) (β +

1 2)

(zα − 1)β− 2 Cαβ (z)

(5.62)

with α = ν, β = 12 , or with the Jacobi function kernel K(z) =

(α + 1) (z − 1)ρ (z + 1)σ Pα(ρ,σ ) (z) + ρ + 1)

2ρ (α

(5.63)

with α = ν, ρ = σ = 0. More general are operators with the Gauss hypergeometric m,n function kernel 2 F1 or Meijer G-function Gp,q (1.41) or Fox–Wright Eα,β function (1.40) kernels (cf. [494]). For studying such operators inequalities for kernel functions are very useful (e.g., [228,528]). Define the first class of generalized operators. Definition 40. Define Gauss–Buschman–Erdélyi operators by formulas 1 F0+ (a, b, c)[f ] =

x 

c−1 

x −1 y

1 , 2c−1 (c)

(5.64)

a+b−c

x +1 y

 2 F1

 1x − f (y) dy, 2 2y

a,b 1 c |

0 2 F0+ (a, b, c)[f ] =

x 

1 , 2c−1 (c)

c−1  y a+b−c y −1 +1 2 F1 x x

(5.65) 

a,b 1 c |

2

−

 1y f (y) dy, 2x

0 1 F− (a, b, c)[f ] =

x  0

1 2c−1 (c)

(5.66)

,

c−1  y a+b−c y −1 +1 2 F1 x x



 1y − f (y) dy, 2 2x

a,b 1 c |

228

Transmutations, Singular and Fractional Differential Equations

2 F− (a, b, c)[f ] =

x 

c−1 

x −1 y

1 2c−1 (c)

(5.67)

,

a+b−c

x +1 y

 2 F1

 1x − f (y) dy, 2 2y

a,b 1 c |

0

d d [f ], 1 F0+ [f ], 4 F0+ [f ] = 2 F0+ dx  dx    d d [f ], 4 F− [f ] = − 3 F− [f ] = 1 F− − 2 F− [f ]. dx dx 3 F0+ [f ] =

(5.68) (5.69)

The symbol 2 F1 in definitions (5.65) and (5.67) means the Gauss hypergeometric function on the natural domain and in (5.64) and (5.66) the main branch of its analytical continuation. Operators (5.64)–(5.67) generalize Buschman–Erdélyi ones (5.5)–(5.8), respectively. They reduce to Buschman–Erdélyi for the choice of parameters a = −(ν + μ), b = 1 + ν − μ, c = 1 − μ. For operators (5.64)–(5.67) the above results are generalized with necessary changes. For example they are factorized via more simple operators (5.68) and (5.69) with a special choice of parameters. Operators (5.68) and (5.69) are generalizations of (5.14) and (5.15). For them the following result is true. Theorem 64. Operators (5.68) and (5.69) may be extended to isometric in L2 (0, ∞) if and only if they coincide with Buschman–Erdélyi operators of zero order smoothness (5.14) and (5.15) for integer values of ν = 12 (b − a − 1). This theorem singles out Buschman–Erdélyi operators of zero order smoothness at least in the class (5.64)–(5.69). Operators (5.64)–(5.67) are generalizations of fractional integrals. Analogously may be studied generalizations to (5.33)–(5.34), (5.43), and (5.46)–(5.47). More general are operators with G-function kernel, 2δ (1 − α)(1 − β) −δ  1+δ−α−β   x  1 α, β x x x 12 × G2 2 −1 +1 − | f (y) dy. y y 2y 2 γ , δ

1 G0+ (α, β, δ, γ )[f ] =

(5.70)

0

Other operators are with different intervals of integration and parameters of the G-function. For α = 1 − a, β = 1 − b, δ = 1 − c, γ = 0 (5.70) reduces to (5.64), for α = 1 + ν, β = −ν, δ = γ = 0 (5.70) reduces to Buschman–Erdélyi operators of ν . zero order smoothness 1 S0+ Further generalizations are in terms of Wright or Fox functions. They lead to Wright–Buschman–Erdélyi and Fox–Buschman–Erdélyi operators. These classes are

Buschman–Erdélyi integral and transmutation operators

229

connected with Sonine–Dimovski and Poisson–Dimovski transmutations [89,93], and also with generalized fractional integrals introduced by Kiryakova [252]. V. Katrakhov found a new approach for boundary value problems for elliptic equations with strong singularities of infinite order. For example for the Poisson equation he studied problems with solutions of arbitrary growth. At singular points he proposed a new kind of boundary condition: the K-trace. His results are based on constant usage of Buschman–Erdélyi transmutations of the first kind for definition of norms, solution estimates, and correctness proofs [225,227]. Moreover in joint papers with I. Kipriyanov he introduced and studied new classes of pseudodifferential operators based on transmutational techniques [248–250]. These results were paraphrased in a reorganized manner in [52].

5.3 Multi-dimensional integral transforms of Buschman–Erdélyi type with Legendre functions in kernels In this section we consider generalizations of Buschman–Erdélyi operators for the multi-dimensional case. This case was studied by O. V. Skoromnik and S. M. Sitnik (see [538]).

5.3.1 Basic definitions Let x = (x1 , x2 , ..., xn ) ∈ Rn+ , t = (t1 , t2 , ..., tn ) ∈ Rn+ , x · t = scalar product, and in particular, x · 1 =

n 

n 

xn tn denotes their

n=1

xn . The expression x > t means that x1 >

n=1

t1 , ..., xn > tn , the nonstrict inequality ≥ has a similar meaning, by N = {1, 2, ...} we denote the set of positive integers, N0 = N {0}, and Nn0 = N0 × N0 × ... × N0 . We denote m = (m1 , m2 , ..., mn ) ∈ Nn0 , m1 = m2 = ... = mn ; n = (n1 , n2 , ..., nn ) ∈ Nn0 , n1 = n2 = ... = nn ; p = (p1 , p2 , ..., pn ) ∈ Nn0 , p1 = p2 = ... = pn ; q = (q1 , q2 , ..., qn ) ∈ Nn0 , q1 = q2 = ... = qn , (0 ≤ m ≤ q, 0 ≤ m ≤ p). We put σ = (σ1 , σ2 , ..., σn ) ∈ Cn ; κ = (κ1 , κ2 , ..., κn ) ∈ Cn ; δ = (δ1 , δ2 , ..., δn ) ∈ Rn ; γ = (γ1 , γ2 , ..., γn ) ∈ Rn , 0 < γ < 1; ai = (ai1 , ai2 , ..., ain ), 1 ≤ i ≤ p, ai1 , ai2 , ..., ain ∈ C, 1 ≤ i1 ≤ p1 , ..., 1 ≤ in ≤ pn ; bj = (bj1 , bj2 , ..., bjn ), 1 ≤ j ≤ q, bj1 , bj2 , ..., bjn ∈ C, 1 ≤ j1 ≤ q1 , ..., 1 ≤ jn ≤ qn ; αi = (αi1 , αi2 , ..., αin ), 1≤i≤p, αi1 , αi2 , ..., αin ∈R1+ , 1 ≤ i1 ≤ p1 , ..., 1 ≤ in ≤ pn ;

230

Transmutations, Singular and Fractional Differential Equations

βj = (βj1 , βj2 , ..., βjn ), 1≤j ≤q, βj1 , βj2 , ..., βjn ∈ R1+ , 1 ≤ j1 ≤ q1 , ..., 1 ≤ jn ≤ qn ; k = (k1 , k2 , ..., kn ) ∈ Nn0 = N0 × ... × N0 , ki ∈ N0 , i = 1, 2, ..., n, is a multi-index with k! = k1 ! · · · kn ! and |k| = k1 + k2 + ... + kn ; for l = (l1 , l2 , ..., ln ) ∈ Rn+ , Dl =

(∂x1

)l1

∂ |l| , · · · (∂xn )ln

dt = dt1 · dt2 · · · dtn , t l = t l1 · · · t ln ; x 2 − t 2 = (x12 − t12 ) · · · (xn2 − tn2 ); f (t) = f (t1 , t2 , ..., tn ). We introduce the function " " !  !  n #   m, n x  (ai ,αi )1,p mk , nk xk  (aik ,αik )1,pk Hpk , qk Hp, q = , t  (bj ,βj )1,q tk  (bjk ,βjk )1,qk

(5.71)

k=1

n which is the product of the H-functions Hm, p, q [z]. Such a function is defined by $  "   1 m,n m,n  (ai ,αi )1,p m,n Hp,q (s)z−s ds, z = 0, = Hp, q [z] ≡ Hp,q z (bj ,βj )1,q 2πi L

where $ m,n m,n Hp, q (s) ≡ Hp, q

m n % % (bj + βj s) (1 − ai − αi s)  " (ai ,αi )1,p  i=1 s = j =1 . p q % % (bj ,βj )1,q  (ai + αi s) (1 − bj − βj s) j =m+1

i=n+1

(5.72) Here L is a specially chosen infinite contour and empty product, if it occurs, being taken to be one. Note that most of the elementary and special functions are special cases of the H-function and one may find its properties in the books by Mathai and Saxena ([362], Chapter 2), Srivastava, Gupta, and Goyal ([499], Chapter 1), Prudnikov, Brychcov, and Marichev ([457], Section 8.3), and Kilbas and Saigo ([239], Chapters 1 and 2). We introduce the function γ

Pδ [z] =

n #

γ

Pδkk [zk ],

k=1 γ

which is the product of the Legendre functions Pδ (z) of the first kind. For complex γ , Re(γ ) < 1, and δ, z ∈ C this function is defined by γ

Pδ (z) =

 γ   1 1−z z+1 2 F 1 + δ; 1 − γ ; , −δ, 2 1 (1 − γ ) z − 1 2

Buschman–Erdélyi integral and transmutation operators

|arg(z − 1)| < π,



1+x 1 −x (1 − γ ) −1 0, and f ∈ Lν,2 . If Re(λ) > (ν − 1 f is represented in the form Re(κ))h − 1, then Hσ,κ &

' 1 Hσ,κ f (x) = hx σ +1−(λ+1)/h $  " ∞ d (λ+1)/h x  (−λ,h),(ai ,αi )1,p m,n+1 × Hp+1,q+1  x t κ−1 f (t)dt, dx t (bj ,βj )1,q , (−λ−1,h) 0

while for Re(λ) < (ν − Re(k))h − 1 it is given by & 1 ' Hσ,κ f (x) = −hx σ +1−(λ+1)/h $  " ∞ d (λ+1)/h x  (ai ,αi )1,p ,(−λ,h) m+1,n × Hp+1,q+1  x t κ−1 f (t)dx. dx t (−λ−1,h),(bj ,βj )1,q 0

(e) If f ∈ Lν,2 and g ∈ L1−ν+Re(κ+σ ),2 , then there the following relation holds: ∞

& 1 ' f (x) Hσ,κ g (x)dx =

0

∞ &

' 2 Hσ,κ f (x)g(x)dx,

(5.83)

0

where &

2 Hσ,κ f

' (x) = x σ

∞ Hm,n p,q 0

$  " dt t  (ai ,αi )1,p κ t f (t) . x  (bj ,βj )1,q x

(5.84)

236

Transmutations, Singular and Fractional Differential Equations

1 5.3.4 Inversion of Hσ,κ 1 are given by the following equalities (for Inversion formulas for the transform Hσ,κ the one-dimensional case, see [5, formulas (5.5.23) and (5.5.24)]):

d −(λ+1)/h x dx $  " ∞ q−m,p−n+1 t  (−λ,h),(1−ai −αi ,αi )m+1,p ,(1−ai −αi ,αi )1,m × Hp+1,q+1 x  (1−bj −βj ,βj )m+1,q ,(1−bj −βj ,βj )1,m (−λ−1,h)

f (x) = −hx (λ+1)/h−κ

0 1 × t −σ (Hσ,κ f )(t)dt

(5.85)

or d −(λ+1)/h x dx $  " ∞ q−m+1,p−n t  (1−ai −αi ,αi )m+1,p ,(1−ai −αi ,αi )1,n , (−λ,h) × Hp+1,q+1 x  (−λ−1,h),(1−bj −βj ,βj )m+1,q ,(1−bj −βj ,βj )1,m

f (x) = hx (λ+1)/h−1

0 1 × t −σ (Hσ,κ f )(t)dt.

(5.86)

Conditions for the validity of these formulas are given by the following assertion (for the one-dimensional case, see [239], Theorem 5.47)). Theorem 66. Let a1∗ = 0, a2∗ = 0, ..., an∗ = 0, α1 < ν1 − Re(κ1 ) < β1 , α2 < ν2 − Re(κ2 ) < β2 , ..., αn < νn − Re(κn ) < βn , α01 < 1 − ν1 + Re(κ1 ) < β01 , α02 < 1 − ν2 + Re(κ2 ) < β02 ,..., α0n < 1 − νn + Re(κn ) < β0n , and let λ ∈ Cn , h > 0. If 1 [ν1 − Re(κ1 )] + Re(μ1 ) = 0, 2 [ν2 − Re(κ2 )] + Re(μ2 ) = 0,..., n [νn − Re(κn )]+Re(μn ) = 0, and f ∈ Lν,2 (ν1 , ν2 , ..., νn ), then the inversion formulas (5.85) and (5.86) are valid for Re(λ) > (1−ν +Re(κ))h−1 and Re(λ) < (1−ν +Re(κ))h− 1, respectively.

5.4 Representations in the form of modified H -transform 5.4.1 Mellin transform of auxiliary functions K1 (x) and K2 (x) We introduce the so-called one-sided functions  −γ /2 K1 (x) = (x 2 − 1)+

Pδγ (x) =

(x 2 − 1)−γ /2 Pδγ (x) 0

for x > 1, for 0 < x < 1,

(5.87)

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237

 −γ /2 K2 (x) = (1 − x 2 )+

Pδγ (x) =

(1 − x 2 )−γ /2 Pδγ (x) 0

for 0 < x < 1, for x > 1.

(5.88)

Using (5.87) and (5.88) we can present transforms (5.74) and (5.75) in respective forms & γ ' Pδ,1 f (x) =

∞ K1

  x & t

' M−γ f (t)dt,

0

' γ Pδ,2 f (x) = x 1−γ

&

∞ RK2

 x & ' M−1 f (t)dt. t

0

The following assertion yields the multi-dimensional Mellin transform formulas of K1 (x) and K2 (x). Lemma 18. Let γ = (γ1 , γ2 , ..., γn ), δ = (δ1 , δ2 , ..., δn ), s = (s1 , s2 , ..., sn ) ∈ Cn . (a) If Re(γ ) < 1, Re(s) < 1 + Re(γ + δ), Re(s) < Re(γ − δ), then & 1+γ +δ−s ' & γ −δ−s ' ' &  γ −1  (5.89) MK1 (s) = 2 & 2 s ' & 1−s2' .  1− 2  2 (b) If Re(γ ) < 1, Re(s) > 0, then ' & ' & & '  2s  s+1 γ −1 2 MK2 (s) = 2 ' & '. &  1 + δ−γ2 +s  1−γ −δ+s 2

(5.90)

Proof. By [457], formula (2.172.9), under the conditions in (a), the following formula holds: & 1+γ1 +δ1 −s1 ' & γ1 −δ1 −s1 ' & '  2γ1 −s1 −1  2 2 MK1 (s) = √ (1 − s1 ) π & 1+γ2 +δ2 −s2 ' & γ2 −δ2 −s2 '  2γ2 −s2 −1  2 2 ... × √ (1 − s2 ) π & 1+γn +δ1 −sn ' & γn −δn −sn '  2γ1 −sn −1  2 2 ... √ (1 − sn ) π & 1+γ +δ−s ' & γ −δ−s '  2γ −s−1  2 2 . (5.91) = √ (1 − s) π Using the duplication formula (1.7) for the gamma function with z = 1−s 2 , from Eq. (5.91) we deduce equality (5.89). If conditions in (b) are satisfied, then according to [457], formula (2.172.), ' √ (s) MK2 (s) = 2γ −s π & 1−γ −δ+s ' &  1+  2

&

δ−γ +s ' 2

.

(5.92)

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Applying equalities (5.92) and (1.7) with z = proved.

s 2

we get equality (5.90). The lemma is

Applying the convolution Mellin formula ([241], (1.4.56)), 

∞   & ' & ' x dt  M K y(t) (s) = MK (s) Mf (s), t t

(x ∈ Rn+ ),

(5.93)

0

& ' & ' being valid for suitable K xt = K xt11 , xt22 , ..., xtnn and y(x). γ

γ

5.4.2 Mellin transform of Pδ,1 (x) and Pδ,2 (x) & γ ' Applying (5.89), for Pδ,1 f (x) we have &

 ∞  x   dt  M K1 M1−γ f (t) (s) t t 0     = MK1 (s) MM1−γ f (s)       (1 + γ + δ − s)/2  (γ − δ − s)/2      Mf (1 − γ + s). = 2γ −1  1 − s/2  (1 − s)/2

' γ MPδ,1 f (s) =

In accordance with (5.72) we obtain        (1 + γ + δ − s)/2  (γ − δ − s)/2 & ' γ     MPδ,1 f (s) = 2γ −1 Mf  1 − s/2  (1 − s)/2 × (1 − γ + s) $& γ −1

=2

1−γ −δ 1 ,2 2

0,2 H2,2

&

' & ,

1 1+ δ−γ 2 ,2

' &1 1'

0, 12 ,

2,2

'  "   s Mf (s + 1 − γ ). 

Therefore, the initial integral transform (5.74) is the modified H -transform (5.73) with σ = 0, κ = 1 − γ : $ & ' & δ−γ 1 ' " ∞ 1−γ −δ 1 & γ ' ,2 1+ , 2 0,2 x  γ −1 & 1 ' & 1 1 '2 2 t −γ f (t)dt. H2,2  Pδ,1 f (s) = 2 t 0, 2 2,2 0

& γ ' Similarly to the above, using Eq. (5.90) for Pδ,2 f (x) we have   & ' γ MPδ,2 f (s) = M x 1−γ

   ' x & ' RK2 M−1 f (t)dt (s) t

∞ & 0

Buschman–Erdélyi integral and transmutation operators

239

  ' x dt  RK2 = M f (t) (s + 1 − γ ) t t 0 & & '' & ' = M RK2 (s + 1 − γ ) Mf (s + 1 − γ ) ' & ' & = MK2 (γ − s) Mf (s + 1 − γ ) & ' & ' ' (γ − s)/2  (γ − s + 1)/2 & γ −1  & ' & ' Mf (1 − γ + s). =2  (1 − δ − s)/2  1 + (δ − s)/2 ∞ &



According to equality (5.72) we obtain & ' & ' ' '  (γ − s)/2  (γ − s + 1)/2 & γ ' & ' Mf (1 − γ + s) MPδ,2 f (s) = 2γ −1 &  (1 − δ − s)/2  1 + (δ − s)/2 $& ' & γ 1 '  " 1−γ 1  , , 1− ,  0,2 2 2 = 2γ −1 H2,2 & 1+δ 1 ' & δ2 12' s Mf (s + 1 − γ ), , , − ,

&

2

2

2 2

& γ ' and hence, the initial transform Pδ,2 f (x) is also a modified H -transform (5.73), with σ = 0, κ = 1 − γ : ' γ Pδ,2 f (s) = 2γ −1

&

∞ 0,2 H2,2 0

$ & ' & '" 1 x  1− γ2 , 12 , 1−γ 2 ,2 & ' & δ 1 ' t −γ f (t)dt. 1 t  1+δ , −2,2 2 2 , γ

5.4.3 Lν, 2 -theory of the transforms Pδ,k f (k = 1, 2) With respect to Lν, 2 -theory of the transforms (5.74) and (5.75), using Theo1 -transform we have a ∗ = a ∗ = ... = a ∗ = 0,  =  = rem 65, for the Hσ,κ 1 2 n 1 2 ... = n = 0, p = (p1 , p2 , ..., pn ) = (2, 2, ..., 2), q = (q1 , q2 , ..., qn ) = (2, 2, ..., 2), αi = (αi1 , αi2 , ..., αin ) = ( 12 , 12 , ..., 12 ), βj = (βj1 , βj2 , ..., βjn ) = ( 12 , 12 , ..., 12 ) (i = 1, ..., p, j = 1, ..., q), μ = γ − 1. As for m, n, and other parameters we obtain m = 0, n = 2, α = −∞, β = min[Re(1 + γ + δ), Re(γ − δ)], n = 0, n = 2, α = −∞, β = Re(γ ), respectively, for the operators (5.74) and (5.75). 0,2 The number 1 − ν does not belong to the exceptional set EH of the H2,2 -function if s = 2m + 1, s = 2l + 2 (l = (l1 , l2 , ..., ln ), m = (m1 , m2 , ..., mn ) ∈ Nn0 ), for Re(s) = 1 − ν.

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From the other side, 1 − ν does not belong to the exceptional set EH of the if

2,0 H2,2 -function

s = −δ + 2m + 1, s = δ + 2l + 2 (l = (l1 , l2 , ..., ln ), m = (m1 , m2 , ..., mn ) ∈ Nn0 ), for Re(s) = 1 − ν. γ From Theorem 65 we deduce the Lν, 2 -theory of the transforms Pδ,k f (k = 1, 2). Theorem 67. Let − ∞ < ν1 − Re(1 − γ1 ) < min[Re(1 + γ1 + δ1 ), Re(γ1 − δ1 )], Re(γ1 − 1) ≤ 0, − ∞ < ν2 − Re(1 − γ2 ) < min[Re(1 + γ2 + δ2 ), Re(γ2 − δ2 )], Re(γ2 − 1) ≤ 0, ..., − ∞ < νn − Re(1 − γn ) < min[Re(1 + γn + δn ), Re(γn − δn )], Re(γn − 1) ≤ 0. The following assertions hold: γ (a) There exists a one-to-one map Pδ,1 ∈ [Lν,2 , Lν−Re(1−γ ),2 ] such that the relation γ (5.74) holds for f ∈ Lν,2 and Re(s) = ν − Re(1 − γ ). If Re(γ − 1) = 0, then Pδ,1 is one-to-one on Lν,2 . γ (b) The transform Pδ,1 f does not depend on ν in the sense that if ν 1 and if the γ γ transforms Pδ,1 f and ) Pδ,1 f are defined in respective spaces Lν 1 ,2 and Lν 2 ,2 by rela+ γ γ tion (5.74), then Pδ,1 f = ) Pδ,1 f for f ∈ Lν 1 ,2 Lν 2 ,2 . γ (c) If Re(γ − 1) < 0, then for f ∈ Lν,2 , Pδ,1 f is given by (5.74). (d) Let λ = (λ1 , λ2 , ..., λn ) ∈ Cn , h = (h1 , ..., hn ) > 0, and f ∈ Lν,2 . If Re(λ) > γ (ν − Re(1 − γ ))h − 1, then Pδ,1 f is represented in the form &

' d γ Pδ,1 f (x) = 2γ −1 hx 1−(λ+1)/h x (λ+1)/h dx $  & 1−γ −δ 1 ' & δ−γ 1 ' " ∞ , , 1+ 2 , 2 0,3 x  (−λ,h), & 1 ' 2& 1 12' × H3,3  t −γ f (t)dt, t 0, 2 , 2 , 2 , (−λ−1,h) 0

while for Re(λ) < (ν − Re(1 − γ ))h − 1 it is given by &

' d γ Pδ,1 f (x) = −2γ −1 hx 1−(λ+1)/h x (λ+1)/h dx $ & " ' & ' ∞ 1−γ −δ 1 1 , 2 , 1+ δ−γ 2 2 , 2 , (−λ,h) 1,2 x  & ' & ' × H3,3  t −γ f (t)dt. t (−λ−1,h), 0, 12 , 12 , 12 0

Buschman–Erdélyi integral and transmutation operators

241

(e) If f ∈ Lν,2 and g ∈ L1−ν+Re(1−γ ),2 , then the following relation holds: ∞

& γ ' f (x) Pδ,1 g (x)dx =

0

∞

& γ ' 2γ −1 P∗ δ,2 f (x)g(x)dx,

0

' γ where P∗ δ,2 f (x) is the transform &

& ∗γ ' P δ,2 f (x) =

∞ &

t

2

'−γ /2 γ − x2 Pδ

  t f (t)dt = g(x) (x > 0). x

x

Theorem 68. Let − ∞ < ν1 − Re(1 − γ1 ) < Re(γ1 ), Re(γ1 − 1) ≤ 0, − ∞ < ν2 − Re(1 − γ2 ) < Re(γ2 ), Re(γ2 − 1) ≤ 0, ..., − ∞ < νn − Re(1 − γn ) < Re(γn ), Re(γn − 1) ≤ 0. The following assertions hold: γ (a) There exists a one-to-one map Pδ,2 ∈ [Lν,2 , Lν−Re(1−γ ),2 ] such that the relation γ (5.75) holds for f ∈ Lν,2 and Re(s) = ν − Re(1 − γ ). If Re(γ − 1) = 0, then Pδ,2 is one-to-one on Lν,2 . γ (b) The transform Pδ,2 f does not depend on ν in the sense that if ν 1 and if the γ γ γ ) )γ transforms Pδ,2 +f and Pδ,2 f are defined in respective spaces Lν 1 ,2 , then Pδ,2 f = Pδ,2 f for f ∈ Lν 1 ,2 Lν 2 ,2 . γ (c) If Re(γ − 1) < 0, then for f ∈ Lν,2 , Pδ,2 f is given by (5.75). γ (d) Let λ ∈ Cn , h > 0, and f ∈ Lν,2 . If Re(λ) > (ν − Re(1 − γ ))h − 1, then Pδ,2 f is represented in the form &

' d γ Pδ,2 f (x) = 2γ −1 hx 1−(λ+1)/h x (λ+1)/h dx $  & γ 1 ' & 1−γ 1 ' " ∞ , 0,3 x  (−λ,h), 1− 2 , 2 , × H3,3  & 1+δ 1 ' & δ 1 ' 2 2 t −γ f (t)dt, t − 2 , 2 , (−λ−1,h) 2 ,2 , 0

while for Re(λ) < (ν − Re(1 − γ ))h − 1 it is given by &

' x γ Pδ,2 f (x) = −2γ −1 hxalign1−(λ+1)/h x (λ+1)/h dx $  & " ' & 1−γ 1 ' ∞ γ 1 2 , 2 , (−λ,h) 1,2 x  1− 2 , 2 , & ' & δ 1 ' t −γ f (t)dt. × H3,3  1 t (−λ−1,h), 1+δ , −2,2 2 2 , 0

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Transmutations, Singular and Fractional Differential Equations

(e) If f ∈ Lν,2 and g ∈ L1−ν+Re(1−γ ),2 , then the following relation holds: ∞

& γ ' f (x) Pδ,2 g (x)dx =

0

∞

& γ ' 2γ −1 P∗ δ,2 f (x)g(x)dx,

0

& γ ' where P∗ δ,2 f is given by &

γ P∗ δ,2 f

' (x) =

∞ &

t 2 − x2

'−γ /2

γ

Pδ

  x f (t)dt = g(x) (x > 0). t

x

γ

5.4.4 Inversion formulas for transforms Pδ,k f (k = 1, 2) Let α0 = 0, β0 = ∞, α0 = 1 + max[Re(δ − 1), Re(−δ − 2)], β0 = ∞, respectively, for the operators (5.74) and (5.75). γ Inversion formulas for Pδ,1 f take the following forms: d −(λ+1)/h x f (x) = −21−γ hx (λ+1)/h−1+γ dx $  & γ +δ 1 ' & γ −δ−1 1 ' " ∞ & γ ' ,2 2 2,1 t  −(λ,h), 2 , 2 , × H3,3  & 1 1 ' & 1 ' Pδ,1 f (t)dt x 0, 2 , (−λ−1,h) 2,2 ,

(5.94)

0

or d f (x) = 21−γ hx (λ+1)/h−1 x −(λ+1)/h dx " $ & ' & γ −δ−1 1 ' ∞ γ +δ 1 ' , 2 , (−λ,h) & γ 2 ,2 , 2 3,0 t  &1 1' & 1' × H3,3  Pδ,1 f (t)dt. x (−λ−1,h), 2 , 2 , 0, 2

(5.95)

0

γ

Inversion formulas for Pδ,4 f take the following forms: d −(λ+1)/h x f (x) = −21−γ hx (λ+1)/h−1+γ dx $  & γ −1 1 ' & γ 1 ' " ∞ & γ ' 2,1 t  (−λ,h), 2 , 2 , 2 , 2 × H3,3  & δ 1 ' & δ+1 1 ' Pδ,4 f (t)dt x − 2 , 2 , 2 , 2 , (−λ−1,h) 0

(5.96)

Buschman–Erdélyi integral and transmutation operators

243

or d f (x) = 21−γ hx (λ+1)/h−1 x −(λ+1)/h dx " $  & ' &γ 1' ∞ γ −1 1 ' & 2 ,2 , 2 , 2 , (−λ,h) 3,0 t  & δ 1 ' & δ+1 1 ' Pγδ,4 f (t)dt. × H3,3  x (−λ−1,h), − 2 , 2 , 2 , 2

(5.97)

0

Theorem 69. Let Re(γ ) = 1, −∞ < ν < min[1, Re(2 + δ), Re(1 − δ)], and let λ ∈ Cn , h > 0. If f ∈ Lν,2 , then the inversion formulas (5.94) and (5.95) are valid for Re(λ) > (1 − ν)h − 1 and Re(λ) < (1 − ν)h − 1, respectively. Theorem 70. Let Re(γ ) = 1, −∞ < ν < min[1, Re(1−δ), Re(2+δ)], and let λ ∈ Cn , h > 0. If f ∈ Lν,2 , then the inversion formulas (5.96) and (5.97) are valid for Re(λ) > (1 − ν)h − 1 and Re(λ) < (1 − ν)h − 1, respectively.

Integral transforms composition method for transmutations

6

In this chapter we study applications of the integral transforms composition method (ITCM) for obtaining transmutations via integral transforms. It is possible to derive a wide range of transmutation operators by this method. Classical integral transforms are involved in the ITCM as basic blocks; among them are Fourier, sine- and cosineFourier, Hankel, Mellin, Laplace, and some generalized transforms. The ITCM and transmutations obtained by it are applied to derive connection formulas for solutions of singular differential equations and more simple nonsingular ones. We consider wellknown classes of singular differential equations with Bessel operators, such as the classical and generalized Euler–Poisson–Darboux equations and the generalized radiation problem of A. Weinstein. Methods of this chapter are applied to more general linear partial differential equations with Bessel operators, such as multivariate Bessel type equations, generalized axially symmetric potential theory (GASPT) equations of A. Weinstein, Bessel type generalized wave equations with variable coefficients, Bultrahyperbolic equations, and others. So with many results and examples the main conclusion of this chapter is illustrated: the ITCM of constructing transmutations is a very important and effective tool also for obtaining connection formulas and explicit representations of solutions to a wide class of singular differential equations, including ones with Bessel operators.

6.1 Basic ideas and definitions of the integral transforms composition method for the study of transmutations To construct transmutation operators, the ITCM, introduced and thoroughly developed in [146,229,230,234,524,533,535], can be used. The essence of this method is to construct the necessary transmutation operator and corresponding connection formulas among solutions of perturbed and nonperturbed equations as a composition of classical integral transforms with properly chosen weighted functions.

6.1.1 Background of ITCM We note that other possible generalizations of considered equations are equations with fractional powers of the Bessel operator considered in [89,252,367,515,516,527,531, 555]. In fractional differential equations theory the so-called “principle of subordination” was proposed (cf. [12,13,118,458]). In the cited literature the principle of subordination is reduced to formulas relating the solutions to equations of various fractional orders. Special cases of the subordination principle are formulas connecting Transmutations, Singular and Fractional Differential Equations With Applications to Mathematical Physics https://doi.org/10.1016/B978-0-12-819781-3.00012-4 Copyright © 2020 Elsevier Inc. All rights reserved.

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Transmutations, Singular and Fractional Differential Equations

solutions of fractional differential equations to solutions of integer order equations. Such formulas are also in fact parameter shift formulas, in which the parameter is the order of the fractional differential equation. So the popular “principle of subordination” may be considered as an example of parameter shift formulas, and consequently is in close connection with transmutation theory and the ITCM developed here. Note that we specially restrict ourselves to linear problems, but of course nonlinear problems are also very important (cf. [469,470] for further references). In transmutation theory explicit operators have been derived based on different ideas and methods, often not connecting altogether. So there is an urgent need in transmutation theory to develop a general method for obtaining known and new classes of transmutations. We give such general method for constructing transmutation operators. It is the integral transforms composition method (ITCM). The method is based on the representation of transmutation operators as compositions of basic integral transforms. The ITCM gives the algorithm not only for constructing new transmutation operators, but also for all now explicitly known classes of transmutations, including Poisson, Sonine, Vekua–Erdélyi–Lowndes, Buschman–Erdélyi, Sonin–Katrakhov, and Poisson– Katrakhov ones (cf. [51–53,56,229,230,524,533,535] as well as the classes of elliptic, hyperbolic, and parabolic transmutation operators introduced by R. Carroll [51–53]).

6.1.2 What is ITCM and how to use it? The formal algorithm of ITCM is the following. Let us take as input a pair of arbitrary operators A, B, and also connecting with them generalized Fourier transforms FA , FB , which are invertible and act by the formulas FA A = g(t)FA , FB B = g(t)FB ,

(6.1)

where t is a dual variable and g is an arbitrary function with suitable properties. It is often convenient to choose g(t) = −t 2 or g(t) = −t α , α ∈ R. Then the essence of the ITCM is to obtain formally a pair of transmutation operators P and S as the method output by the following formulas: S = FB−1

1 FA , w(t)

P = FA−1 w(t)FB ,

(6.2)

with arbitrary function w(t). When P and S are transmutation operators intertwining A and B, SA = BS,

P B = AP .

(6.3)

A formal checking of (6.3) can be obtained by direct substitution. The main difficulty is the calculation of compositions (6.2) in an explicit integral form, as well as the choice of domains of operators P and S. Also, we should note that the formulas in (6.2) are formal and the situation is possible when one operator, for example P , exists and is generated by the formula P = FA−1 w(t)FB , but its inverse operator S cannot

Integral transforms composition method for transmutations

247

1 be constructed by the formula FB−1 w(t) FA since this integral, for example, diverges. In this case if it is needed to construct an inverse operator for P it is necessary to use some regularization methods. Let us list the main advantages of the ITCM.

• Simplicity – many classes of transmutations are obtained by explicit formulas from elementary basic blocks, which are classical integral transforms. • The ITCM gives by a unified approach all previously explicitly known classes of transmutations. • The ITCM gives by a unified approach many new classes of transmutations for different operators. • The ITCM gives a unified approach to obtain both direct and inverse transmutations in the same composition form. • The ITCM directly leads to estimates of norms of direct and inverse transmutations using known norm estimates for classical integral transforms on different functional spaces. • The ITCM directly leads to connection formulas for solutions to perturbed and unperturbed differential equations. Some obstacle to apply ITCM is the following one. We know classical integral transforms usually act on standard spaces like L2 , Lp , C k , variable exponent Lebesgue spaces [465], and so on. But for the application of transmutations to differential equations we usually need some more conditions to hold, say, at zero or at infinity. For these problems we may first construct a transmutation by the ITCM and then expand it to the needed functional classes. Let us stress that formulas of the type (6.2) of course are not new for integral transforms and their applications to differential equations. But the ITCM is new when applied to transmutation theory! In other fields of integral transforms and connected differential equations theory compositions (6.2) for the choice of the classical Fourier transform leads to famous pseudodifferential operators with symbol function w(t). For the choice of the classical Fourier transform and the function w(t) = (±it)−s we get fractional integrals on the whole real axis, for w(t) = |x|−s we get the Riesz potential, for w(t) = (1 + t 2 )−s in (6.2) we get the Bessel potential, and for w(t) = (1 ± it)−s we obtain modified Bessel potentials [494]. The choice for the ITCM algorithm A = B = Bν , FA = FB = Hν , g(t) = −t 2 , w(t) = jν (st)

(6.4)

leads to generalized translation operators of Delsart [315,319,321]. For this case we have to choose in the ITCM algorithm defined by (6.1)–(6.2) the above values (6.4) in which Bν is the Bessel operator (1.87), Fν is the Hankel transform (1.56), and jν is the normalized (or “small”) Bessel function (1.19). In the same manner other families of operators commuting with a given one may be obtained by the ITCM for the choice A = B, FA = FB with arbitrary functions g(t), w(t) (generalized translation commutes with the Bessel operator). In the case of the choice of differential operator A as quantum oscillator and the connected integral transform FA as fractional or quadratic Fourier transform [437], we may obtain by the ITCM transmutations also

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Transmutations, Singular and Fractional Differential Equations

for this case [230]. It is possible to apply the ITCM instead of classical approaches for obtaining fractional powers of Bessel operators [230,515,516,527,531]. Direct applications of the ITCM to multi-dimensional differential operators are obvious; in this case t is a vector and g(t), w(t) are vector functions in (6.1)–(6.2). Unfortunately for this case we know and may derive some new explicit transmutations just for simple special cases. But among them are well-known and interesting classes of potentials. In the case of using the ITCM by (6.1)–(6.2) with Fourier transform when w(t) is a positive definite quadratic form, we come to elliptic Riesz potentials [475,494]; when w(t) is an indefinite quadratic form we come to hyperbolic Riesz potentials [426,475,494]; when w(x, t) = (|x|2 − it)−α/2 we come to parabolic potentials [494]. In the case of using the ITCM by (6.1)–(6.2) with Hankel transform and when w(t) is a quadratic form we come to elliptic Riesz B-potentials [206,344] or hyperbolic Riesz B-potentials [503]. For all abovementioned potentials we need to use distribution theory and consider for the ITCM convolutions of distributions; for inversion of such potentials we need some cutting and approximation procedures (cf. [426,503]). For this class of problems it is appropriate to use Schwartz and/or Lizorkin spaces for probe functions and dual spaces for distributions. So we may conclude that the ITCM we consider in this chapter for obtaining transmutations is effective, it is connected to many known methods and problems, it gives all known classes of explicit transmutations, and it works as a tool to construct new classes of transmutations. Application of the ITCM requires the following three steps. • Step 1. For a given pair of operators A, B and connected integral transforms FA , FB , define and calculate a pair of transmutations P , S by basic formulas (6.1)–(6.2). • Step 2. Derive exact conditions and find classes of functions for which transmutations obtained by step 1 satisfy proper intertwining properties. • Step 3. Apply now correctly defined transmutations by steps 1 and 2 on proper classes of functions to derive connection formulas for solutions of differential equations. Based on this plan the next part of the chapter is organized as follows. First we illustrate step 1 of the above plan and apply the ITCM for obtaining some new and known transmutations. For step 2 we prove a general theorem for the case of Bessel operators; it is enough to complete strict definitions of necessary transmutations and start to solve problems using them. After that we give an example to illustrate step 3 of applying transmutations obtained by ITCM to derive formulas for solutions of a model differential equation.

6.2 Application of the ITCM to derive transmutations connected with the Bessel operator The topic of this section is the application of the ITCM to obtain different classes of transmutations connected with the Bessel operator.

Integral transforms composition method for transmutations

249

6.2.1 Index shift for the Bessel operator Here we derive connection formulas of Bessel operators with the indices μ and ν. Such relations are called parameter shift formulas and such operator is called index shift transmutation. Aforesaid formulas arise when the classical wave equation is solved by the mean values method. The descent parameter in this case is the space dimension. Essentially, such parameter shift formulas define transmutation operators which are responsible for connection formulas among solutions of perturbed and nonperturbed equations. In this subsection we apply the ITCM to obtain integral representations for index shift transmutations. This corresponds to step 1 of the above plan for the ITCM algorithm. Let us look at the operator T transmuting the operator Bν defined by (1.87) into the same operator but with another parameter Bμ . To find such a transmutation we use the ITCM with Hankel transform. Applying the ITCM we obtain an interesting and important family of transmutations, including index shift transmutations, “descent” operators, classical Sonine and Poisson type transmutations, explicit integral representations for fractional powers of the Bessel operator, generalized translations of Delsart, and others. (ϕ) So we are looking for an operator Tν, μ such that Tν,(ϕ)μ Bν = Bμ Tν,(ϕ)μ ,

(6.5)

or in the form factorized by the ITCM,   Tν,(ϕ)μ = Hμ−1 ϕ(t)Hν ,

(6.6)

where Hν is a Hankel transform (1.59). Assuming ϕ(t) = Ct α , C ∈ R does not depend (ϕ) (α) on t, and Tν, μ = Tν, μ , we derive the following theorem. Theorem 71. Let f ∈ L2 (0, ∞), Re (α + μ + 1) > 0,

  μ−ν Re α + < 0. 2 (α)

Then for the transmutation operator Tν, μ obtained by the ITCM and such that Tν,(α)μ Bν = Bμ Tν,(α)μ , the following integral representation is true:     2α+3  α+μ+1 2   × Tν,(α)μ f (x) = C ·  μ+1 2 ⎡ x  

−1−μ−α ν + 1 y2 α+μ+1 α ⎣x  f (y)2 F1 , + 1; ; 2 y ν dy+ 2 2 2 x  − α2 0

250

Transmutations, Singular and Fractional Differential Equations

  

 μ+1 2



2 F1

ν+1 2



∞

   ν−μ−α 2

f (y)× x

  μ + 1 x2 α+μ+1 α+μ−ν , + 1; ; 2 y ν−μ−α−1 dy , 2 2 2 y

(6.7)

where 2 F1 is the Gauss hypergeometric function. Proof. We have 



Tν,(α)μ f (x) = C · Hμ−1 t α Hν [f ](t) (x) =

C·

C·

21−μ  

2 2

μ+1 2

ν−μ 2 +2

 

C·

2

 C·

2

 C·

2







μ+1 2

ν−μ 2 +2







μ+1 2

ν−μ 2 +2







μ+1 2

ν−μ 2 +2



∞





μ+1 2

∞ j μ−1 (xt) t

μ+α

j ν−1 (ty)f (y)y ν dy =

dt

2

2

0 ν+1 2





ν+1 2





2

0

x

0 1−μ 2

∞ y

ν+1 2

∞ t α+1+

f (y)dy

μ−ν 2

J μ−1 (xt)J ν−1 (ty)dt = 2

0





ν+1 2

∞

∞ 1−μ 1−ν μ+α 2 (xt) J μ−1 (xt)t dt (ty) 2 J ν−1 (ty)f (y)y ν dy = 2



ν+1 2

0

x

1−μ 2

x y

ν+1 2

∞ t α+1+

f (y)dy

μ−ν 2

J μ−1 (xt)J ν−1 (ty)dt+ 2

0

 x

1−μ 2

2

0

2

0

∞ y

ν+1 2

∞ t α+1+

f (y)dy

μ−ν 2

J μ−1 (xt)J ν−1 (ty)dt. 2

x

2

0

Using formula (2.12.31.1) from [456], p. 209, of the form

∞ t β−1 Jρ (xt)Jγ (yt) dt = 0

  ⎧    γ +ρ+β 2 ⎪ 2 ⎪ β−1 −γ −β γ   2 F1 γ +ρ+β , γ −ρ+β ; γ + 1; y 2 ⎪ x y 0 < y 0, Re β < 2

Integral transforms composition method for transmutations

and putting β = α + have

∞ t α+1+

μ−ν 2

μ−ν 2

+ 2, ρ =

γ =

ν−1 2

we obtain the formula (6.7). We

J μ−1 (xt)J ν−1 (ty)dt = 2

2

0

⎧ μ−ν ν−1 α+1+ 2 ⎪ 2 y 2 ⎪ ⎪ ⎪ ⎨ x α+2− 1−μ 2 μ−ν μ−1 ⎪ α+1+ 2 ⎪ 2 x 2 ⎪ ⎪ ⎩ μ+α− ν−3

y

μ−1 2 ,

251

2

  

α+μ+1 2







2 α+μ+1 α   , 2 + 1; ν+1 ; yx 2 F 2 2 ν+1 α 2 1  2  −2      α+μ+1 2 2    2 F1 α+μ+1 , α+μ−ν + 1; μ+1 ; x 2  2 2 2 y  ν−μ−α  μ+1 2 2

Re (α + μ + 1) > 0,



μ−ν Re α + 2

0 < y < x, 0 < x < y,



0 we obtain the operator  

x   2 μ+1 μ 2 (−μ)  μ x 1−μ f (y)(x 2 − y 2 ) 2 −1 dy, (6.9) T0, μ f (x) = √ π 2 0

such that (−μ)

(−μ)

T0, μ D 2 = Bμ T0, μ

(6.10)

(−μ)

and T0, μ 1 = 1, Proof. From Theorem 71 we obtain 

(−μ) T0, μ f



√  

x 23−μ π μ 1 y2 1  f (y)2 F1 (x) = C · , 1 − ; ; 2 dy.   2 2 2 x x μ2  μ+1 2 0

Using formula (6.8) we get  2 F1

μ 1 y2 1 ,1 − ; ; 2 2 2 2 x



  μ −1 μ y2 2 = x 2−μ (x 2 − y 2 ) 2 −1 = 1− 2 x

and 

(−μ) T0, μ f



√ x μ x 1−μ 23−μ π  f (y)(x 2 − y 2 ) 2 −1 dy. (x) = C ·    μ2  μ+1 2 0

It is easy to see that

x x

(x − y )

1−μ

2

2

μ 2 −1

1 dy = {y = xz} =

0

0

1 {z = t} = 2

1 (1 − t)

2

μ 2 −1

0

and taking C =

μ

(1 − z2 ) 2 −1 dz =



μ+1 2 2−μ 2 π

2

t

− 12

 π μ2   dt = 2 μ+1 2 √

 (−μ)

we get T0, μ 1 = 1. This completes the proof.

The operator (6.9) is the well-known Poisson operator (3.120). We will use the μ conventional symbol Px for it:

x Pxμ f (x) = C(μ)x 1−μ 0

μ

f (y)(x 2 − y 2 ) 2 −1 dy,

(6.11)

Integral transforms composition method for transmutations

253

  2 μ+1 2  . C(μ) = √ π μ2

Pxμ 1 = 1,

Remark 11. It is easy to see that if u = u(x, t), x, t ∈ R, and ut (x, 0) = 0,

u(x, 0) = f (x), then μ Pt u(x, t)|t=0

  ∂ μ = 0. Pt u(x, t) ∂t t=0

= f (x),

(6.12)

Indeed, we have μ Pt u(x, t)|t=0

t = C(μ)t

u(x, y)(t − y )

1−μ

2

0

1 C(μ)

2

μ 2 −1

  dy 

= t=0

μ

u(x, ty)|t=0 (1 − y 2 ) 2 −1 dy = f (x)

0

and 

1  μ ∂ μ = C(μ) ut (x, ty)|t=0 (1 − y 2 ) 2 −1 dy = 0. Pt u(x, t) ∂t t=0 0

Statement 15. Let f ∈ L2 (0, ∞), α = 0, μ = 0. In this case for μ > 0 we obtain the operator 

 (0) Tν, 0 f (x) =

√

∞ ν 21−γ π    f (y)(y 2 − x 2 ) 2 −1 dy ν+1 ν  2  2 x

(6.13)

with the intertwining property (0)

(0)

Tν, 0 Bν = D 2 Tν, 0 . Proof. Using formula (6.8) we obtain    

∞    ν+1 ν 1 x2 1 (0) 3  2 f (y)2 F1 , 1 − ; ; 2 y ν−1 dy = Tν, 0 f (x) = C · 2  2 2 2 y  12  ν2 x     ν −1

∞  ν+1 x2 2 3  2 f (y) 1 − 2 y ν−1 dy = C ·2 ν 1 y   2

2

x

254

Transmutations, Singular and Fractional Differential Equations



 C · 23 √ Putting C =



ν+1

∞ 2  f (y)(y 2 π ν2 x

π , 2γ +2  2 ν+1 2

ν

− x 2 ) 2 −1 dy.

we get (6.13).

Statement 16. For f ∈ L2 (0, ∞), α=ν−μ, −1 0,

k ∈ R,

(7.4) (7.5)

in the compact integral form via generalized translation and spherical mean operators for all values of the parameter k, including also exceptional odd negative values, which have not been studied before.

Differential equations with Bessel operator

277

2 , x ∈ Rn . Then for the case k > n + |γ | − 1 the Theorem 75. Let f = f (x) ∈ Cev + unique solution to (7.4)–(7.5) is    2n  k+1 k−n−|γ |−1 2 ty 2 u(x, t; k) =  [ γ Tx f (x)](1−|y|2 ) y γ dy    n k−n−|γ |+1 γi +1   2 B + (n) 2

 

= 

2

1

i=1

k+1 2

k−n−|γ |+1 2



1

   |  n+|γ 2

(1 − r 2 )

k−n−|γ |−1 2

Mtr [f (x)]r n+|γ |−1 dr. γ

0

(7.6) The unique solution of the problem (7.4)–(7.5) for k=n+|γ |−1 is the weighted spherγ ical mean Mt [f (x)] (see (3.183)). Proof. Using Theorem 36 and the property (3.189) we obtain that the weighted spherγ ical mean Mt [f (x)] satisfies the general Euler–Poisson–Darboux equation γ

γ

(Bk )t Mt [f (x)] = (γ )x Mt [f (x)], and initial conditions γ

M0 [f (x)] = f (x),

  γ Mt [f (x)]

k = n + |γ | − 1

= 0. t=0 γ

It means the weighted spherical mean Mt [f (x)] is the solution of the problem (7.4)–(7.5) for k = n + |γ | − 1. In order to obtain the solution of (7.4)–(7.5) for k > n + |γ | − 1, we will use the method of descent. First, we will seek a solution of the Cauchy problem (7.4)–(7.5) for the case k > n + |γ |. n+1  ), γ   Let γ  = (γ1 , ..., γn , γn+1 n+1 > 0, x = (x1 , ..., xn+1 ) ∈ R+ , and  (γ  )x  = (Bγ1 )x1 + ... + (Bγn )xn + (Bγn+1 )xn+1 .

Consider the equation of type (7.4) (Bk )t u = (γ  )x  u,

u = u(x  , t; k),

x  ∈ Rn+1 + ,

t >0

with the initial conditions u(x  , 0; k) = f1 (x  ),

ut (x  , 0; k) = 0.

 , the weighted spherical mean M [f (x  )] is a When k = n + |γ  | = n + |γ | + γn+1 1 t solution of this Cauchy problem: γ

u(x  , t; k) = 1 |S1+ (n + 1)|γ 

 S1+ (n+1)





n+1 [ γ1 Tx1 1 ... γn Txn n γn+1 Txn+1 f1 (x)](y  )γ dSy  ,

ty

ty

ty

(7.7)

278

Transmutations, Singular and Fractional Differential Equations

y  = (y1 , ..., yn , yn+1 ) ∈ Rn+1 + ,     γ  +1     n n   |+1  γi 2+1  n+12  γi 2+1  k−n−|γ 2 i=1 i=1   . |S1+ (n + 1)|γ  =  n+1+|γ |+γ   = n  k+1 n+1 n 2 2  2 2 Let us put f1 (x1 , ..., xn , 0) = f (x1 , ..., xn ), where f is the function which appears in initial conditions (7.5). In this way, u defined by (7.7) becomes a function only of x1 , ..., xn which satisfies Eq. (7.4) and initial conditions (7.5). We have u(x, t; k) =

1 |S1+ (n + 1)|γ 





[ γ Tx f (x)](y  )γ dSy  , ty

 γn+1 = k−n−|γ |.

S1+ (n+1)

Now we rewrite the integral over the part of the sphere S1+ (n + 1) as an integral over n  the part of the ball B1+ (n)={y∈Rn+ : yi2 ≤ 1}. We write the surface integral as a i=1

multiple integral: 



[ γ Tx f (x)](y  )γ dSy  = ty

S1+ (n+1)



ty

[ γ Tx f (x)](1 − y12 − ... − yn2 )

 −1 γn+1 2

y γ dy

B1+ (n)



=

ty

[ γ Tx f (x)](1 − |y|2 )

k−n−|γ |−1 2

y γ dy,

B1+ (n)

where B1+ (n) is a projection of S1+ (n + 1) on the equatorial plane xn+1 = 0. We have    2n  k+1 k−n−|γ |−1 2 ty 2 [ γ Tx f (x)](1−|y|2 ) y γ dy. u(x, t; k)= n      γi +1 k−n−|γ |+1  2  2 B + (n) 1

i=1

(7.8) Although (7.8) was obtained as the solution of the problem (7.4)–(7.5) for the case k>n+|γ |, the integral on its right side converges for k > n + |γ | − 1. We can verify by direct substitution of (7.8) in (7.4)–(7.5) that (7.8) satisfies the differential equation (7.4) and the initial conditions (7.5) for all values of k which are greater than (n + |γ | − 1). Let us show this. Changing coordinates from y to y/t and using that y y (Bγi )xi γi Txii = (Bγi )yi γi Txii (see (3.146)), we obtain  I = (γ )x

ty

[ γ Tx f (x)](1 − |y|2 ) B1+ (n)

k−n−|γ |−1 2

y γ dy =

Differential equations with Bessel operator n 



ty

[ γ Tx f (x)](1 − |y|2 )

(Bγi )xi

i=1

t 1−k

279

k−n−|γ |−1 2

y γ dy =

B1+ (n)

 n 

y

k−n−|γ |−1 2

y

k−n−|γ |−1 2

[(Bγi )xi γ Tx f (x)](t 2 − |y|2 )

y γ dy =

i=1 + Bt (n)

t

1−k

 n 

[(Bγi )yi γ Tx f (x)](t 2 − |y|2 )

y γ dy,

(7.9)

i=1 + Bt (n)

where Bt+ (n)={y∈Rn+ :

n  i=1

yi2 ≤ t}.

2 ( + ) integrable over  + , we have formula (1.101). By For the functions w, v∈Cev applying formula (1.101) to the right side of relation (7.9), we get

I = t 1−k



n   i=1 + St (n)

k−n−|γ |−1 ∂ γ y 2 Tx f (x) (t 2 − |y|2 ) cos( ν , e i ) y γ dS, ∂yi

where e i is the direction of the axis Oyi , i = 1, ..., n, and thus cos( ν , e i ) = yti . Now, by using the fact that the direction of the outward normal to the boundary of a ball with center the origin coincides with the direction of the position vector of the point on the ball, we obtain the relation   k−n−|γ |−1 1 ∂ k∂ γ ty 2 Tx f (x) (1 − |y|2 ) y γ dy. I= k t t ∂t ∂t B1+ (n)

Given that

1 ∂ k ∂ t t k ∂t ∂t



= (Bk )t and (7.9) we have ty

[ γ Tx f (x)](1 − |y|2 )

(γ )x B1+ (n)



(Bk )t



γ

k−n−|γ |−1 2

y γ dy =

k−n−|γ |−1 ty 2 Tx f (x) (1 − |y|2 ) y γ dy.

B1+ (n)

It means that u(x, t; k) defined by formula (7.8) indeed satisfies Eq. (7.4) for k>n + |γ | − 1. In order to obtain the representation (7.6) it is necessary to use spherical coordinates y = rθ. Validity of the first and second initial conditions follows from the γ property (3.189) for Mtr [f (x)] and from the formula     | |+1 1  n+|γ  k−n−|γ k−n−|γ |−1 2 2 2   (1 − r 2 ) r n+|γ |−1 dr = . k+1 2 2 0

280

Transmutations, Singular and Fractional Differential Equations

Remark 12. It is easy to see that the solution of (7.4)–(7.5) can be obtained using the first “descent” operator (6.14) when μ = k, ν = n + |γ | − 1, ν < μ applied by the variable t:   (n+|γ |−1−k) γ u(x, t; k)= Tn+|γ |−1, k Mt [f (x)] (t). 

n+|γ |−k 2



+2

. Then the solution of (7.4)–(7.5) for k < Theorem 76. Let f = f (x) ∈ Cev n + |γ | − 1, k = −1, −3, −5, ... is 

∂ m k+2m−1 u(x, t; k) = t 1−k (t u(x, t; k + 2m)), (7.10) t∂t where m is a minimum integer such that m ≥ solution of the Cauchy problem

n+|γ |−k−1 2

and u(x, t; k + 2m) is the

(Bk+2m )t u = (γ )x u, u(x, 0; k + 2m) =

(7.11)

f (x) , (k + 1)(k + 3)...(k + 2m − 1)

ut (x, 0; k + 2m) = 0. (7.12)

Proof. In order to proof that (7.10) is the solution of (7.4)–(7.5) when k 0,

k ∈ R,

(7.17) (7.18)



Theorem 78. If g = g(x) ∈ Cev , then the solution u = u(x, t; k) of (7.17)– (7.18) for k < 1 is given by     1−k q   3−k+2q  Cqs t 1−k+2s 2 2  ×    u(x, t; k) =  | | 3−k s  3−k+2q−n−|γ  n+|γ s=0 2  2 2 2 +s 1 (1 − r ) 2

0

1−k+2q−n−|γ | 2

r

n+|γ |−1

1 ∂ t ∂t

s

γ

Mtr g(x)dr

(7.19)

Differential equations with Bessel operator

283

if n + |γ | + k is not an odd integer and   

q Cqs  1−k  2 1 ∂ s γ 1−k+2s  t Mt g(x) u(x, t; k) = t ∂t s+1  3−k + s s=0 2 2

(7.20)

if n + |γ | + k is an odd integer, where q ≥ 0 is the smallest positive integer such that 2 − k + 2q ≥ n + |γ | − 1. Proof. Let  q≥0 be the smallest positive integer number such that 2−k+2q≥n+|γ |−1, n+|γ |+k−1 i.e., q= , and let u(x, t; 2 − k + 2q) be a solution of (7.17) when we take 2 2 − k + 2q instead of k such that u(x, 0; 2 − k + 2q) = g(x),

ut (x, 0; 2 − k + 2q) = 0.

(7.21)

By the recurrent formula (1.109) we obtain that u(x, t; k − 2q) = t 1−k+2q u(x, t; 2 − k + 2q) is a solution of the equation (γ )x u =

∂ 2 u k − 2q ∂u + . t ∂t ∂t 2

Further, applying q times the formula (1.110) we obtain that

 

1 ∂ q 1−k+2q 1 ∂ q u(x, t; k − 2q) = (t u(x, t; 2 − k + 2q)) t ∂t t ∂t is a solution of (7.17). Let us consider u(x, t; k) =

2−q  (1 − k)

 

3−k 2



3−k+2q 2



1 ∂ t ∂t

q (t 1−k+2q u(x, t; 2 − k + 2q)). (7.22)

We have shown that (7.22) satisfies Eq. (7.17). Now we will prove that u(x, t; k) given by (7.22) satisfies the conditions (7.21). q For u ∈ Cev (+ ) we have the formula (see [564], p. 9) 

1 ∂ q 1−k+2q (t u(x, t; 2 − k + 2q)) = t ∂t   

q 2q−s C s  1−k + q + 1  q 2 1 ∂ s   u(x, t; 2 − k + 2q). (7.23) t 1−k+2s t ∂t  1−k + s + 1 s=0

2

284

Transmutations, Singular and Fractional Differential Equations

Taking into account formula (7.23) we obtain u(x, 0; k) = 0 and lim t k ut (x, t; k) =  

 2−q  3−k 2 ∂ 1 ∂ q 1−k+2q   lim t k (t u(x, t; 2 − k + 2q)) = t→0 ∂t t ∂t (1 − k) 3−k+2q 2     3−k −q q 2q−s C s  1−k + q + 1 2  2  q 2 ∂     lim t k × 3−k+2q t→0 ∂t 1−k (1 − k)  2 +s +1 s=0 2 s

1−k+2s 1 ∂ u(x, t; 2 − k + 2q) = t t ∂t  1 ∂  1−k lim t k t u(x, t; 2 − k + 2q) = 1 − k t→0 ∂t   1 lim t k (1 − k)t −k u(x, t; 2 − k + 2q) + t 1−k ut (x, t; 2 − k + 2q) = 1 − k t→0 1 lim ((1 − k)u(x, t; 2 − k + 2q) + tut (x, t; 2 − k + 2q)) = g(x). 1 − k t→0

t→0

Now let us obtain the representation of u(x, t; k) through the integral. Using formula (7.6) we get u(x, t; 2 − k + 2q) =   1 2 3−k+2q 1−k+2q−n−|γ | 2 γ 2     (1 − r 2 ) r n+|γ |−1 Mtr g(x)dr. 3−k+2q−n−|γ | n+|γ |   2 2 0 If 2 − k + 2q > n + |γ | − 1, then applying (7.22) and (7.23) we obtain   2−q  3−k 2  × u(x, t; k) = (1 − k) 3−k+2q 2   1−k q−s s 

q 2 Cq  2 + q + 1  1 ∂ s 1−k+2s   t u(x, t; 2 − k + 2q) = t ∂t  3−k s=0 2 +s   

q  3−k  Cqs t 1−k+2s 2 1 ∂ s   u(x, t; 2 − k + 2q) = 3−k 1−k t ∂t s s=0 2  2 +s     1−k q  3−k+2q   Cqs t 1−k+2s 2 2   ×    | n+|γ | s  3−k + s  3−k+2q−n−|γ 2  s=0 2 2 2

Differential equations with Bessel operator

1 (1 − r ) 2

1−k+2q−n−|γ | 2

r

n+|γ |−1

285

1 ∂ t ∂t

s

γ

Mtr g(x)dr.

0 γ

If 2 − k + 2q = n + |γ | − 1, then u(x, t; 2 − k + 2q) = Mt g(x) and   

 2−q  3−k 2 1 ∂ q  n+|γ |−2 γ   Mt g(x) = t u(x, t; k) = t ∂t (1 − k) 3−k+2q 2     

q 2q−s C s  3−k + q 2−1−q  1−k  q 2 2 1 ∂ s γ 1−k+2s     Mt g(x) = t t ∂t  3−k+2q  3−k s=0 2 2 +s   

q Cqs  1−k  2 1 ∂ s γ   t 1−k+2s Mt g(x). t ∂t 2s+1  3−k + s s=0

2

This completes the proof. Remark 13. A solution of the problem close to (7.4)–(7.5) was obtained in [147] (see also [56], p. 243) when k = −1, −3, −5, ... in terms of the Lauricella function (1.37). More precisely, in [147] the solution of the problem   n λi ∂ 2 v k ∂v  ∂ 2 v + + v = 0, v = v(t, x), (7.24) − t ∂t ∂t 2 ∂xi2 xi2 i=1  ∂v  v(0, x) = T (x), =0 (7.25) ∂t t=0 has the form  v(t, x) =

 n 2

π 

k+1 2





 

k−n+1 2 |x−ξ |=|t|

|t|1−k (t 2 − |x − ξ |2 )

 k−n+1 F(n) , ..., a , b , ..., b ; , ..., z ; z a 1 n 1 n 1 n dSξ , γ 2 where √ √ 1 − 4λ1 1 + 1 − 4λn , ..., an , a1 = 2√ √2 1 − 1 − 4λ1 1 − 1 − 4λn b1 = , ..., bn = , 2 2 t 2 − |x − ξ |2 t 2 − |x − ξ |2 z1 = , ..., zn = . 2x1 ξ1 2xn ξn 1+

k−n−1 2

T (ξ )×

(7.26)

286

Transmutations, Singular and Fractional Differential Equations

γ1 γn   γ If λk = γ2k 1 − γ2k , i = 1, ..., n, and u = x 2 v = x12 ...xn2 v, then we get our problem (7.79)–(7.80). As we see, expression (7.81) gives a much more convenient formula for solving the problem (7.79)–(7.80). Also note that in [513] in two different ways including k = −1, −3, −5, ....

Now we concentrate on the case when x is one-dimensional. Then problems and solutions constructed above are simplified. For these problems we consider below some illustrative examples with explicit solution representations (see Section 7.1.4). In this case we have the first Cauchy problem ∂ 2 u γ ∂u ∂ 2 u k ∂u + = 2 + , x ∂x t ∂t ∂x 2 ∂t  ∂u(x, t; k)  = 0, u(x, 0; k) = f (x),  ∂t t=0

(7.27) 2 f (x) ∈ Cev .

(7.28)

When k > γ > 0 the solution of (7.27)–(7.28) is given by the formula (see (7.8))   1 2 k+1 k−γ −2 2    (1 − y 2 ) 2 γ Txty f (x)y γ dy. u(x, t; k)=  (7.29) k−γ γ +1  2  2 0 When k < γ the solution of (7.27)–(7.28) is found by formula (7.13), (7.14), or (7.15). As for the second Cauchy problem for k < 1, γ > 0, and 2 − k > γ , ∂ 2 u γ ∂u ∂ 2 u k ∂u + = 2 + , x ∂x t ∂t ∂x 2 ∂t lim u(x, t; k) = 0,

t→0

lim t k

t→0

(7.30) 

∂u(x, t; k) = g(x), ∂t

k+γ 2

g(x) ∈ Cev



.

(7.31)

The condition 2 − k > γ means that we can take q = 0 in (7.19). The solution of (7.30)–(7.31) for 2 − k > γ is defined by   1  1−k t 1−k k+γ 2    (1 − r 2 )− 2 γ Txtr f (x) r γ dr. u(x, t; k) =  2−k−γ γ +1   2 2 0

7.1.3 The singular Cauchy problem for the generalized homogeneous Euler–Poisson–Darboux equation In this subsection, we solve the singular Cauchy problem for a generalized form of a homogeneous Euler–Poisson–Darboux equation with constant potential, where the Bessel operator acts instead of each second derivative. In the classical formulation, the Cauchy problem for this equation is not correct. However, S. A. Tersenov [564] observed that, considering the form of a general solution of the classical Euler–Poisson– Darboux equation, the derivative in the second initial condition must be multiplied by

Differential equations with Bessel operator

287

a power function whose degree is equal to the index of the Bessel operator acting on the time variable. The first initial condition remains in the usual formulation. With the chosen form of the initial conditions, the considering equation has a solution. The obtained solution is represented as the sum of two terms. The first term is an integral containing the normalized Bessel function and the weighted spherical mean. The second term is expressed in terms of the derivative of the square of the time variable from the integral, which is similar in structure to the first term. We study the initial value problem   n    ∂2 γi ∂ k ∂ ∂2 + + (7.32) u = c2 u, − Lu = t ∂t ∂t 2 ∂xi2 xi ∂xi i=1

u(x, 0; k) = ϕ(x),

lim t k ut (x, t; k) = ψ(x),

t→+0

u = u(x, t; k),

(7.33)

where γi >0, xi >0, i=1, ..., n, k ∈ R, t>0. Eq. (7.32) is called the generalized Euler– Poisson–Darboux equation. Using the terminology from the book [56], a problem for the equation of the type A(t)

∂u ∂ 2u + B(t) + C(t)u = Gu, 2 ∂t ∂t

u = u(t, x),

x = (x1 , ..., xn ),

where G is a linear operator, acting only by variables x1 , ..., xn , is called singular if at least one of the operator coefficients tends to infinity in some sense as t → 0. In [56] five general techniques were given for the solution of the singular Cauchy problem n  ∂ 2u i=1

∂xi2

=

∂ 2 u k ∂u + , t ∂t ∂t 2

u(x, 0; k) = ϕ(x),

u = u(x, t; k),

ut (x, 0; k) = 0.

(7.34) (7.35)

These methods are: 1. 2. 3. 4. 5.

Fourier transform method in a distribution space, spectral technique in a Hilbert space, transmutation method, studying related simpler differential equations, and energy methods.

Some of these methods were successfully applied to the generalized Euler–Poisson– Darboux equation (7.32) and to Lu = 0 in other papers. Namely, using the Hankel transform instead of Fourier solutions to Lu = 0, (7.32) with conditions (7.35) was obtained in [514] and [508], accordingly. The third and the closely connected forth method were used to solve Lu = 0 in [514] and [509]. In [532] a transmutation method was used for obtaining new integral initial conditions for the Euler–Poisson– Darboux equation (7.34). Abstract differential equations with Bessel operator of Euler–Poisson–Darboux type were studied in [56] (cf. also [184]). In [577] the problem (7.34)–(7.33) was solved using “descent” operators, which are special cases of

288

Transmutations, Singular and Fractional Differential Equations

Buschman–Erdélyi transmutation operators (see [533,535]). Here as a main result we obtain a solution to the problem (7.32)–(7.33). We will be concerned with the solutions of the following singular initial value hyperbolic problem:  n     ∂2 γi ∂ k ∂ ∂2 Lu = + + (7.36) u = c2 u, − t ∂t ∂t 2 ∂xi2 xi ∂xi i=1

u(x, 0; k) = ϕ(x),

lim t k ut (x, t; k) = ψ(x),

t→+0

u = u(x, t; k).

We will call (7.36) the generalized Euler–Poisson–Darboux equation. In [508] the following theorem was proved. 2 (Rn+1 ) to Theorem 79. The solution u ∈ Cev +   (γ )x − (Bk )t u = c2 u, c > 0, u(x, 0; k) = ϕ(x), ut (x, 0; k) = 0

u = u(x, t; k),

(7.37) (7.38)

for k > n + |γ | − 1 is unique and defined by the formula t u(x, t; k) = A(n, γ , k) t

(t 2 − r 2 )

1−k

k−n−|γ |−1 2

   j k−n−|γ |−1 c t 2 − r 2 × 2

0

r n+|γ |−1 Mr [ϕ(x)]dr, γ

where

(7.39)

  2 k+1 2   . A(n, γ , k) =  | k−n−|γ |+1  n+|γ  2 2

It is proved in Chapter 1 that if u(x, t; k) is a solution to (7.37), then the following two fundamental recursion formulas (1.109) and (1.110) hold: u(x, t; k) = t 1−k u(x, t; 2 − k), 

n+|γ |−k 2

u(x, t; k)t = tu(x, t; 2 + k). +2

. Then the solution of (7.37)–(7.38) for Theorem 80. Let ϕ = ϕ(x), ϕ ∈ Cev k ≤ n + |γ | − 1, k = −1, −3, −5, ..., is 

 ∂ m  k+2m−1 1−k u(x, t; k) = t u(x, t; k + 2m) , (7.40) t t∂t where m is a minimum integer such that m ≥ n+|γ |−k−1 and u(x, t; k + 2m) is the 2 solution of the Cauchy problem   (γ )x − (Bk+2m )t u = c2 u, c > 0, (7.41)

Differential equations with Bessel operator

u(x, 0; k + 2m) =

289

ϕ(x) , (k + 1)(k + 3)...(k + 2m − 1)

ut (x, 0; k + 2m) = 0. (7.42)

Proof. In order to proof that (7.40) is a solution of (7.37)–(7.38) when k≤n + |γ |−1, k=−1, −3, −5, ..., we will use the recursion formulas (1.109) and (1.110). Let us choose the minimum integer m such that k + 2m > n + |γ | − 1. Now we can write the solution of the Cauchy problem   (γ )x − (Bk+2m )t u = c2 u, c > 0, u(x, 0; k + 2m) = g(x),

ut (x, 0; k + 2m) = 0,

2 g ∈ Cev

by (7.39). We have t u(x, t; k + 2m) = A(n, γ , k + 2m) t

(t 2 − r 2 )

1−k−2m

k+2m−n−|γ |−1 2

0

× j k+2m−n−|γ |−1 2

where

   γ c t 2 − r 2 r n+|γ |−1 Mr [g(x)]dr,

  2 k+2m+1 2   . A(n, γ , k + 2m) =  | |+1  n+|γ  k+2m−n−|γ 2 2

Considering (1.109), it is easy to see that t k+2m−1 u(x, t; k + 2m) = u(x, t; 2 − k − 2m). Applying (1.110) to the last formula m times we get

 ∂ m k+2m−1 (t u(x, t; k + 2m) = u(x, t; 2 − k). t∂t Applying again (1.109) we can write 

 ∂ m  k+2m−1 u(x, t; k) = t 1−k u(x, t; k + 2m) , t t∂t

(7.43)

which gives the solution of (7.41). Now we obtain the function g such that (7.42) is true. From (7.43) it follows that u(x, t; k) = (k + 1)(k + 3)...(k + 2m − 1)u(x, t; k + 2m) + C t u(x, t; k + 2m) + O(t 2 ), when t → 0, where C is a constant. Evidently, if

290

Transmutations, Singular and Fractional Differential Equations

g(x) =

ϕ(x) , (k + 1)(k + 3)...(k + 2m − 1)

then u(x, t; k) defined by (7.40) satisfies the initial conditions (7.38). Let us recall that for u(x, t; k + 2m) to be a solution of (7.41)–(7.42) it is sufficient 2 . In order to be able to carry out the construction (7.43), it is sufficient to that f ∈ Cev  n+|γ |−k 2

require that f ∈ Cev



+2

.

n+|γ |+k−1 2



Theorem 81. Let ψ ∈ Cev   (γ )x − (Bk )t u = c2 u, u(x, 0; k) = 0,

. The solution u = u(x, t; k) to c > 0,

(7.44)

lim t ut (x, t; k) = ψ(x) k

(7.45)

t→+0

for k < 1 is defined by the formula

1 ∂ u(t, x; k) = B(n, γ , k, q) t ∂t

q  t (t 2 − r 2 )

1−k+2q−n−|γ | 2

×

0

j 1−k+2q−n−|γ | 2

where

    γ c t 2 − r 2 r n+|γ |−1 Mr [ψ(x)]dr ,

(7.46)

  2−q  1−k 2   . B(n, γ , k, q) =  n+|γ | 2−k+2q−n−|γ |+1   2 2

Proof. Let q ≥ 0 be the smallest positive integer such that 2 − k + 2q > n + |γ | − 1, , and let u(x, t; 2 − k + 2q) be a solution to (7.44) when we take i.e., q > n+|γ |+k−3 2 2 − k + 2q instead of k such that u(x, 0; 2 − k + 2q) = ψ(x),

ut (x, 0; 2 − k + 2q) = 0.

By property (1.109) we obtain that u(t, x; k − 2q) = t 1−k+2q u(t, x; 2 − k + 2q) is a solution to the equation (γ )x u −

∂ 2 v k − 2q ∂v − = c2 u. t ∂t ∂t 2

Further, applying q times formula (1.110) we obtain that

   1 ∂ q 1 ∂ q  1−k+2q u(t, x; k − 2q) = u(t, x; 2 − k + 2q) t t ∂t t ∂t

(7.47)

Differential equations with Bessel operator

291

is a solution to (7.44). In order to get a solution to (7.44) satisfying the conditions (7.45) we use the multiplier

u(t, x; k) =

2−q  (1 − k)

2−q  3−k 2 .  (1−k) 3−k+2q 2

 

3−k 2



3−k+2q 2



Let

1 ∂ t ∂t

q 

 t 1−k+2q u(t, x; 2 − k + 2q) . (7.48)

We have shown that (7.48) satisfies Eq. (7.44). Now we will prove that u(t, x; k) satisfies the conditions (7.47). Using formula (1.13) from [564], p. 9, we obtain 

 1 ∂ q  1−k+2q u(t, x; 2 − k + 2q) = t t ∂t   

q 2q−s C s  1−k + q + 1  q 2 1 ∂ s 1−k+2s   t u(t, x; 2 − k + 2q), t ∂t  1−k + s + 1 s=0

2

and u(0, x; k) = 0 for k < 1. For the second condition in (7.47) we get lim t k ut (t, x; k) =  

  2−q  3−k 2 ∂ 1 ∂ q  1−k+2q k   lim t u(t, x; 2 − k + 2q) = t t→0 ∂t t ∂t (1 − k) 3−k+2q 2     q 2q−s C s  1−k + q + 1 2−q  3−k  q 2 2 ∂     lim t k t 1−k+2s × 3−k+2q t→0 ∂t 1−k (1 − k)  2 +s +1 s=0 2 s

1 ∂ u(t, x; 2 − k + 2q) = t ∂t  ∂  1−k 1 1 lim t k t u(t, x; 2 − k + 2q) = lim t k × 1 − k t→0 ∂t 1 − k t→0   (1 − k)t −k u(t, x; 2 − k + 2q) + t 1−k ut (t, x; 2 − k + 2q) = t→0

1 lim ((1 − k)u(t, x; 2 − k + 2q) + tut (t, x; 2 − k + 2q)) = 1 − k t→0 lim u(t, x; 2 − k + 2q) = ψ(x).

t→0

Now we write the representation of u(t, x; k) through the integral. Using formula (7.39) we get t u(x, t; 2 − k + 2q) = A(n, γ , 2 − k + 2q) t

(t 2 − r 2 )

k−1−2q 0

1−k+2q−n−|γ | 2

×

292

Transmutations, Singular and Fractional Differential Equations

   γ j 1−k+2q−n−|γ | c t 2 − r 2 r n+|γ |−1 Mr [ψ(x)]dr. 2

Considering (7.48) we write u(t, x; k) = A(n, γ , 2 − k + 2q)

1 ∂ t ∂t

q t (t 2 − r 2 )

1−k+2q−n−|γ | 2

2−q  (1 − k)

 

3−k 2



3−k+2q 2

×

   γ j 1−k+2q−n−|γ | c t 2 − r 2 r n+|γ |−1 Mr [ψ(x)]dr. 2

0

Simplifying we get (7.46), and this completes the proof. The union of Theorems 80 and 81 gives the following statement. 

n+|γ |−k 2

+2



n+|γ |+k−1 2

Theorem 82. Let ϕ = ϕ(x), ϕ ∈ Cev , ψ = ψ(x), ψ ∈ Cev solution of   (γ )x − (Bk )t u = c2 u, c > 0, u(x, 0; k) = ϕ(x),

lim t ut (x, t; k) = ψ(x), k

t→+0



. Then the (7.49) (7.50)

for k ≤ min{n + |γ | − 1, 1}, k = −1, −3, −5, ..., is given by the formula u(x, t; k) = u1 (x, t; k) + u2 (x, t; k), where u1 (x, t; k) is found by Theorem 80 and u1 (x, t; k) is found by Theorem 81.

7.1.4 Examples Example 1. We are looking for the solution of ∂ 2 u γ ∂u ∂ 2 u k ∂u + = 2 + , x ∂x t ∂t ∂x 2 ∂t u(x, 0; k) = j γ −1 (x), ut (x, 0; k) = 0. 2

1) For k > γ > 0 using (7.29) we obtain   1 2 k+1 k−γ −2 2    (1 − y 2 ) 2 γ Txty j γ −1 (x)y γ dy. u(x, t; k)=  2  k−γ  γ +1 2 2 0 Using (3.152) and formula (2.12.4.6) from [456] of the form a x ν+1 (a 2 − x 2 )β−1 Jν (cx)dx = 0

2β−1 a β+ν (β)Jβ+ν (ac), cβ

(7.51)

Differential equations with Bessel operator

a > 0,

293

Re ν > −1,

Re β > 0,

we obtain u(x, t; k)=j γ −1 (x) t

1−γ 2

2

2



γ +1 2

 



k+1 2

k−γ 2



1



(1 − y 2 )

k−γ −2 2

J γ −1 (ty) y

γ +1 2

dy,

2

0

(7.52)

u(x, t; k)=j γ −1 (x)j k−1 (t). 2

2

Example 2. The solution for all k ∈ R such that k = −1, −3, −5, ... is given by (7.13) and it is easy to check that u(x, t; k) = j γ −1 (x)j k−1 (t). 2

2

Example 3. When k = −1, −3, −5, ..., we have u(x, t; −1) = j γ −1 (x) 2

and for k = −3, −5, ... − k+1 2

u(x, t; k) = j γ −1 (x) + 2

⎛

 h=1

Bγh j γ −1 (x)

t 2h (k + 1)...(k + 2h − 1) 2 · 4 · .... · 2h 2

− k+1 2

 ⎜ = j γ −1 (x) ⎝1 + 2

h=1

t 2h

(−1)h

(k + 1)...(k + 2h − 1) 2 · 4 · .... · 2h

⎞ ⎟ ⎠.

Example 4. Now let us find a solution to ∂ 2 u γ ∂u ∂ 2 u k ∂u + = 2 + , x ∂x t ∂t ∂x 2 ∂t lim t k ut (x, t; k) = j γ −1 (x). lim u(x, t; k) = 0,

t→0

t→+0

For k < 1 we have



2



t 1−k j γ −1 (x) 1 k+γ   2  (1 − ξ 2 )− 2 j γ −1 (tξ )ξ γ dξ u(x, t; k) = 2−k−γ γ +1 2 (1 − k)  2 2 0 2

=

3−k 2

t 1−k j γ −1 (x)j 1−k (t). 2 1−k 2

Example 5. Let us consider an example   (Bγ )x − (Bk )t u = c2 u, c > 0,

(7.53)

u = u(t, x; k),

(t, x) ∈ R2+ , (7.54)

294

Transmutations, Singular and Fractional Differential Equations

u(x, 0; k) = jγ (ax),

ut (x, 0; k) = 0,

(7.55)

where n=1, γ − 2≤k≤γ , k = −1, 0 0, Re ν > −l − 1, Re μ > −m − 1,

we obtain t (t 2 − r 2 )

k−γ 4

0

t

k+1 2

a

γ −1 2

c

k−γ 2

   γ −1 J k−γ c t 2 − r 2 r 2 +1 J γ −1 (ar)dr = 2

2

 k+1 ( a 2 + c2 )− 2 J k+1 2

  t a 2 + c2

(7.57)

Differential equations with Bessel operator

295

and u(x, t; k + 2) =   k+1   2 2  k+3 k+1  k+1 2 j γ −1 (ax) t − 2 ( a 2 + c2 )− 2 J k+1 t a 2 + c2 = 2 2 k+1   1 j γ −1 (ax)j k+1 t a 2 + c2 . 2 k+1 2 Then the solution of (7.54)–(7.55) is  ∂  k+1 t u(x, t; k + 2) = t∂t   ∂  k+1 t 1−k j γ −1 (ax) t j k+1 t a 2 + c2 = 2 k+1 2 t∂t   k+1 k+3   2 2 t 1−k  2 ∂  k+1 2 + c2 2 J k+1 t a j (ax) t = γ −1 √ k+1 2 2 t∂t (k + 1)( a 2 + c2 ) 2   k−1     2 2  k+1 1−k 2 j (ax)t 2 J k−1 t a 2 + c2 = j γ −1 (ax)j k−1 t a 2 + c2 . √ k−1 γ −1 2 2 2 2 ( a 2 + c2 ) 2

u(x, t; k) = t 1−k

As might be seen from (1.23) and (1.24),   j γ −1 (ax) lim j k−1 t a 2 + c2 = j γ −1 (ax), 2 2 t→0 2   ∂ j γ −1 (ax) lim j k−1 t a 2 + c2 = j γ −1 (ax) = 0, 2 2 2 t→0 ∂t     2 (Bγ )x j γ −1 (ax)j k−1 t a 2 + c2 = −a j γ −1 (ax)j k−1 t a 2 + c2 , 2 2 2 2     2 2 2 2 (Bk )t j γ −1 (ax)j k−1 t a + c = −(a + c )j γ −1 (ax)j k−1 t a 2 + c2 , 2

2

2

2

which shows that the function   u(t, x; k) = j γ −1 (ax)j k−1 t a 2 + c2

(7.58)

2

2

satisfies (7.54)–(7.55). Example 6. Consider the problem  (Bγ )x − (Bk )t u = c2 u,



u(x, 0; k) = 0,

c > 0,

u = u(t, x; k),

lim t k ut (x, t; k) = jγ (bx),

t→+0

(t, x) ∈ R2+ , (7.59) (7.60)

296

Transmutations, Singular and Fractional Differential Equations

where n = 1, k < 1, 0 < γ < 3, b ∈ R. When q = 1 and considering (7.56) we obtain

 t k+γ 1 ∂ (t 2 − r 2 )1− 2 × u(t, x; k) = B(1, γ , k, 1) j γ −1 (bx) 2 t ∂t 0     γ j1− k+γ c t 2 − r 2 j γ −1 (br)r dr = 2 2     γ +1

 t B(1, γ , k, 1) 2  2 − k+γ 2−k−γ 2 1 ∂ j γ −1 (bx) (t 2 − r 2 ) 4 × k+γ k−1 γ −1 2 t ∂t 2 2 b 2 c1− 2 0     γ −1 J1− k+γ c t 2 − r 2 J γ −1 (br)r 2 +1 dr , 2

2

where    1−k 2    B(1, γ , k, 1) = γ +1 2 2  2 −

k+γ 2

.

Applying formula (2.12.35.2) from [456] of the form t

 μ (t 2 − x 2 )m+ 2 x ν+1+2l Jμ (c t 2 − x 2 )Jν (hx)dx =

0

t μ+ν−m−l+1 cμ hν [(c2 + h2 )−

∂ c∂c

μ+ν+m+l+1 2

m

l ×

 Jμ+ν+m+l+1 (t c2 + h2 )],

Re ν > −l − 1,

t > 0,

∂ h∂h

Re μ > −m − 1,

we obtain t (t 2 − r 2 )

2−k−γ 4

   γ −1 J1− k+γ c t 2 − r 2 J γ −1 (br)r 2 +1 dr = 2

2

0

b

γ −1 2

c

1− k+γ 2

(b + c ) 2

2

k−3 4

t

3−k 2

J 3−k 2

  2 2 t b +c

and

  γ +1 k+γ B(1, γ , k, 1)  2− × 2 2

  3−k 1 ∂ j γ −1 (bx) t 2 J 3−k t b2 + c2 = 2 2 t ∂t

u(t, x; k) = 2 (b2 + c2 )

k−3 4

1−k 2

Differential equations with Bessel operator

2−

k+1 2

t 1−k 1−k

297

   k−1 1−k 1−k  2 ( b + c2 ) 2 j γ −1 (bx) t 2 J 1−k t b2 + c2 = 2 2 2   j γ −1 (bx) j 1−k t b2 + c2 .



2

2

Taking into account (1.23) and (1.24) it is easy to check that   t 1−k j γ −1 (bx) j 1−k t b2 + c2 = 2 1−k 2   1−k t j γ −1 (bx) j 1−k t b2 + c2 , −b2 2 1−k 2   t 1−k (Bk )t j γ −1 (bx) j 1−k t b2 + c2 = 2 1−k 2   1−k t j γ −1 (bx) j 1−k t b2 + c2 , −(b2 + c2 ) 2 1−k 2   1−k  t lim j γ −1 (bx) j 1−k t b2 + c2 = 0, 2 2 t→0 1 − k (Bγ )x

and j γ −1 (bx) lim t k 2

t→0

∂ ∂t

  t 1−k j 1−k t b2 + c2 = j γ −1 (bx), 2 1−k 2

which confirms that the function u(t, x; k) =

  t 1−k j γ −1 (bx) j 1−k t b2 + c2 2 1−k 2

(7.61)

satisfies (7.59)–(7.60). Example 7. From Examples 5 and 6 it is plain to see that the solution of  (Bγ )x − (Bk )t u = c2 u,



u(x, 0; k) = jγ (ax),

c > 0,

u = u(t, x; k),

(t, x) ∈ R2+ ,

lim t k ut (x, t; k) = jγ (bx),

t→+0

where n = 1, 0 < γ < 1, γ − 2 ≤ k ≤ γ , k = −1, a, b ∈ R, is   t 1−k   j γ −1 (bx) j 1−k t b2 + c2 . u(t, x; k) = j γ −1 (ax)j k−1 t a 2 + c2 + 2 2 2 1−k 2

298

Transmutations, Singular and Fractional Differential Equations

7.2 Hyperbolic and ultrahyperbolic equations with Bessel operator in spaces of weighted distributions 7.2.1 The generalized Euler–Poisson–Darboux equation and the singular Klein–Gordon equation In this subsection we apply the Hankel transform method to solve the initial value problem 

 (γ )x − (Bk )t u = c2 u,

u(x, 0; k) = f (x),

(7.62)

ut (x, 0; k) = 0,

u = u(x, t; k),

(7.63)

where γi > 0, xi > 0, i = 1, ..., n, t > 0. We will call (7.62) the generalized Euler– Poisson–Darboux equation. We obtain the distributional solution of (7.62)–(7.63) in convenient space. Besides, we give formulas for regular solution of (7.62)–(7.63) in the particular case of k and of Cauchy for the singular Klein–Gordon equation.  (Rn ) × C 2 (0, ∞) of (7.62)–(7.63). The We are looking for the solution u ∈ Sev + ev n  2  (Rn ) by varinotation u ∈ Sev (R+ ) × Cev (0, ∞) means that u(x, t; k) belongs to Sev + 2 able x and belongs to Cev (0, ∞) by variable t. Here we use methods of weighted generalized function, see [501].  (Rn ) × C 2 (0, ∞) of (7.62)–(7.63) for k = Theorem 83. The solution u ∈ Sev + ev −1, −3, −5, ... is unique and defined by the formula

u(x, t; k) =  

k−n−|γ |−1 1 1−k 2 2 2 2 2 2 j k−n−|γ |−1 (t − |x| )+ · c ∗ f (x) , C(n, γ , k) t (t − |x| )+ 2

γ

(7.64) where 2n 

C(n, γ , k) =

 



k−n−|γ |+1 2

k+1 2  n



i=1

 

γi +1 2

.

In the case when k < 0 of (7.62)–(7.63) is not unique, when k < 0 and k = −1, −3, −5, ..., the difference between two arbitrary solutions is always of the form At 1−k u(t, x; 2 − k),

A = const,

where u(t, x; 2 − k) is a solution of the Cauchy problem

(7.65)

Differential equations with Bessel operator

299



 (γ )x − (B2−k )t u = c2 u,

u(x, 0; 2 − k) = ψ(x),

ut (x, 0; 2 − k) = 0

 . When k = and ψ(x) is an arbitrary function or distribution belonging to Sev −1, −3, −5, ..., a nonunique solution of the Cauchy problem (7.62)–(7.63) will contain a term (7.65) and

  1 

e± 2 πni  n+|γ |−k+1 k−n−|γ |−1 2 1−k 2 2 2 2 ∗ f (x) . (t − |x| − c ± i0)γ    t n γi +1 γ 2n  1−k  2 2 i=1

Proof. Applying the multi-dimensional Hankel transform to (7.62) with respect to the variables x1 , ..., xn only and using (1.95), we obtain

 ∂2 k ∂ |ξ | + c + 2 +  u(ξ, t; k) = 0, t ∂t ∂t  u(ξ, 0; k) = f(ξ ),  ut (ξ, 0; k) = 0, 2

2

(7.66) (7.67)

where ξ = (ξ1 , ξ2 , ..., ξn ) ∈ Rn+ corresponds to x = (x1 , ..., xn ) ∈ Rn+ , |ξ |2 = ξ12 + ξ22 + ... + ξn2 ,   u(ξ, t; k) = (Fγ )x [u(x, t; k)](ξ ) =

u(x, t; k) jγ (x; ξ )x γ dx,

Rn+

and f(ξ ) = Fγ [f ](ξ ). k (ξ, t) of the Cauchy problem The solution G

 ∂2 k ∂ k |ξ | + c + 2 + G (ξ, t) = 0, t ∂t ∂t kt (ξ, 0) = 0 k (ξ, 0) = 1, G G 2

2

was obtained in [38]. We have different solutions for nonnegative and negative values of k, specifically: 1. for k ≥ 0,  k (ξ, t) = j k−1 ( |ξ |2 + c2 t), G 2

(7.68)

300

Transmutations, Singular and Fractional Differential Equations

2. for k < 0, k = −1, −3, −5, ...,   Gk (ξ, t) = At 1−k j 1−k ( |ξ |2 + c2 t) + j k−1 ( |ξ |2 + c2 t), 2

(7.69)

2

where A is an arbitrary complex number which depends on ξ and c, and 3. for k = −1, −3, −5, ...,  k 1−k  G (ξ, t) = Bt j 1−k ( |ξ |2 + c2 t)− 2

1−k  k−1  2 π2 2 2 2   |ξ | + c t Y 1−k ( |ξ |2 + c2 t), 2  1−k 2

(7.70)

where B denotes an arbitrary complex number which depends on ξ and c. From (7.68)–(7.70) we conclude that the problem (7.62)–(7.63) has a unique solution for k ≥ 0 only. Besides, we can see that the difference between two different solutions (7.69) is always of the form  At 1−k j 1−k ( |ξ |2 + c2 t). (7.71) 2

  k k Now let us find Gk (x, t) = (F−1 γ )ξ G (ξ, t) (x). We call G (x, t) the fundamen  k tal solution of problem (7.62)–(7.63). The inverse transform (F−1 γ )ξ G (ξ, t) (x) is most easily found by considering c as an additional independent variable. Setting k (ξ, t) = j k−1 (|ξ  | t) for k ≥ 0 and find an inverse ξ  = (ξ1 , ..., ξn , c), we can write G 2 Hankel transform of j k−1 (|ξ  | t) by variable ξ  using (4.94). We obtain 2



   2n+1  k+1 k−n−|γ  |−2 2   1−k 2  2 2  j k−1 (|ξ | t) (x )= (F−1 t ) (t −|x | ) ,  ξ γ +,γ    n+1   2 k−n−|γ  |  γi +1   2 2 i=1

where γ  = (γ1 , .., γn , γn+1 ), γn+1 is an arbitrary positive number, x  = (x, σ ), and σ ∈ R+ is dual to the variable c. Now in order to find Gk (x, t) we need to apply a direct Hankel transform only on a one-dimensional variable σ . We have   2n+1  k+1 2 1−k Gk (x, t) =    t × n+1 k−n−|γ  |  γi +1   2 2 i=1

k−n−|γ  |−2 2 +,γ 

(Fγn+1 )σ (t 2 − x 2 − σ 2 )



2n 

(c) =

 



k−n−|γ |+1 2

k+1 2  n



i=1

 

γi +1 2

×

Differential equations with Bessel operator k−n−|γ |−1 2

t 1−k (t 2 − x 2 )+

301



1 j k−n−|γ |−1 (t 2 − |x|2 )+2 · c . 2

Therefore, the solution of (7.62)–(7.63) for k ≥ 0 is given by u(x, t; k) = (Gk (x, t) ∗ f (x))γ .

(7.72)

It is easy to see that (7.72) still gives one of the possible solutions of (7.62)–(7.63) for k < 0, k = −1, −3, −5, .... Moreover, in consideration of (7.71) the difference between two solutions for k < 0 has the form

 −1 1−k 2 2 (Fγ )ξ At j 1−k ( |ξ | + c t) (x). (7.73) 2

Consequently, the difference between two arbitrary solutions for k < 0 is always of the form A(t 1−k G2−k (t, x) ∗ ψ(x))γ = At 1−k u(t, x; 2 − k),

(7.74)

 , u(t, x; 2 − k) is where ψ(x) is an arbitrary function or distribution belonging to Sev the solution of the Cauchy problem

  (γ )x − (B2−k )t u = c2 u, u(x, 0; 2 − k) = ψ(x),

ut (x, 0; 2 − k) = 0,

and G2−k (t, x) is the corresponding fundamental solution. Finally we consider the case k = −1, −3, −5, .... In this case the solution will be of a different character than the solutions for other values of k and will always contain a term k−1

π2 2  F−1 Gk (t, x) =  γ  1−k 2

 

 1−k 2

|ξ |2 + c2 t

  Y 1−k ( |ξ |2 + c2 t) . 2

It is clear that k−1

iπ2 2  F−1 G (t, x) =  γ 1−k  2 k

 

 1−k |ξ |2

+ c2 t

2

  2 2 H 1−k ( |ξ | + c t) (1) 2

and k−1

iπ2 2  F−1 G (t, x) = −  γ 1−k  2 k

 

 1−k |ξ |2

+ c2 t

2

  2 2 H 1−k ( |ξ | + c t) (2) 2

302

Transmutations, Singular and Fractional Differential Equations

will also be a fundamental solution of our problem. Then using (4.102) and (4.103) we obtain   1 e± 2 πni  n+|γ |−k+1 k−n−|γ |−1 2 1−k 2 2 2 2 Gk (t, x) = (t − |x| − c ± i0) . t γ     n γi +1 1−k n 2  2  2 i=1

Since (t 2 − |x|2 )λ+,γ has its support in the interior of the part of the sphere when x1 ≥ 0, ..., xn ≥ 0, we may conclude that in the case k = −1, −3, −5, ...  . However, in the case the generalized convolutions exist for arbitrary ϕ(x) ∈ Sev k = −1, −3, −5, ... the fundamental solution is no longer concentrated within the part of the sphere S1+ (n). S1+ (n)

 (Rn ) × C 2 (0, ∞) of Corollary 1. The solution u ∈ Sev + ev   (γ )x − (Bk )t u = 0, u(x, 0; k) = f (x), ut (x, 0; k) = 0, u = u(x, t; k)

for k = −1, −3, −5, ... is unique and defined by the formula

 k−n−|γ |−1 u(x, t; k) = C(n, γ , k) t 1−k (t 2 − |x|2 )+ 2 ∗ f (x) , γ

where 2n 

C(n, γ , k) =

 



k−n−|γ |+1 2

k+1 2  n



i=1

 

γi +1 2

.

In the case when k < 0 of (7.62)–(7.63) is not unique, when k < 0 and k = −1, −3, −5, ..., the difference between two arbitrary solutions is always of the form At 1−k u(t, x; 2 − k),

A = const,

(7.75)

where u(t, x; 2 − k) is solution of the Cauchy problem   (γ )x − (B2−k )t u = 0, u(x, 0; 2 − k) = ψ(x), ut (x, 0; 2 − k) = 0,  . When k = where ψ(x) is an arbitrary function or distribution belonging to Sev −1, −3, −5, ..., a nonunique solution of the Cauchy problem (7.62)–(7.63) will contain terms (7.75) and   1 

e± 2 πni  n+|γ |−k+1 k−n−|γ |−1 2 1−k 2 2 2 − |x| ± i0) ∗ f (x) . t (t γ   n   γi +1 1−k  γ n 2  2  2 i=1

Differential equations with Bessel operator

303

 (Rn ) × C 2 (0, ∞) of the initial value problem for Corollary 2. The solution u ∈ Sev + ev the singular Klein–Gordon equation

∂2 (7.76) (γ )x − 2 v = c2 v, c > 0, v = v(x, t), x ∈ Rn+ , t > 0, ∂t  f (x) ∈ Sev (7.77) v(x, 0) = f (x), vt (x, 0) = 0,

is v(x, t) =

 

√ 2n π  n

1−n−|γ | 2

− n+|γ2 |+1

t (t 2 − |x|2 )+,γ

 

i=1

γi +1 2

×

 

1 j− n+|γ |+1 (t 2 − |x|2 )+2 · c ∗ f (x) . 2

γ

This solution was obtained by letting k tend to 0 in (7.64). The Klein–Gordon equation

∂2 z − 2 v = c2 v, v = v(z, t), z ∈ RN , ∂t

(7.78)

is the most frequently used wave equation for the description of particle dynamics in relativistic quantum mechanics. When function v is radially symmetric by some groups of variables z1 , ..., zN in (7.78), we obtain (7.76) with a smaller number of spatial variables. In this case numbers γi , i = 1, .., n, in (7.76) will be integer. Corollary 3. In the case k > n + |γ | − 1 the integral in (7.64) converges in the usual sense and we obtain the unique classical solution of (7.62)–(7.63) u(x, t; k) = t A(n, γ , k) t

(t 2 − r 2 )

1−k

k−n−|γ |−1 2

   γ j k−n−|γ |−1 c t 2 − r 2 r n+|γ |−1 Mr [f (x)]dr, 2

0

  2 k+1 2   . A(n, γ , k) =  n+|γ | k−n−|γ |+1   2 2 Proof. For k > n + |γ | − 1, passing to spherical coordinates, we obtain u(x, t; k) = C(n, γ , k) t 1−k ×    k−n−|γ |−1 1 y 2 (t 2 − |y|2 ) j k−n−|γ |−1 (t 2 − |y|2 ) 2 · c γ Tx f (x) y γ dy = 2

Bt+ (n)



C(n, γ , k)

(1 − |y|2 )

B1+ (n)

k−n−|γ |−1 2

  1 ty j k−n−|γ |−1 (1 − |y|2 ) 2 · tc γ Tx f (x) y γ dy = 2

304

Transmutations, Singular and Fractional Differential Equations

1 (1 − r 2 )

C(n, γ , k)

k−n−|γ |−1 2

  1 j k−n−|γ |−1 (1 − r 2 ) 2 · tc r n+|γ |−1 dr× 2

0

 S1+ (n)

  2 k+1 2 γ trθ   × Tx f (x) θ γ dS =  n+|γ | |+1   k−n−|γ 2 2

1 (1 − r 2 ) 0

 

2

k−n−|γ |−1 2

  1 γ j k−n−|γ |−1 (1 − r 2 ) 2 · tc r n+|γ |−1 Mrt [f (x)]dr = 2

k+1 2



t

t 1−k

   | k−n−|γ |+1  n+|γ  2 2

(t 2 − r 2 )

k−n−|γ |−1 2

  1 j k−n−|γ |−1 (t 2 − r 2 ) 2 · c × 2

0

r n+|γ |−1 Mr [f (x)]dr. γ

Corollary 4. In the case k > n + |γ | − 1, the solution of  (γ )x − (Bk )t u = 0,



k ∈ R,

u = u(x, t; k),

n x ∈ R+ ,

t > 0,

ut (x, 0; k) = 0

u(x, 0; k) = f (x),

(7.79) (7.80)

is unique and is given by t u(x, t; k) = A(n, γ , k) t

(t 2 − r 2 )

1−k



k−n−|γ |−1 2

r n+|γ |−1 Mr [f (x)]dr, γ

0

 2   , A(n, γ , k) =  | k−n−|γ |+1  n+|γ  2 2 k+1 2

which coincides with (7.6). Example 1. Let us consider the Cauchy problem for k > n + |γ | − 1, 

 (γ )x − (Bk )t u = c2 u, ut (x, 0; k) = 0.

u(x, 0; k) = jγ (x; ξ ),

In this case the solution is unique and is gives by u(x, t; k) = A(n, γ , k) t 1−k × t (t 2 − r 2 ) 0

k−n−|γ |−1 2

   γ j k−n−|γ |−1 c t 2 − r 2 r n+|γ |−1 Mr jγ (x; ξ )dr, 2

(7.81)

Differential equations with Bessel operator

305

  2 k+1 2   . A(n, γ , k) =  | k−n−|γ |+1  n+|γ  2 2 γ

For Mr jγ (x; ξ ) we have formula (3.190), so we get   2 k+1 2    t 1−k jγ (x; ξ ) × u(x, t; k) =  n+|γ | k−n−|γ |+1   2 2 t (t 2 − r 2 ) 0

2

k−1 2

 

k−n−|γ |−1 2

c t

k−n−|γ |−1 2

2

2

k+1 2

|ξ |

(t 2 − r 2 )

   r n+|γ |−1 j k−n−|γ |−1 c t 2 − r 2 j n+|γ | −1 (r|ξ |)dr =



n+|γ | 2 −1

k−n−|γ |−1 4

t 1−k jγ (x; ξ ) ×

r

n+|γ | 2

   J k−n−|γ |−1 c t 2 − r 2 J n+|γ | −1 (r|ξ |)dr. 2

2

0

Applying formula (2.12.35.2) from [456] of the form t

 μ (t 2 − x 2 )m+ 2 x ν+1+2l Jμ (c t 2 − x 2 )Jν (hx)dx =

0

t

   ∂ m ∂ l 2 2 − μ+ν+m+l+1 2 c h [(c +h ) Jμ+ν+m+l+1 (t c2 +h2 )], c∂c h∂h Re ν > −l − 1, Re μ > −m − 1,

μ+ν−m−l+1 μ ν

t > 0,

we have k = m = 0, ν = t (t 2 − r 2 )

n+|γ | 2

k−n−|γ |−1 4

r

− 1, μ =

n+|γ | 2

k−n−|γ |−1 , 2

   J k−n−|γ |−1 c t 2 − r 2 J n+|γ | −1 (r|ξ |)dr = 2

0

t

k−1 2

k−n−|γ |−1

n+|γ |

2 c |ξ | 2  k−1 ( c2 + |ξ |2 ) 2

−1

 J k−1 (t c2 + |ξ |2 ). 2

Therefore,  u(x, t; k) = jγ (x; ξ )j k−1 (t c2 + |ξ |2 ). 2

h = |ξ |, and

2

306

Transmutations, Singular and Fractional Differential Equations

7.2.2 Iterated ultrahyperbolic equation with Bessel operator The classical ultrahyperbolic equation has the form x u = y u,

u = u(x, y),

x ∈ Rp ,

y ∈ Rq .

(7.82)

Eq. (7.82) was studied by many authors (see [23,29,30,76,156,271,272,435,466]). For p = 1 or q = 1, (7.82) is the usual wave equation describing the dynamic development of many processes of classical and quantum physics. To equations of the form (7.82) for p = q = 2 lead, for example, the Hilbert problems of determining in a threedimensional Cartesian space all metrics whose geodesics are straight lines (see [436]); the inverse diffraction problem in the study of the heterogeneity of the distribution of grains of polycrystalline materials; and the hyperspherical X-ray transformation, namely, the density functions of the crystallographic poles satisfy the ultrahyperbolic equation with the Laplace–Beltrami operator (see [418]). The case when in (7.82) p > 2 and q > 2 is important from a mathematical point of view thanks to the Asgeirsson theorem about the spherical mean (see [9], [75], p. 475, [155], p. 84, [162], p. 318, [170] I, p. 183). This theorem is a generalization of the mean value theorem for harmonic functions, as well as a generalization of the Green formula for a linear wave equation with constant coefficients. Generally speaking, the initial problem for the ultrahyperbolic equation (7.82) is incorrect. In particular, in the general case, the solution of the initial problem for it either does not exist or is not unique, and if it is possible to find some solution, then the solution is unstable. However, in the article [76] it was shown that the initial problem for an ultrahyperbolic equation with a nonlocal constraint on codimensional hyperspaces has a unique global solution in the Sobolev space H m . Thus, in this case, the initial problem for (7.82) is correct. We will consider a generalization of Eq. (7.82) to the case when instead of every second derivative with respect to each variable the Bessel operator acts. Let n = p + q, p and q are natural, γ = (γ  , γ  ), γ  = (γ1 , ..., γp ), p q  γ = (γp+1 , ..., γp+q ), γi > 0, i = 1, ..., n, x  ∈ R+ , x  ∈ R+ , x = (x  , x  ) ∈ Rn+ , p q Rn+ = R+ × R+ . The B-ultrahyperbolic equation or singular ultrahyperbolic equation has the form γ u = 0,

u = u(x),

(7.83)

where γ is a homogeneous linear differential operator of the form γ = (γ  )x  − (γ  )x  = Bγ1 + ... + Bγp − Bγp+1 − ... − Bγp+q , (γ  )x  =

p 

(Bγi )xi , (γ  )x  =

i=1

p+q  j =p+1

(Bγj )xj , Bγi =

∂2 + γi ∂ , ∂xi2 xi ∂xi

i = 1, ..., n.

The iterated B-ultrahyperbolic equation we will call the equation of the form kγ u = f, where k∈N and f = f (x) is a suitable function.

(7.84)

Differential equations with Bessel operator

307

In this subsection we find the fundamental solution to the equation kγ u = f using the results obtained for weighted generalized functions. Let x ∈ Rn+ , n = p + q, p, q ∈ N. The fundamental solution to Eq. (7.84) is the weighted generalized function u such that kγ u = δγ .

(7.85)

Note that the fundamental solutions for the hyperbolic and ultrahyperbolic equations with the Bessel operator applied only by one variable are obtained in [246,247]. Theorem 84. Except when n + |γ | = 2, 4, 6, ... and k ≥ ized function

u = (−1)k

e±i

π(q+|γ  |) 2

 

n+|γ | 2

4k (k − 1)!|S1+ (n)|γ 



−k

n+|γ | 2

n+|γ | 2 ,

the weighted general-

 | − n+|γ 2 +k

−1

 (P ± i0)γ

(7.86)

is the fundamental solution to the equation kγ u = f in the sense (7.85). If n + |γ | = 2, 4, 6, ... and k ≥ | − n+|γ 2 +k

(P + i0)γ kγ u = 0.

n+|γ | 2 ,

| − n+|γ 2 +k

=(P − i0)γ

then the weighted generalized function

is a solution to a homogeneous equation

Proof. Using (4.61) we obtain

  n+|γ | n+|γ | λ+ ... λ+ +k−1 (P +i0)λγ . 2 2

k kγ (P +i0)λ+k γ =4 (λ+1)...(λ+k)

| in the last equality and using formula (4.74) for Tending to the limit at λ → − n+|γ 2 k = 0, we obtain − n+|γ | +k

= kγ (P + i0)γ 2   

n + |γ | n + |γ | n+|γ | ... k − (k − 1)! lim (P + i0)λγ = λ+ 4k 1 − n+|γ | 2 2 2 λ→− 2  

n + |γ | n + |γ | ... k − (k − 1)! res (P + i0)λγ = 4k 1 − | 2 2 λ=− n+|γ 2  

π(q+|γ  |) n + |γ | n + |γ | ... k − (k − 1)!e−i 2 |S1+ (n)|γ δγ (x). 4k 1 − 2 2 If n+|γ | is even and k ≥

n+|γ | 2 , then among the multipliers − n+|γ | +k



1−

n+|γ | 2



 ... k −

− n+|γ | +k

n+|γ | 2



there is a zero and therefore kγ (P +i0)γ 2 =0 and u=(P +i0)γ 2 is a solution to a homogeneous equation kγ u = 0. For all other values n + |γ | and k the

308

Transmutations, Singular and Fractional Differential Equations

weighted generalized function u = (−1)k

ei

π(q+|γ  |) 2

 

n+|γ | 2

4k (k − 1)!|S1+ (n)|γ 



−k



n+|γ | 2

| − n+|γ 2 +k

 (P + i0)γ

(7.87)

−1

is the fundamental solution in the sense (7.85) to Eq. (7.84). In (7.87) it was used that

   

n + |γ | n + |γ | n + |γ | k n + |γ | 1− ... k − = (−1) − 1 ... −k 2 2 2 2   |  n+|γ 2 −1 . = (−1)k  |  n+|γ − k 2 Similarly, using (4.75) it can be shown that if the number n+|γ | is even and k ≥ | − n+|γ 2 +k

n+|γ | 2 ,

then u=(P − i0)γ is a solution to a homogeneous equation kγ u = 0. For all other values n + |γ | and k the weighted generalized function   π(q+|γ  |) | e−i 2  n+|γ − k 2 − n+|γ | +k   (P − i0)γ 2 u = (−1)k | 4k (k − 1)!|S1+ (n)|γ  n+|γ 2 −1 is the fundamental solution in the sense (7.85) to Eq. (7.84).

7.2.3 Generalization of the Asgeirsson theorem In this subsection we present the results generalizing the Asgeirsson theorem to the case of the B-ultrahyperbolic equation (7.83) (see [349,354]). 2 (Rn ), x = (x , ..., x  ), y = (y , ..., y  ). Consider the spherical Let u(x, y) ∈ Cev 1 1 m m + weighted means (3.183) taken on parts of surfaces of unit spheres S1+ (m ) and S1+ (m )  m m in Rm + and R+ by each of the groups and variables x and y centered at y ∈ R + and  z ∈ Rm + , respectively. For these weighted spherical means we introduce the notation  1  γ γ γ Tx rξ u(x, y) ξ γ dSξ , (Mr u)(x, r; y) = Mu (x, r; y) = +  |S1 (m )|γ  S1+ (m )

γ 

γ 

(Ms u)(x; y, s) = Mu (x; y, s) =

γ

1 |S1+ (m )|γ 



γ 



Ty sζ u(x, y) ζ γ dSζ .

S1+ (m ) 

So Mu (x, r; y) is the weighted spherical mean of the function u in Rm + at constant 

γ 

y = (y1 , . . . , ym ) ∈ R m + , and Mu (x; y, s) is the weighted spherical mean of the   at constant x = (x1 , . . . , xm ) ∈ R m function u in Rm + +.

Differential equations with Bessel operator

309

Let us define also the general weighted spherical mean of function u by (x, y) of the form γ

γ 

(Mr Ms u)(x, r; y, s) = U (x, r; y, s) =   1 γ  θ dS(θ ) |S1+ (m )|γ  |S1+ (m )|γ  S1+ (m )

γ





γ γ Trξ,sζ x,y u(x, y)ξ ζ dS(ξ ).

S1+ (m )

It is obvious that γ

Mu (x, r; y) = U (x, r; y, 0),

γ 

Mu (x; y, s) = U (x, 0; y, s).

(7.88)

Theorem 85. If m + |γ  | = m + |γ  |

(7.89)

and the function u(x  , x  ) satisfies the singular ultrahyperbolic equation γ u = 0,

(7.90)

then γ

γ 

γ

γ 

M r Ms u = M s Mr u .

(7.91)

Proof. Each of the weighted spherical means μγ  (y, z; r) and νγ  (y, z; r) of u satisfies Eq. (7.83). This gives two equalities for ωγ  ,γ  (y, z; r, s): 

γ  ωγ  ,γ  =

m 

Byi ωγ  ,γ  =

i=1

∂ 2 ωγ  ,γ  m + |γ  | − 1 ∂ωγ  ,γ  + r ∂r ∂r 2

(7.92)

∂ 2 ωγ  ,γ  m + |γ  | − 1 ∂ωγ  ,γ  + . s ∂s ∂s 2

(7.93)

and 

γ  ωγ  ,γ  =

m 

Bzi ωγ  ,γ  =

i=1

By virtue of property 5 of the weighted spherical average we get      γ  γ  γ γ  γ  ωγ  ,γ  = γ  Mr Ms u (y, r; z, s) = Mr γ  Ms u (y, r; z, s) . (7.94) γ 

In (7.94) operator γ  and Ms are also permutable, since they act on different (nonintersecting) groups of variables. Therefore,    γ γ  γ  ωγ  ,γ  = Mr Ms γ  u (y, r; z, s) .

310

Transmutations, Singular and Fractional Differential Equations

Similarly, we find    γ γ  γ  ωγ  ,γ  = Mr Ms γ  u (y, r; z, s) . According to (7.90), γ  u = γ  u. This implies the equality of the right parts in (7.92) and (7.93) are equal. Consequently, the double weighted spherical mean of the solution of a B-ultrahyperbolic equation satisfies the singular differential equation ∂ 2 ωγ  ,γ  ∂ 2 ωγ  ,γ  m + |γ  | − 1 ∂ωγ  ,γ  m + |γ  | − 1 ∂ωγ  ,γ  + + = . 2 2 s ∂s r ∂r ∂s ∂r (7.95) Eq. (7.95) is Eq. (3.145). Besides, ωγ  ,γ  (y, z; r, s)|s=0 = μγ  (y, z; r) = f (r) and

  ∂ = 0. ωγ  ,γ  (y, z; r, s) ∂s s=0

That gives ωγ  ,γ  (y, z; r, s) = m+|γ |−1

m+|γ |−1

Trs f (r) =

m+|γ |−1

Tsr f (s) = ωγ  ,γ  (y, z; s, r), (7.96)

Trs f (r) =

(m + |γ | − 1)   2m+|γ |−3  2 m+|γ2 |−1

π

 f ( r 2 − 2rs cos ϕ + s 2 ) sinm+|γ |−2 ϕdϕ.

0

The equality m+|γ |−1 Trs f (r) = m+|γ |−1 Tsr f (s) follows from property 5 of the generalized translation. That gives ωγ  ,γ  (y, z; r, s) = ωγ  ,γ  (z, y; s, r) or (7.91). Corollary 5. The weighted spherical mean of the function u taken at constant y by z by the part of a sphere of radius r is equal to the weighted spherical mean of the function u taken at constant z by y by the part of a sphere of radius r: μγ (y, z, r) = νγ (y, z, r). Proof. Since μγ (y, z, r) = ωγ (y, z; r, 0),

νγ (y, z, r) = ωγ (y, z; 0, r),

the equality μγ (y, z, r) = νγ (y, z, r) follows from (7.91).

Differential equations with Bessel operator

311

As a corollary of Theorem 85 follows a generalization of the classical Asgeirsson theorem (see [9,75,155] on the B-ultrahyperbolic equation (7.90)). Theorem 86. Let the function u = u(x, y) ∈ C 2ev , (n = m + m ) be a solution of the B-ultrahyperbolic equation (7.90) and let the condition (7.89) be valid. Then the  m weighted spherical mean of the function u(x, y) taken at constant x∈R m + in R+ by the part of the sphere of radius r is equal to the weighted spherical mean of the  m function u(x, y) taken at constant y∈R m + in R+ by the part of the sphere of radius r: γ

γ 

(Mu )x (x, y, r) = (Mu )y (x, y, r).

(7.97)

The inverse statement to this theorem is also true. It is the inverse Asgeirsson theorem for the B-ultrahyperbolic equation. 



2 (Rm ×Rm ), n=m +m and let for every point Theorem 87. Let u(x, y)∈Cev + +   m m (x, y)∈R+ ×R+ and for any nonnegative r and s condition (7.97) be true. Then if (7.89) is valid, then the function u(x, y) satisfies the B-ultrahyperbolic equation (7.83)  m in Rm + × R+ .

In [170], p. 222, a clarification of the Asgeirsson theorem is given. This clarification is generalized to the case of the B-ultrahyperbolic equation. 







m m m 2 Theorem 88. Let x∈Rm + , y∈R+ , let u=u(x, y) ∈ Cev (R+ ×R+ ) be a continuous   m m in some neighborhood of the set K={θ ∈R+ , ω∈R+ : |θ |+|ω|=r} solution of the Bultrahyperbolic equation (γ  )x u = (γ  )y u, and let m + |γ  | = m + |γ  | ≥ 3. Then

1 |S1 (m )|γ





u(rθ ; 0) S1+ (m )

m !

γ θi i

i=1

1 dSθ = |S1 (m )|γ





u(0; rω) S1+ (m )

m !

ν

ωi i dSω .

i=1

7.2.4 Descent method for the general Euler–Poisson–Darboux equation Using the generalized Asgeirsson relations (7.97) found in the previous subsection, the method of descent of the solution of the general Euler–Poisson–Darboux equation is presented. Theorem 89. Let 0 < δ < n + |γ |, and consider the natural number m ≥ 1 and the multi-index (γ2 , . . . , γm ), γi > 0, such that n + |γ | = m + δ + γ2 + . . . + γm 2 (R+ ) is the solution to the problem exist. Then u ∈ Cev n+1 

δ ∂ ∂ + u(x, t) = γ u(x, t), δ > 0, ∂t t ∂t

(7.98)

312

Transmutations, Singular and Fractional Differential Equations

u(x, 0) = f (x),

ut (x, 0) = 0,

(7.99)

where f ∈ Cev (R+ n ) is given by the formula 

u(x, t) = B s

2 n+|γ | δ−n−|γ |+1 2 , 2



1 d t δ dt

t s (t 2 − s 2 )

δ−n−|γ |−1 2

ds×

0

r n+|γ |−1 Mr f (x)dr. γ

(7.100)

0

Proof. We use the traditional “descent method,” appealing to a singular ultrahyperbolic equation. Namely, if the function u(x, t) satisfies Eq. (7.98), then the function u(x, y), extended from u(x, t) as a constant in the direction t2 , . . . tm , where y = (t, t2 , . . . tm ) obviously, also satisfies the equation (γ  )y u(x, y) = (γ )x u(x, y) ,

y = (t, t2 , . . . , tm ) ∈ R+ m,

(7.101)

where γ  = (δ, γ2 , ..., γm ), γi > 0. Let m and γ  = (δ, γ2 , ..., γm ) such that n + |γ | = m + |γ  |, where |γ  | = δ + γ2 + ... + γm . Then for u(x, y) we have (7.97) 1 + |S1 (n)|γ

 γ

Txrξ u(x, y) ξ γ dS(ξ )=

S1+ (n)

1 + |S1 (m)|γ 



γ



Tyrζ u(x, y) ζ γ dS(ζ ),

S1+ (m)

(7.102) + + where S1+ (n) = {ξ ∈ R+ n : |ξ | = 1} and S1 (m) = {ζ ∈ Rm : |ζ | = 1}. Since the function u(x, y) (y = (t, t2 , . . . , tm )) is a function u(x, t) extended as a constant by variables t2 , . . . , tm , we have γ

rζ1

Tyrζ u(x, y) = δ Tt

rζ1

u(x, t, t2 , . . . , tm ) = δ Tt rζ

u(x, t) · 1(t2 . . . , tm ).

rζ

We agree to write further δ Tt 1 u(x, t) instead of δ Tt 1 u(x, t) · 1(t2 . . . , tm ). Taking into account that T0t u(x, 0) = Tt0 u(x, t) = u(x, t), we obtain γ

Tyrζ u(x, y1 )|y1 =0 = δ Tyrζ1 1 u(x, y1 )|y1 =0 = u(x, rζ1 ).

Therefore, setting t = 0 in (7.102) and using the first condition in (7.85), we obtain 1 + |S1 (n)|γ

 γ S1+ (n)

Txrξ f (x) ξ γ dS =

1 + |S1 (m)|γ 





u(x, rζ1 ) ζ γ dS. S1+ (m)

Differential equations with Bessel operator

313

Therefore, to find the unknown function u = u(x, t) it remains to solve the integral equation  1  γ u(x, rζ1 ) ζ γ dS = Mr f (x). (7.103) + |S1 (m)|γ  S1+ (m)

Multiplying both sides of equality (7.103) by r n+|γ |−1 and integrating over r from 0 to s, we obtain 1 + |S1 (m)|γ 

s r

n+|γ |−1

 dr

u(x, rζ1 ) ζ

γ

s dS =

S1+ (m)

0

r n+|γ |−1 Mr f (x)dr. γ

0

Consider the left side of this equality, which we denote by J . Returning to the rectangular Cartesian coordinates by the formula z = rζ , we obtain ⎛ ⎞  s 1  γ J= + u(x, z1 ) zγ dz , ⎝= r n+|γ |−1 Mr f (x)dr ⎠ , |S1 (m)|γ  Bs+ (m)

0

+. where Bs+ (m) is a part of a ball of radius s with center in the origin belonging to Rm Given the continuity of the integrand, we can write it in the repetitive form

1 J= + |S1 (m)|γ 

s

 u(x, z1 ) z1δ

γ

γ

z22 . . . zmm dz2 . . . dzm ,

dz1 +

0

" #   2 < s 2 − z 2 ∈ R+ . where + = (z2 , . . . , zm ) : z22 + . . . + zm m−1 1 The integral by the (m − 1)-dimensional domain + can be easily calculated by transition to spherical coordinates z2 =ρ θ1 , . . . , zm =ρ θm−1 . We have 

 s 2 −z12 γ z22

γ . . . zmm dz2 ...dzm

+



=

ρ 0

m+γ2 +...+γm −2

 dρ

γ

γ

m θ1 2 ...θm−1 dS.

S1+ (m−1)

(7.104) The inner integral on the right side of equality (7.104) can be calculated by formula (1.107):       γ2 +1 γm +1   ...   2 2 γ2 γm  (7.105) θ1 ...θm−1 dS =   . m−2  m−1+γ2 +...+γm 2 + 2 S1 (m−1)

314

Transmutations, Singular and Fractional Differential Equations

The external integral on the right side of equality (7.104) is easily found by the formula 

s 2 −z12



ρ

m+γ2 +...+γm −2

0

 −1 m+γ2 +...+γm

2 (s 2 − z12 ) dρ = .  m + γ2 + ... + γm − 1

(7.106)

Since m + γ2 + ... + γm = n + |γ | − δ, formulas (7.105) and (7.106) can be rewritten in the forms       γ2 +1 γm +1  ...  2 2 γ2 γm   (7.107) θ1 ...θm−1 dS = n+|γ |−δ−1 2m−2  + 2 S1 (m−1)

and 

s 2 −z12



ρ

n+|γ |−δ−1

m+γ2 +...+γm −2

0

(s 2 − z12 ) 2 dρ = . n + |γ | − δ − 1

(7.108)

Substituting (7.107) and (7.108) into (7.104), we get       γ2 +1 γm +1  ...   n+|γ |−δ−1 2 2 1 γ2 γm   (s 2 −z12 ) 2 z2 ...zm dz2 ...dzm = . n+|γ |−δ−1 n+|γ |−δ−1 2m−2  +

2

(7.109) Using formula (1.107), we obtain |S1+ (m)|γ  :         γ2 +1 γm +1  δ+1  ... 2 2 2 +    . |S1 (m)|γ = n+|γ | 2m−1  2

(7.110)

Applying (7.109) and (7.110), we get  1  u(x, z1 ) zγ dζ = |S1+ (m)|γ  Bs+ (m)



2 (n + |γ | − δ − 1)

n+|γ | 2



δ+1 2



    n+|γ |−δ−1 2

So we obtain s (s 2 − z12 ) 0

n+|γ |−δ−1 2

u(x, z1 )z1δ dz1 =

s (s 2 − z12 ) 0

n+|γ |−δ−1 2

u(x, z1 )z1δ dz1 .

Differential equations with Bessel operator

 

315

   s  n+|γ |−δ+1 2 γ   r n+|γ |−1 Mr f (x)dr. n+|γ |  2 0

δ+1 2

(7.111)

In (7.111) the formula z(z) = (z + 1) was used. Equality (7.111) is the Abel equation with respect to u(x, t) (see [494]). Multiplyδ−n−|γ |−1 ing both parts of (7.111) by 2s(t 2 − s 2 ) 2 and integrating by s from 0 to t, we obtain t 2s(t − s ) 2

0



2

2

δ−n−|γ |−1 2

s (s 2 − z12 )

ds

n+|γ |−δ−1 2

u(x, z1 )z1δ dz1 =

0

   t s  n+|γ |−δ+1 δ−n−|γ |−1 2 γ   s(t 2 − s 2 ) 2 ds r n+|γ |−1 Mr f (x)dr. n+|γ |  2 0 0

δ+1 2

In the left part, we change the order of integration: t

t (t 2 − s 2 )

u(x, z1 )z1δ dz1 0



2

δ−n−|γ |−1 2

(s 2 − z12 )

n+|γ |−δ−1 2

2sds =

z1

   t s  n+|γ |−δ+1 δ−n−|γ |−1 2 γ 2 2 2   s (t − s ) ds r n+|γ |−1 Mr f (x)dr. n+|γ |  2 0 0

δ+1 2

(7.112) In the inner integral on the left side of (7.112) replacing s 2 by h, we get t t 2 δ−n−|γ |−1 n+|γ |−δ−1 δ−n−|γ |−1 n+|γ |−δ−1 (t 2 −s 2 ) 2 (s 2 −z12 ) 2 2sds = (t 2 −h) 2 (h−z12 ) 2 dh. z1

z12

In the resulting integral, we introduce a new variable τ by the formula h = ζ12 + τ (t 2 − ζ12 ): dh = (t 2 − ζ12 )dτ, h = ζ12 ,

τ = 0;

t 2 − h = (1 − τ )(t 2 − ζ12 ), h = t 2,

h − ζ12 = τ (t 2 − ζ12 ),

τ = 1.

We have t (t 2 − s 2 ) z1

δ−n−|γ |−1 2

(s 2 − z12 )

n+|γ |−δ−1 2

1 2sds =

τ 0

n+|γ |−δ−1 2

(1 − τ )

δ−n−|γ |−1 2

dτ =

316

Transmutations, Singular and Fractional Differential Equations



n + |γ | − δ + 1 2





 δ − n − |γ | + 1 . 2

Returning to (7.112), we can write

  t n + |γ | − δ + 1 δ − n − |γ | + 1 u(x, z1 )z1δ dz1 =  2 2 0     t n+|γ |−δ+1  s 2 δ+1  δ−n−|γ |−1 2 2 γ 2 2   s (t − s ) 2 ds r n+|γ |−1 Mr f (x)dr n+|γ |  2 0 0



or t u(x, z1 )z1δ dz1 = 0

  t s 2 δ+1 δ−n−|γ |−1 2 γ 2 2     s (t − s ) 2 ds r n+|γ |−1 Mr f (x)dr. n+|γ | δ−n−|γ |+1   2 2 0 0 (7.113)

Differentiating both sides of equality (7.113) by t and dividing by t δ we get u(x, t) =

  t s 2 δ+1 δ−n−|γ |−1 2 1 d γ 2 2     s (t − s ) 2 ds r n+|γ |−1 Mr f (x)dr. n+|γ | δ−n−|γ |+1 t δ dt   2 2 0 0 (7.114)

Application in (7.114) of the formula for the Euler beta function (1.9) completes the proof.

7.3 Elliptic equations with Bessel operator 7.3.1 Weighted homogeneous distributions Here, following the approach of Gelfand and Shapiro [178] (see also [170,177]), we study weighted homogeneous generalized functions. The research of homogeneous distributions is important because the fundamental solutions of many differential operators are homogeneous distributions. If a differential operator contains a Bessel operator, it is natural to use weighted instead of ordinary distributions. We also note that when we consider analytic continuation of weighted homogeneous distributions,

Differential equations with Bessel operator

317

we thereby give a method for analytic continuation for more general distributions, since many functions in the neighborhood of singular points can be approximated by homogeneous ones. Further, we will apply these results to finding the fundamental solution to elliptic equations with Bessel operator. a and x a on R are For all a > −1, functions x+ − " a x > 0, x a x+ = 0 x ≤0 and

" a = x−

x ≥ 0, x < 0.

0 |x|a

a is locally integrable with the weight x γ , and therefore determines the Function x+ weighted distribution

∞ a (x+ , ϕ)γ

=

x a ϕ(x)x γ dx. 0

Similarly, 0 a , ϕ)γ (x−

= (−1)

a+γ

x a ϕ(x)x γ dx. −∞

Functions

a x+

and

a x−

are connected by the equality

a a (x− , ϕ)γ = (x+ , ϕ) ˘ γ,

ϕ(x) ˘ = ϕ(−x).

a is homogeneous of degree a for a > −1, i.e., the following equality for Function x+ t > 0 is valid:

∞ a (x+ , ϕ)γ

ϕt (x) = t

= 0 1+γ

∞ x ϕ(x)x dx = t a

γ

a x a ϕ(tx)t 1+γ x γ dx = t a (x+ , ϕt )γ ,

a 0

ϕ(tx).

Let us find (x ± i0)a : (x ± i0)a = ea ln(x±i0) = ea ln(x±i0) = ea(ln |x±i0|+iarg (x±i0)) = " a x ≥ 0, x a(ln |x|+iarg (x±i0)) a a = x+ e = + e±iπa x− . e±iπa |x|a x < 0 Therefore, a a + e±iπa x− . (x ± i0)a = x+

(7.115)

318

Transmutations, Singular and Fractional Differential Equations

Definition 42. Let function u ∈ L1loc,γ (Rn+ ) be homogeneous of degree a, i.e., u(tx) = t a u(x) for t > 0. The weighted distribution u is homogeneous of degree a in Rn+ if the following equality is valid: (u, ϕ)γ = t a (u(x), ϕt (x))γ ,

(7.116)

◦

n + where ϕt (x) = t n+|γ | ϕ(tx), ϕ ∈ C ∞ ev (Rn ). If u is a weighted distribution in R + and ◦

+ (7.116) is valid for all ϕ ∈ C ∞ ev (Rn ), then u is said to be homogeneous of degree a in n R +.

Let us comment on Definition 42. If the function u ∈ L1loc,γ (Rn+ ) is homogeneous of degree a, i.e., u(tx) = t a u(x), for t > 0, then  (u(y), ϕ(y))γ = u(y)ϕ(y)y γ dy = {y = tx, t > 0} Rn+

= t n+|γ |+a

 u(x)ϕ(tx)x γ dx = t a (u(x), ϕt (x))γ ,

Rn+ ◦

+ where ϕt (x) = t n+|γ | ϕ(tx), ϕ ∈ C ∞ ev (Rn ). And vice versa, from the relation a (u(y), ϕ(y))γ = t (u(x), ϕt (x))γ it follows that u is homogeneous. If a > −n − |γ |, then u is integrable with weight x γ in some neighborhood of zero, since in polar coordinates x = rω, |ω| = 1, we have dx = r n+|γ |−1 ωγ drdω and

ε

 |u(rω)|x dx = γ

r a+n+|γ |−1 dr

=

|u(ω)|ωγ dω

{|ω|=1}+

0

Uε (0)





ε a+n+|γ | a + n + |γ |

|u(ω)|ωγ dω < ∞. {|ω|=1}+

The problem which we will discuss is the extension of weighted homogeneous disn tributions from R+ n to R + . First we prove the following theorem about homogeneity conditions for weighted homogeneous distributions. Theorem 90. The homogeneity conditions for weighted homogeneous distributions u of degree a (u, ϕ)γ = t a (u, ϕn,γ ,t )γ ,

ϕn,γ ,t (x) = t n+|γ | ϕ(tx),

◦

+ ϕ(x) ∈ C ∞ ev (Rn )

(7.117) and (u, ψ)γ = 0,

ψ

◦

+ ∈C∞ ev (Rn ),

∞ 0

r a+n+|γ |−1 ψ(rx)dr = 0

(7.118)

Differential equations with Bessel operator

319

are equivalent. In addition, for weighted homogeneous distributions u of degree a the formula n 

xk

k=1

∂u = au ∂xk

(7.119)

is valid. Proof. Differentiating equality (7.117) by t we get at a−1 (u(x), t n+|γ | ϕ(tx))γ + 

a n+|γ |−1 n+|γ |−1 dϕ(tx) ϕ(tx) + t = 0. t u(x), (n + |γ |)t dt γ

(7.120)

Since dϕ(tx)  ∂ϕ(tx) d(txk )  ∂ϕ(tx) xk , = = dt ∂xk dt ∂xk n

n

k=1

k=1

equality (7.120) can be written in the form at a−1 (u(x), t n+|γ | ϕ(tx))γ +  t

a

u(x), (n + |γ |)t

n+|γ |−1

ϕ(tx) + t

n+|γ |−1

n  k=1

∂ϕ(tx) xk ∂xk

 = 0.

(7.121)

γ

Putting t = 1 in (7.121), we obtain (a + n + |γ |)(u, ϕ)γ + (u, λϕ)γ = 0, where λ =

n  k=1

(7.122)

xk ∂x∂ k .

Let us consider the equation (a + n + |γ |)ϕ(x) +

n  k=1

xk

∂ ϕ(x) = ψ(x) ∂xk

(7.123)

◦

+ and show that it has a solution in C ∞ ev (Rn ). Using in (7.123) spherical coordinates x = rω, we get

∂ a+n+|γ | ϕ(rω)) = ψ(rω)r a+n+|γ |−1 . (r ∂r Indeed, ∂ a+n+|γ | ∂ϕ(rω) ϕ(rω)) = (a + n + |γ |)r a+n+|γ |−1 ϕ(rω) + r a+n+|γ | (r = ∂r ∂r

320

Transmutations, Singular and Fractional Differential Equations

(a + n + |γ |)r a+n+|γ |−1 ϕ(rω) + r a+n+|γ |

n  ∂ϕ(rω) ∂(rωk ) k=1

(a + n + |γ |)r a+n+|γ |−1 ϕ(rω) + r a+n+|γ |−1

∂(rωk )

n 

rωk

k=1

 r a+n+|γ |−1 (a + n + |γ |)ϕ(x) +

n  k=1

∂r

=

∂ϕ(rω) = ∂(rωk )

 ∂ xk ϕ(x) = r a+n+|γ |−1 ψ(x). ∂xk

Consequently, for ψ = (a + n + |γ |)ϕ + λϕ,

(7.124)

the equality ∞ r

a+n+|γ |−1

∞ ψ(rx)dr =

0

0

∞  ∂ a+n+|γ | ϕ(rω))dr = r a+n+|γ | ϕ(rω) = 0 (r ∂r 0

◦

+ is true and ψ ∈ C ∞ ev (Rn ). So, from (7.124) and (7.122) equality (7.118) follows. Now let us prove (7.119). For (u, λϕ)γ we have   n n    ∂ϕ(x) ∂ϕ(x) γ xk = u(x)xk x dx = (u, λϕ)γ = u(x), ∂xk ∂xk k=1

n   k=1

k=1Rn +

γ

⎡∞ ⎤  ∂ϕ(x) γk +1 ⎦ u(x) x dxk . ∂xk k

γk−1 γk+1 γ γ x1 1 ...xk−1 xk+1 ...xnn dx1...dxk−1 dxk+1 ...dxn⎣

0

Rn−1 +

(7.125) We apply the integration formula by parts to the integral over xk : ∞ u(x) 0

" # ∂ϕ(x) γk +1 ∂ϕ(x) γ +1 xk dxk = U = u(x)xk k , dV = dxk = ∂xk ∂xk ∞

γ +1  u(x)ϕ(x)xk k 

∞ − 0

0

∞ ∂u γ − + (1 + γk )u(x) ϕ(x)xk k dxk = xk ∂xk 0

∂u γ xk ϕ(x)xk k dxk − (1 + γk ) ∂xk

∞

γ

u(x)ϕ(x)xk k dxk . 0

Summing by k from 1 to n and returning to the integral (7.125), we get (u, λϕ)γ = − (λ, ϕ)γ − (n + |γ |)(u, ϕ)γ .

(7.126)

Differential equations with Bessel operator

321

Substituting (7.126) in (7.122), we obtain  n   ∂u xk , ϕ(x) −(n+|γ |)(u(x), ϕ(x))γ = 0 (a +n+|γ |)(u(x), ϕ(x))γ − ∂xk k=1

or a(u(x), ϕ(x))γ =

 n  k=1

γ

 ∂u xk , ϕ(x) , ∂xk γ

which gives (7.119).

7.3.2 Extension of the weighted homogeneous distributions  (Rn ) be weighted homogeneous distributions u of degree a. Theorem 91. Let u ∈ Dev +  (R n ) If a = k, k ∈ Z, k ≤ −n − |γ |, then u has a unique weighted extension u∗ ∈ Dev + ∗ ∗ homogeneous of degree a. If a = 1 − n − |γ |, then (Bγj u) = Bγj u . The map u → u∗ is continuous.  (R n ), homoProof. We first prove the existence of a weighted distribution u∗ ∈ Dev + n  geneous of degree a, which is an extension of u ∈ Dev (R+ ). ◦

n If u is a function and ϕ ∈ C ∞ ev (R+ ), then using spherical coordinates x = rω we get

∞

 (u, ϕ)γ =



u(ω)ϕ(rω)r a+n+|γ |−1 ωγ drdω.

u(x)ϕ(x)x dx = γ

0 {|ω|=1}+

Rn+

Based on this equality, we introduce the one-dimensional distribution a+n+|γ |−1

(Ra ϕ)(x) = (t+

, ϕ(tx)),

◦

n ϕ∈C∞ ev (R + ).

(7.127)

The function Ra ϕ is homogeneous of degree −n − |γ | − a, i.e., (Ra ϕ)(bx) = b−n−|γ |−a (Ra ϕ)(x). Indeed, a+n+|γ |−1 , ϕ(btx)) = (Ra ϕ)(bx) = (t+

∞

t a+n+|γ |−1 ϕ(btx)dt = {bt = y} =

0

b−n−|γ |−a

∞

y a+n+|γ |−1 ϕ(yx)dy = b−n−|γ |−a (Ra ϕ)(x).

0 ◦

◦

n ∞ From [170] it follows that Ra is a continuous map from C ∞ ev (K) to C ev (R+ ) for every compact set K ⊂ Rn+ .

322

Transmutations, Singular and Fractional Differential Equations ◦

n We choose a fixed function ψ ∈ C ∞ ev (R+ ) such that

∞ ψ(tx)

dt = 1, t

x = 0.

0 ◦

n Then ψRa ϕ ∈ C ∞ ev (R+ ) and

∞ Ra (ψRa ϕ)(x) =

t a+n+|γ |−1 ψ(tx)(Ra ϕ)(tx)dt =

0

∞ (Ra ϕ)(x)

ψ(tx)

dt = (Ra ϕ)(x). t

0 ◦

n So, u(ψRa ϕ) is always independent of ψ and u(ψRa ϕ) = u(ϕ) if ϕ ∈ C ∞ ev (R+ ). Thus

(u∗ , ϕ)γ = (u, ψRa ϕ)γ ,

◦

n ϕ∈C∞ ev (R + )

defines a distribution u∗ in R n+ which extends u. The map u → u∗ is continuous. Since a+n+|γ |−1

(Ra ϕn,γ ,t )(x) = (r+

, t n+|γ | ϕ(rtx)) = t −a Ra ϕ(x),

that gives homogeneity of u∗ . Finally we note that (Bγj u)∗ − Bγj u∗ is homogeneous of degree a − 2 and supported by 0, so it must be zero. This completes the proof of the theorem.

7.3.3 Weighted fundamental solution of the Laplace–Bessel operator Weighted fundamental solutions are very important in the study of existence and regularity of solutions of differential equations with Bessel operators.  (Rn ) is called a weighted fundamental soluDefinition 43. A distribution E ∈ Dev + m  ai Bγi with constant tion of the differential operator with Bessel operators L =

(complex) coefficients if LE = δγ .

i=1

 (Rn ) all be homogeneous of degree 2 − n − |γ | in Theorem 92. Let u1 , ..., un ∈ Dev + n  n Bγj uj = 0. Then R+ and satisfy the condition j =1

Differential equations with Bessel operator n 

323

Bγj u∗j = cδγ ,

j =1

where c is some constant. n  Proof. The distribution Bγj u∗j is homogeneous of degree −n − |γ | and supported j =1

by 0, so n 

Bγj u∗j = cδγ

j =1

for some constant c. Lemma 19. Let x ∈ Rn+ , n > 1, and ⎧ 1 ⎪ n + |γ | = 2, ⎨ |S1+ (n)|γ ln |x| E(x) = |x|2−n−|γ | ⎪ ⎩ n + |γ | > 2, (2−n−|γ |)|S + (n)| 1

γ

where |S1+ (n)|γ is defined by (1.107). Then for |x| > ε ∀ε > 0 γ E(x) = 0. Proof. Let us consider n + |γ | > 2. We have γ E(x) =

n 

Bγj E(x) =

j =1

n  1 γ x j j =1 j

∂ γj ∂ x E(x) = ∂xj j ∂xj

n  1 1 ∂ γj ∂ x |x|2−n−|γ | = γ + (2 − n − |γ |)|Sn |γ j =1 xj j ∂xj j ∂xj n  1 1 ∂ γj (2 − n − |γ |) −n−|γ | x 2xj = |x| γ + 2 (2 − n − |γ |)|Sn |γ j =1 xj j ∂xj j

1

n  1

|Sn+ |γ j =1 n 

γ xj j

|Sn+ |γ

xj j

1

1

1 γ

j =1 n 

∂ 1+γ |x|−n−|γ | xj j = ∂xj

(−n − |γ |) −n−|γ |−2 2+γj −n−|γ | γj 2xj + (1 + γj )|x| xj = |x| 2

[(−n − |γ |)|x| |Sn+ |γ j =1 1 |Sn+ |γ

−n−|γ |−2 2 xj

+ (1 + γj )|x|−n−|γ | ] =

[(−n − |γ |)|x|−n−|γ | + (n + |γ |)|x|−n−|γ | ] = 0.

324

Transmutations, Singular and Fractional Differential Equations

We now consider the case n + |γ | = 2. We get γ E(x) =

n 

Bγj E(x) =

j =1 n  1

1 |Sn+ |γ 1 |Sn+ |γ 1 |Sn+ |γ 1 |Sn+ |γ

γ x j j =1 j n 

1

γ x j j =1 j n 

n  1 γ x j j =1 j

∂ γj ∂ x E(x) = ∂xj j ∂xj

n ∂ γj ∂ 1  1 ∂ 1+γ xj ln |x| = + |x|−2 xj j = γ ∂xj ∂xj |Sn |γ j =1 xj j ∂xj 2+γj

[−2|x|−4 xj

γ

+ (1 + γj )|x|−2 xj j ] =

[−2|x|−4 xj2 + (1 + γj )|x|−2 ] =

j =1

[−2|x|−2 + (n + |γ |)|x|−2 ] = 0.

Theorem 93. Let x ∈ R+ n , n > 1, and ⎧ 1 ⎪ n + |γ | = 2, ⎨ |S + |γ ln |x| 2 E(x) = |x|2−n−|γ | ⎪ ⎩ n + |γ | > 2. (2−n−|γ |)|S + | n γ

Then Bγj E ∈ L1loc,γ (Rn+ ) and γ E = δγ . ◦

n Proof. First let us prove that Bγj E ∈ L1loc,γ (Rn+ ). For ϕ ∈ C ∞ ev (R + ), we have

 

 (Bγj E, ϕ)γ =

Bγj E(x) ϕ(x)x dx = γ

Rn+



Rn+

 ∂ γj ∂ x E(x) ϕ(x)x γ dx = ∂xj j ∂xj

1 γ

xj j

∞

γj −1 γj +1 γ γ x1 1...xj −1 xj +1 ...xnn dx1 ...dxj −1 dxj +1 ...dxn

 ∂ γj ∂ xj E(x) ϕ(x)dxj. ∂xj ∂xj

0

Rn−1 +

Integrating by parts by xj we obtain  ∞ ∞ 1 ∂ γj ∂ γj γ (Bγj E(x)) ϕ(x)xj dxj = xj E(x) ϕ(x)xj j dxj = γ ∂xj xj j ∂xj 0

∞ 0

0

 " # ∂ γj ∂ ∂ γj ∂ x E(x) ϕ(x)dxj = U = ϕ(x), dV = x E(x)dxj = ∂xj j ∂xj ∂xj j ∂xj

Differential equations with Bessel operator

γ xj j

325

∞ ∞ 

 ∂ ∂ ∂ γj  E(x)ϕ(x) − xj E(x) ϕ(x)dxj = ∂xj ∂x ∂x j j 0 0

∞ − "

 ∂ γ ∂ E(x) xj j ϕ(x)dxj = ∂xj ∂xj

0

# ∂ ∂ ϕ(x), dV = E(x)dxj = U ∂xj ∂xj ∞ ∞   ∂ γj ∂ γj ∂ ϕ(x)E(x) + xj ϕ(x) E(x)dxj = −xj ∂xj ∂xj ∂xj 0 γ = xj j

∞

0

1 γ

0

xj j



∂ γj ∂ γ x ϕ(x) E(x)xj j dxj = ∂xj j ∂xj

∞ γ (Bγj ϕ(x)) E(x)xj j dxj . 0

Therefore,  (Bγj E, ϕ)γ = (E, Bγj ϕ)γ = lim ⎡ ⎢ lim ⎣

ε→0 {|x|>ε}+



 (Bγj ϕ(x)) E(x)x γ dx −

ε→0

(Bγj ϕ(x)) E(x)x γ dx = ⎤ ⎥ (Bγj ϕ(x)) E(x)x γ dx ⎦ .

{|x|≤ε}+

Rn+

Using formula (1.99) we obtain 

 (Bγj E, ϕ)γ = (Bγj ϕ(x)) E(x)x dx − lim γ

ε→0 {|x|=ε}+

R+ n

∂ϕ(x) E(x) cos( ν , e j )x γ dS, ∂xj

where ν is the direction of the outer normal to the boundary {|x| = ε}+ and e j is the x direction of axis Oxj . Since cos( ν , e j ) = |x|j , we have 

 (Bγj E, ϕ)γ =

(Bγj ϕ(x)) E(x)x γ dx − lim

Rn+

Moreover, we have  lim

ε→0 {|x|=ε}+

xj ∂ϕ(x) E(x) x γ dS = 0 ∂xj |x|

ε→0 {|x|=ε}+

xj ∂ϕ(x) E(x) x γ dS. ∂xj |x|

326

Transmutations, Singular and Fractional Differential Equations

and thus



(Bγj E, ϕ)γ =

(Bγj E(x)) ϕ(x)x γ dx,

Rn+

i.e., the weighted distribution Bγj E defined by a function Bγj E(x) locally integrable with the weight x γ . Since for |x| > ε, ∀ε > 0, the equality γ E(x) = 0 holds, using again formula (1.99) we obtain  (γ E, ϕ)γ = (E, γ ϕ)γ = lim

ε→0 {|x|>ε}+

 lim

ε→0 {|x|>ε}+



lim

ε→0 {|x|=ε}+

E(x)(γ ϕ(x))x γ dx =

[E(x)(γ ϕ(x)) − (γ E(x))ϕ(x)]x γ dx = 

∂E(x) ∂ϕ(x) − ϕ(x) x γ dS = ϕ(0). E(x) ∂ ν ∂ ν

This result also follows from Theorem 92.

7.3.4 The Dirichlet problem for an elliptic singular equation In this subsection we consider the second order singular elliptic equations of the form   n  ∂ 2 u γi ∂u + (7.128) = b2 u. ∂xi2 xi ∂xi i=1

For b > 0 we find an analytical solution to the Dirichlet problem for this equation. A procedure for obtaining its solution is proposed based on the use of a modified Bessel function of the second kind. Elliptic equations with the Bessel operator occur in modern models of mathematical physics. Methods for solving problems for elliptic equations with the Bessel operator have been sufficiently developed by A. Weinstein [592,594] and I. A. Kipriyanov [242]. In the paper of M. B. Kapilevic [217] the theory of degenerate elliptic differential equations of Bessel class were considered. Here following [216] we obtain the solution to the Dirichlet problem for Eq. (7.128) and get some properties of that solution. 2 be a bounded function, γ < 1. The Theorem 94. Let τ (x  )=τ (x1 , ..., xn−1 )∈Cev n solution to the Dirichlet problem

(γ )x u = b2 u,

b > 0,

(7.129)

Differential equations with Bessel operator

327

u(x1 , x2 , ..., xn−1 , 0) = τ (x1 , x2 , ..., xn−1 ), is

uxi (x)|xi =0 = 0

(7.130)

  γn −n−|γ  |  x y  u(x)=C(n,γ ) [ γ Txn τ ](x  ) (1+|y  |) 2 K n+|γ  |−γn (bxn 1+|y  |)(y  )γ dy , 2

Rn−1 +

(7.131) 

where (y  )γ =

n−1  i=1

γ

yi i ,

   2n−1  n+|γ2|−γn C(n, γ ) =   n−1  , 1−γn  γi +1   2 2

Kν (r) =

21−ν ν r Kν (r), (ν)

i=1

such that Kν (0) = 1 when ν > 0. Proof. Let us show that a solution to Eq. (7.129) is 1−γn

k n+|γ  |−γn (b|x|) =

2  n+|γ  |−γn γn −n−|γ  | n + |γ  | − γn 1−γ 2 2  + 1 xn n |x| 2 K n+|γ  |−γn (b|x|) = 2 2

 n+|γ  |−γn γ −n−|γ  | n + |γ  | − γn n 1−γ 2 2  + 1 xn n (x12 + ... + xn2 ) 4 × 2

  K n+|γ  |−γn b x12 + ... + xn2 . u = xn

2

Taking into account that (see [591], p. 96, formula (6)) ∂ −ν z Kν (z) = −z−ν Kν+1 (z), ∂z we can easily check that u is a solution to (7.129). When i = 1, ..., n − 1, we have for ∂u γi ∂u ∂2u 2 ∂xi , xi ∂xi , and ∂xi

γn −n−|γ  | 1 ∂u 1−γ = −bxi C(n, γ )xn n (x12 + ... + xn2 ) 4 − 2 K n+|γ  |−γn +1 × ∂x 2

i  b x12 + ... + xn2 , γn −n−|γ  | 1 γi ∂u 1−γ = −bγi C(n, γ )xn n (x12 + ... + xn2 ) 4 − 2 K n+|γ  |−γn +1 × x ∂x 2

i  i  b x12 + ... + xn2 ,

328

Transmutations, Singular and Fractional Differential Equations | 1 ∂ 2u 1−γn 2 2 γn −n−|γ −2 4 = −bC(n, γ )x (x + ... + x ) K n+|γ  |−γn +1 × n n 1 2 ∂xi2 

 b x12 + ... + xn2 +

  | 1−γn 2 2 2 2 γn −n−|γ −1 2 2 4 K n+|γ  |−γn +2 b x1 + ... + xn . b xi C(n, γ )xn (x1 + ... + xn ) 2

Considering that Bγi u = n−1 

∂2u ∂xi2

+

γi ∂u xi ∂xi

and summing by i from 1 to n − 1, we obtain

Bγi u = −b(n − 1 + |γ  |)C(n, γ )×

i=1

  2 2 b x1 + ... + xn + 2

  | 1−γn 2 2  2 2 γn −n−|γ −1 2 2 4 b |x | C(n, γ )xn (x1 + ... + xn ) K n+|γ  |−γn +2 b x1 + ... + xn . 1−γ xn n (x12



γn −n−|γ | 1 + ... + xn2 ) 4 − 2 K n+|γ  |−γn +1

2

Now let us find

∂u γn ∂u ∂xn , xn ∂xn ,

and

∂2u : ∂xn2

  γn −n−|γ  | ∂u −γ = C(n, γ )(1−γn )xn n (x12 + ...+xn2 ) 4 K n+|γ  |−γn b x12 + ...+xn2 − ∂xn 2

  | 1 γ −n−|γ n 2−γ C(n, γ )bxn n (x12 + ... + xn2 ) 4 − 2 K n+|γ  |−γn +1 b x12 + ... + xn2 , 2

γn ∂u = xn ∂xn

  2 2 K n+|γ  |−γn b x1 + ... + xn − 2

  γn −n−|γ  | 1 1−γn 2 2 −2 2 2 4 C(n, γ )bγn xn (x1 + ... + xn ) K n+|γ  |−γn +1 b x1 + ... + xn ,

−1−γn 2 (x1 C(n, γ )γn (1 − γn )xn

γn −n−|γ + ... + xn2 ) 4

|

2

∂ 2u = ∂xn2

  K n+|γ  |−γn b x12 + ...+xn2 − 2

  | 1 γ −n−|γ n 1−γ C(n, γ )b(1 − γn )xn n (x12 + ...+xn2 ) 4 − 2 K n+|γ  |−γn +1 b x12 + ...+xn2 − 2

  | 1 γ −n−|γ n 1−γ C(n, γ )b(2 − γn )xn n (x12 + ...+xn2 ) 4 − 2 K n+|γ  |−γn +1 b x12 + ...+xn2 + 2

  | γ −n−|γ n 2 3−γn 2 2 −1 2 2 C(n, γ )b xn (x1 + ...+xn ) 4 K n+|γ  |−γn +2 b x1 + ...+xn . −1−γn

C(n, γ )(−γn )(1−γn )xn

(x12 + ...+xn2 )

γn −n−|γ  | 4

2

Differential equations with Bessel operator

Considering that Bγn u =

∂2u ∂xn2

+

329

γn ∂u xn ∂xn ,

we obtain

Bγn u =

  K n+|γ  |−γn +1 b x12 + ...+xn2 + 2

  | γ −n−|γ n 3−γ C(n, γ )b2 xn n (x12 + ... + xn2 ) 4 −1 K n+|γ  |−γn +2 b x12 + ... + xn2 . 1−γn

(γn −3)C(n, γ )bxn

(x12 + ...+xn2 )

γn −n−|γ  | 1 −2 4

2

Using the recurrence formula Kν+1 (z) − γ u =

n−1 

2ν z Kν (z) = Kν−1 (z)

(see [591]), we get

Bγi u + Bγn u = −b(n + |γ  | − γn + 2)C(n, γ )xn

1−γn

×

i=1

  2 2 b x1 + ... + xn + 2

  | 2 1−γn 2 2 γn −n−|γ 2 2 C(n, γ )b xn (x1 + ... + xn ) 4 K n+|γ  |−γn +2 b x1 + ... + xn = 2

  | γ −n−|γ n 1−γn 2 2 2 2 2 4 K n+|γ  |−γn b x1 + ... + xn = b C(n, γ )xn (x1 + ... + xn ) 2

  1−γ b2 xn n k n+|γ  |−γn b x12 + ... + xn2 . (x12



γn −n−|γ | 1 + ... + xn2 ) 4 − 2 K n+|γ  |−γn +1

2

1−γn

So u = xn Since

k n+|γ  |−γn (br) is a solution to (7.129). 2

 γ  Ty ( ) γ x x

y



1−γn γ  y  [ Tx  k n+|γ  |−γn ](b|x|)

= (γ )x γ Tx  , xn

is also a solu-

2

tion to (7.129). Moreover, if g(x  ) = g(x1 , ..., xn−1 ) is a twice continuously differentiable, bounded function such that gxi (x  )|xi =0 = 0 for i = 1, ..., n − 1, then  y  1−γ xn n g(y  )[ γ Tx  k n+|γ  |−γn ](b|x|)(y  )γ is also a solution to (7.129). Integrating by 2

y  , the function 1−γ u(x) = xn n

1−γ xn n







y



g(y  )[ γ Tx  k n+|γ  |−γn ](b|x|)(y  )γ dy  = 2

Rn−1 +

  y  [ γ Tx  g](x  )k n+|γ  |−γn (b |y  |2 + xn2 )(y  )γ dy  2

Rn−1 +

satisfies Eq. (7.129). Changing variables y  → xn y  , we get   γn −n−|γ  |  x y  [ γ Txn g](x  ) (1 + |y  |2 ) 2 K n+|γ  |−γn (bxn 1 + |y  |2 )(y  )γ dy  . u(x) = 2

Rn−1 +

(7.132)

330

Transmutations, Singular and Fractional Differential Equations

Putting xn = 0, we obtain  γn −n−|γ  |    τ (x ) = g(x ) (1 + |y  |2 ) 2 (y  )γ dy  = {y  = σρ} = Rn−1 +

∞ γn −n−|γ  |  g(x ) (1 + ρ 2 ) 2 ρ n+|γ |−2 dρ







σ γ dS =

S1+ (n−1)

0

   γi 2+1 ∞ γn −n−|γ  | 1  i=1    (1 + ρ 2 ) 2 ρ n+|γ |−2 dρ = g(x ) g(x  )  n−1+|γ | C(n, γ ) n−2 2  2 0 n−1 

or

g(x  )

=

C(n, γ )τ (x  ),

where C(n, γ ) =

   2n−1  n+|γ 2|−γn ,  n−1    γi +1  n  1−γ  2 2

γn < 1. Substituting

i=1

the expression for g(x  ) into equality (7.132) gives (7.131). Since Kν (r) is exponentially decaying when r → ∞, the integral (7.131) is uniformly and absolutely convergent. 2 be a bounded function, γ < 1, Theorem 95. Let τ (x  ) = τ (x1 , ..., xn−1 ) ∈ Cev n

   2n−1  n+|γ2|−γn C(n, γ ) =   n−1  , 1−γn  γi +1   2 2

Kν (r) =

21−ν ν r Kν (r). (ν)

i=1

The function  u(x)=C(n,γ )



x y

[ γ Txn τ ](x  ) (1+|y  |)

γn −n−|γ  | 2

  K n+|γ  |−γn (bxn 1+|y  |)(y  )γ dy  2

Rn−1 +

(7.133) 2 bounded at the orthant Rn and vanishes when x → ∞. is a function from Cev n +

Proof. Let |τ (x  )| ≤ M for 0 < xi ≤ ∞, i = 1, ..., n − 1. Then using property (3.149) of generalized translation, we obtain   γn −n−|γ  |  |u(x)| ≤ M · C(n, γ ) (1 + |y  |) 2 K n+|γ  |−γn (bxn 1 + |y  |)(y  )γ dy  . 2

Rn−1 +

Differential equations with Bessel operator

331

Passing to spherical coordinates y  = σρ, we get   γn −n−|γ  |  (1 + |y  |2 ) 2 K n+|γ  |−γn (bxn 1 + |y  |2 )(y  )γ dy  = 2

Rn−1 +

∞  γn −n−|γ  |  (1 + ρ 2 ) 2 K n+|γ  |−γn (bxn 1 + ρ 2 )ρ n+|γ |−2 dρ 2





σ γ dS =

S1+ (n−1)

0

  n+|γ  |−γn  γi 2+1 21− 2 n+|γ  |−γn i=1 2     (bx ) × n | n+|γ  |−γn 2n−2  n−1+|γ  2 2 n−1 

∞  γn −n−|γ  |  (1 + ρ 2 ) 4 K n+|γ  |−γn (bxn 1 + ρ 2 )ρ n+|γ |−2 dρ. 2

0

Let us find the integral using formula (2.13.1.2) from [456]: ∞  γn −n−|γ  |  I = (1 + ρ 2 ) 4 K n+|γ  |−γn (bxn 1 + ρ 2 )ρ n+|γ |−2 dρ = 2

0

∞  γn −n−|γ  | n+|γ  |−3 { 1 + ρ 2 = z} = z 2 +1 K n+|γ  |−γn (bxn z)(z2 − 1) 2 dz = 2

1

(bxn )

1−n−|γ  | 2

2

n+|γ  |−3 2



 n − 1 + |γ  | K 1−γn (bxn ). 2 2

So n−1  i=1

|u(x)| ≤ M · C(n, γ ) 2 2 

M 

γn +1 2

1−γn 2

 (bxn )

 

1−γn 2

n− γn2+3

 

γi +1 2



n+|γ  |−γn 2

 (bxn )

K 1−γn (bxn ) = M K 1−γn (bxn ), 2

2

which gives |u(x)| ≤ M since K 1−γn (bxn ) ≤ 1 for xn > 0. Considering that 2

Kν (r) =

.

1−γn 2

π −r e [1 + O(r −1 )] 2r

K 1−γn (bxn ) = 2

332

Transmutations, Singular and Fractional Differential Equations

2 follows from the for large r, we get u(x) → 0 when xn → ∞. The fact that u ∈ Cev  2 properties of integral (7.133) and from the fact that τ (x ) ∈ Cev . So the theorem is proved.

A classical solution to the Dirichlet problem for a singular second order linear elliptic partial differential equation has been obtained. This equation contains, for example, the Tricomi equation in the upper half-plane, which arises in the study of aerodynamics (see [385]). The Keldysh equation arising in modeling weak shock reflection at a wedge is a particular case of the studied equation (see [171,434]). One more particular case of (7.128) is the equation of Weinstein generalized axially symmetric potential theory, which arises in the study of fluid dynamics and elasticity [592, 594]. Finally, Eq. (7.128) generalizes the Schrödinger equation with a singular potential, which arises in quantum mechanics (see [24]). There are a lot of open problems for Eq. (7.128), for example, the principle of extremum, the principle of Hopf, the principle of Zaremba–Giraud, and other quality properties of the solution to (7.128).

Applications of transmutations to different problems

8

8.1 Inverse problems and applications of Buschman–Erdélyi transmutations 8.1.1 Inverse problems In this subsection we would like to pay attention to deep connection of inverse problems and transmutation theory. From historical perspectives transmutation methods formed a basis to solve initial inverse problems for Sturm–Lioville operators (see [322,323,327]). In this case the inverse problem by spectral function data or the inverse problem by scattering data leads by applying transmutations to similar integral equations, namely famous Marchenko and Gelfand–Levitan equations (see [52,60,376]). These are equations for transmutation kernels, and after finding them transmutation kernels on diagonals recover potentials from initial equations by simple formulas. The same technique based on transmutations also works for perturbed Bessel equations, Dirac system, and some other general problems (see [60,376,567,590]). Although Marchenko and Levitan–Gelfand equations lead to many important results and applications in different fields these equations have some restrictions which were by and by recognized. In short the restrictions are due to 1) computational difficulties in gathering data and numeric solving of these equations; 2) lower effectiveness for multidimensional problems. So for many years another competing methods for solving inverse problems for Sturm–Liouville equations were proposed, which avoid the direct usage of Marchenko and Levitan–Gelfand equations. Among them are: • • • •

method of M. G. Krein, cf. [296–301]; method of A. N. Tikhonov, cf. [8,181,566,567]; method of spectral mappings of Z. L. Leybenzon, cf. [310–312,603,604], method of V. V. Kravchenko, cf. [279,280].

The above mentioned methods together with Marchenko and Levitan–Gelfand equations form a powerful ground for solving different theoretical and applied inverse problems. For the connection of Riesz potential theory to inverse problem see [162,449]. Namely, hyperbolic Riesz potential is the negative real power of the hyperbolic operator (10.44). Such operator inversion problem is closely related to the determination of a function from its weighted integral over Lorentzian spheres or Lorentzian weighted spherical mean (for non-weighted case see [164]). A problem of finding weighted γ spherical mean (Mr )f (see formula (3.183)) from f is well posed, if it has a unique γ solution and small variations in f lead to small variations in (Mr )f . Otherwise the Transmutations, Singular and Fractional Differential Equations With Applications to Mathematical Physics https://doi.org/10.1016/B978-0-12-819781-3.00014-8 Copyright © 2020 Elsevier Inc. All rights reserved.

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Transmutations, Singular and Fractional Differential Equations γ

problem is ill-posed (see [302]). For the weighted spherical mean (Mr )f we have γ γ very simple formula for finding function f if we know (Mr )f . It is f = lim (Mr )f r→0

(see [350]). But for the Lorentzian weighted spherical mean the inverse problem is not simple at all. In [164] for the determination of a function from its integral over Lorentzian spheres the hyperbolic Riesz potential was used. In this chapter we obtain an inverse operator for the mixed hyperbolic Riesz B-potential which generalizes the classical, used in [164]. Thus, it will be possible to solve the problem of finding a function from its Lorentzian weighted spherical mean.

8.1.2 Copson lemma Consider the partial differential equation with two variables on the plane ∂ 2 u(x, y) 2α ∂u(x, y) ∂ 2 u(x, y) 2β ∂u(x, y) + + = x ∂x y ∂y ∂x 2 ∂y 2 (this is the Euler–Poisson–Darboux equation or the B-hyperbolic one in Kipriyanov’s terminology) for x>0, y>0, and β>α>0 with boundary conditions on the characteristics u(x, 0) = f (x),

u(0, y) = g(y),

f (0) = g(0).

It is supposed that the solution u(x, y) is continuously differentiable in the closed first quadrant and has second derivatives in this open quadrant, and boundary functions f (x), g(y) are differentiable. Then if the solution exists, the following formulas hold true: ∂u ∂u = 0, y = 0, = 0, x = 0, ∂y ∂x   1   β−1 1 2 1−β β f (xt)t α+β+1 1 − t 2 P−α t dt = 2  β+ 2

(8.1)

(8.2)

0

 2α  α +

1 2

 1

  α−1 2 1−α g(xt)t α+β+1 1 − t 2 P−β t dt.

0

Therefore,

  y 2 β + 12 1−2β  y x 2α−1 f (x)(y 2 − x 2 )β−α−1 x dx, g(y) =  1  α + 2  (β − α) 0 μ

(8.3)

where Pν (z) is the Legendre function of the first kind [532]. So the main conclusion from the Copson lemma is that the data on characteristics cannot be taken arbitrary; these functions must be connected by the Buschman– Erdélyi operators of the first kind (for a more detailed consideration, cf. [532]).

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8.1.3 Norm estimates and embedding theorems in Kipriyanov spaces Consider a set of functions D(0, ∞) such that if f (x)∈D(0, ∞), then f (x)∈C ∞ (0, ∞) and f (x) tends to zero at infinity. On this set, define the seminorms α f L2 (0,∞) , f hα2 = D−   1 d α α = x f L2 (0,∞) , − f  α h2 x dx

(8.4) (8.5)

α is the Riemann–Liouville fractional integro-differentiation, the operator in where D− (8.5) is defined by   1 d β −β − = 2β I−; 2, 0 x −2β , (8.6) x dx −β

I−;2, 0 is the Erdélyi–Kober operator, and α ∈ R. For β = n ∈ N0 expression (8.6) reduces to classical derivatives. Theorem 96. Let f (x) ∈ D(0, ∞). Then the following formulas are valid:   1 d α α−1 α α f = 1 S− x − f, D− x dx   1 d α α xα − f = 1 P−α−1 D− f. x dx

(8.7) (8.8)

So the Buschman–Erdélyi transmutations of zero order smoothness for α ∈ N link differential operators in seminorm definitions (8.4) and (8.5). Theorem 97. Let f (x)∈D(0, ∞). Then the following inequalities hold true for seminorms: √ f hα2 ≤ max(1, 1 + sin πα)f  (8.9) hα , 2

f  hα ≤ 2

1

f hα2 , √ min(1, 1 + sin πα)

(8.10)

where α is any real number except α = − 12 + 2k, k ∈ Z. The constants in inequalities (8.9) and (8.10) are not greater than 1, which will be used below. If sin πα = −1 or α = − 12 + 2k, k ∈ Z, then the estimate (8.10) is not valid. Define on D(0, ∞) the Sobolev norm f W2α = f L2 (0,∞) + f hα2 .

(8.11)

Define one more norm, f W  α = f L2 (0,∞) + f  hα . 2

2

(8.12)

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 α as closures of D(0, ∞) in (8.11) or (8.12), respectively. Define the spaces W2α , W 2  α is continuously imbedded in W α ; moreTheorem 98. (a) For all α ∈ R the space W 2 2 over, f W2α ≤ A1 f W α ,

(8.13)

2

√ with A1 = max(1, 1 + sin πα). (b) Let sin πα = −1 or α = − 12 + 2k, k ∈ Z. Then the inverse embedding of W2α in  α is valid; moreover, W 2 f W  α ≤ A2 f W2α ,

(8.14)

2

√ with A2 = 1/ min(1, 1 + sin πα).  α are isomorphic with equivalent (c) Let sin πα = −1. Then the spaces W2α and W 2 norms. (d) The constants in embedding inequalities (8.13) and (8.14) are sharp. In fact this theorem is a direct corollary of the results on boundedness and norm estimates in L2 of the Buschman–Erdélyi transmutations of zero order smoothness. In the same manner, from the unitarity of these operators the following theorem follows. Theorem 99. The norms f W2α = f W α 2

s  j =0

j

D− f L2 ,

(8.15)

  s  1 d j j = f x − x dx j =0

(8.16) L2

are equivalent for integer s ∈ Z. Moreover, each term in (8.15) equals an appropriate term in (8.16) of the same index j . I. Kipriyanov introduced in [243] function spaces which essentially influenced the theory of partial differential equations with Bessel operators and, in more general sense, the theory of singular and degenerate equations. These spaces are defined in the following way. First we consider a subset of even functions in D(0, ∞) with all zero derivatives of odd orders at x = 0. We denote this set as Dc (0, ∞) and equip it with a norm s

2 f W

s = f L2,k + Bk L2,k ,

(8.17)

2,k

s/2

where s is an even natural number and Bk is an iteration of the Bessel operator. We define the Kipriyanov spaces for even s as a closure of Dc (0, ∞) in the norm (8.17). It is a known fact that a norm equivalent to (8.17) may be defined by [243]   s 1 d s f W f (8.18) −

s = f L2,k + x . 2,k x dx L2,k

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337

s may be defined for all s. Essentially this approach is the same as in So the norm W 2, k [243]. Another approach is based on usage of the Hankel transform. Below we adopt

s . the norm (8.18) for the space W 2,k We define the weighted Sobolev norm by s s = f L f W2,k 2,k + D− f L2,k

(8.19)

s as a closure of D (0, ∞) in this norm. and a space W2, c k

s is continuously embedded Theorem 100. (a) Let k = −n, n ∈ N. Then the space W 2, k s into W2, k , and there exists a constant A3 > 0 such that s ≤ A3 f  s . f W2,k W 2,k

(8.20)

(b) Let k + s = −2m1 − 1, k − s = −2m2 − 2, m1 ∈ N0 , m2 ∈ N0 . Then the inverse s into W

s , and there exists a constant A4 > 0 such that embedding holds true of W2, k 2, k s . f W

s ≤ A4 f W2,k 2,k

(8.21)

(c) If the abovementioned conditions are not valid, then the embedding theorems under consideration fail. Corollary 15. Let the following conditions hold true: k = −n, n ∈ N, k + s = −2m1 − 1, m1 ∈ N0 , k − s = −2m2 − 2, m2 ∈ N0 . Then the Kipriyanov spaces may be defined as closure of Dc (0, ∞) in the weighted Sobolev norm (8.19). Corollary 16. The sharp constants in embedding theorems (8.20) and (8.21) are s−1 L2,k ), A4 = max(1, 1 P−s−1 L2,k ). A3 = max(1, 1 S−

It is obvious that the theorem above and its corollaries are direct consequences of estimates for the Buschman–Erdélyi transmutations. The sharp constants in embedding theorems (8.20) and (8.21) are also direct consequences of estimates for the Buschman–Erdélyi transmutations of zero order smoothness. Estimates in Lp,α allow to consider embedding theorems for the general Sobolev and Kipriyanov spaces. So by applying the Buschman–Erdélyi transmutations of zero order smoothness, we received an answer to a problem which for a long time was discussed in “folklore”: Are the Kipriyanov spaces isomorphic to power weighted Sobolev spaces or not? Of course we investigated just the simplest case; the results can be generalized to other seminorms, higher dimensions, and bounded domains, but the principal idea is clear. All that disparages neither the essential role nor the necessity of applications of the Kipriyanov spaces in the theory of partial differential equations in any sense.

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Transmutations, Singular and Fractional Differential Equations

The importance of Kipriyanov spaces is a special case of the following general principle of L. Kudryavtsev: “Every equation must be investigated in its own space!” The embedding theorems proved in this section may be applied to direct transfer of known solution estimates for B-elliptic equations in Kipriyanov spaces (cf. [242,243]) to new estimates in weighted Sobolev spaces. It is a direct consequence of boundedness and transmutation properties of the Buschman–Erdélyi transmutations.

8.1.4 Other applications of Buschman–Erdélyi operators First let us show how to apply Buschman–Erdélyi operators to the Radon transform. It was proved by Ludwig in [342] that the Radon transform in terms of spherical harmonics acts in every harmonics at radial components as Buschman–Erdélyi operators. Let us formulate this result. Theorem 101. (Ludwig theorem, [162,342]) Let the function f (x) be expanded in Rn by spherical harmonics f (x) =



(8.22)

fk,l (r)Yk,l (θ ).

k,l

Then the Radon transform of this function may be calculated as another series in spherical harmonics, Rf (x) = g(r, θ ) =



(8.23)

gk,l (r)Yk,l (θ ),

k,l

∞ gk,l (r) = A(n)

s2 1− 2 r

 n−3 2

n−2 2

Cl

s  r

fk,l (r)r n−2 ds,

(8.24)

r n−2  where A(n) is some known constant and Cl 2 rs is the Gegenbauer function (see [19]). The inverse formula is also valid of representing values fk,l (r) via gk,l (r).

The Gegenbauer function may be easily reduced to the Legendre function (see [19]). So the Ludwig formula (8.24) reduces the Radon transform in terms of spherical harmonics series and up to unimportant power and constant terms to Buschman– Erdélyi operators of the first kind. Exactly, this formula in dimension two was developed by Cormack as the first step to the Nobel Prize. Special cases of Ludwig’s formula proved in 1966 are for any special spherical harmonics and in the simplest case on pure radial functions; in

Applications of transmutations to different problems

339

this case it is reduced to Sonine–Poisson–Delsarte transmutations of Erdélyi–Kober type. Besides the fact that such formulas are known for about half a century they are rediscovered still... As consequences of the above connections, the results may be proved for integral representations, norm estimates, and inversion formulas for the Radon transform via Buschman–Erdélyi operators. In particular, it makes clear that different kinds of inversion formulas for the Radon transform are at the same time inversion formulas for the Buschman–Erdélyi transmutations of the first kind, and vice versa. A useful reference for this approach is [80]. Now let us consider an application of the Buschman–Erdélyi transmutations for estimation of generalized Hardy operators. Unitarity of the shifted Hardy operators (5.30) was proved in [305]. It is interesting that the Hardy operators naturally arise in transmutation theory. We use Theorem 7 with integer parameter, which guarantees the unitarity for finding more unitary in L2 (0, ∞) integral operators of very simple form. Theorem 102. The following are pairs of unitary mutually inverse integral operators in L2 (0, ∞): x

dy 1 f (y) , U4 f = f + y x

U3 f = f +

∞ f (y) dy, x

0

x U5 f = f + 3x

dy 3 f (y) 2 , U6 f = f − 2 y x

0

∞

∞ yf (y) dy, U8 f = f − 3x

x

1 2

yf (y) dy,

0

3 U7 f = f + 2 x U9 f = f +

x

x 

f (y)

dy , y2

x

15x 2 y3



−

3 f (y) dy, y

0

1 U10 f = f + 2

∞

 15y 2 3 − f (y) dy. x x3

x

Next we consider an application of the Buschman–Erdélyi transmutations in the works of V. Katrakhov. Namely, he found a new approach for boundary value problems for elliptic equations with strong singularities of infinite order. For example, for the Poisson equation he studied problems with solutions of arbitrary growth. At singular point he proposed a new kind of boundary condition: the K-trace. His results are based on the constant usage of Buschman–Erdélyi transmutations of the first kind for the definition of norms, solution estimates, and correctness proofs [225, 227].

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Transmutations, Singular and Fractional Differential Equations

Finally, we briefly discuss applications of the Buschman–Erdélyi transmutations to Dunkl operators. In recent years the Dunkl operators were thoroughly studied. These are difference–differentiation operators consisting of combinations of classical derivatives and finite differences. In higher dimensions, the Dunkl operators are defined by symmetry and reflection groups. For this class there are many results on transmutations which are of Sonine–Poisson–Delsarte and Buschman–Erdélyi types (cf. [560] and references therein).

8.2 Applications of the transmutation method to estimates of the solutions for differential equations with variable coefficients and the problem of E. M. Landis 8.2.1 Applications of the transmutations method to the perturbed Bessel equation with a potential The problem of constructing an integral formula for solutions to the differential equation with certain asymptotics is considered, Bα g(x) − q(x)g(x) = λ2 g(x),

(8.25)

where Bα is a Bessel operator, which in this subsection is convenient for us to define in the following form: Bα g = g (x) +

2α g (x), x

α > 0.

(8.26)

This problem is solved by the transmutation method. To do this, it is enough to construct a pair of mutually inverse transformation operators, the first of which is Sα , of the form ∞ Sα h(x) = h(x) +

S(x, t)h(t) dt,

(8.27)

x

which intertwines the operators Bα − q(x) and Bα by the formula Sα (Bα − q(x))h = Bα Sα h.

(8.28)

The second operator Pα , inverse to the first, should be constructed as integral with the kernel P (x, t), ∞ Pα h(x) = h(x) +

P (x, t)h(t) dt, x

and act by the formula

(8.29)

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341

Pα Bα h = (Bα − q(x))Pα h, where h ∈ C 2 (0, ∞) and such that integrals in (8.27) and (8.28) converge. As a result, on solutions to the differential equation (8.25) the function Sα u = v will be expressed through solutions of the unperturbed equation obtained by discarding the term with potential in (8.25), that is, in fact, through the Bessel functions. The function u = Pα v will be a solution to the initial perturbed equation (8.25). In this case, the integral representation will be obtained for the solution (8.29) with an explicit kernel description P (x, t). This technique reflects one of the main applications of transmutations. It is the expression of solutions of more complex differential equations through similar simpler ones, which has already been noted several times. Also note that the same pair of mutually inverse transmutations allows one to obtain both representations of solutions to the differential equation (8.25) with spectral parameter. Besides, we can present the solution in the homogeneous equation case Bα h(x) − q(x)h(x) = 0. Moreover, if we consider the problem of finding a representation for the solutions to the perturbed equation (8.25), then we can skip the construction of the direct transmutation operator and go straight to the construction of the inverse and find the integral representation for the desired solution of the form (8.29). An original technique for solving such problems was developed by V. V. Stashevskaya [557,558], which allowed her to include singular potentials with an extremely accurate estimate at zero |q(x)| ≤ cx −3/2+ε , ε > 0, for entire α. This technique, based on the application of generalized Paley–Wiener theorems, has been widely developed and recognized. The case of continuous q for α > 0 was considered in papers of A. S. Sokhin [546–549] and also in the paper [586] of V. Ya. Volk. Moreover, Povzner type transformation operators with integration over a finite interval were constructed in the works of V. V. Stashevskaya and V. Ya. Volk and Levin types with integration over an infinite interval were constructed in the works of A. S. Sokhin. Further, we propose a new modified method that allows one to combine both of these approaches. Among many works on obtaining representations of solutions for the perturbed Bessel equation (8.25)–(8.26), we note those in which the solution is sought in the form of series of a special form. These are the works of A. Fitouhi et al. [66,145] and works of V. V. Kravchenko et al. [45,57,278,281,282,285,287–292]. A critical analysis of a number of results on this problem was recently presented in [169]. However, in many mathematical and physical problems it is necessary to consider strongly singular potentials, for example, admitting an arbitrary power singularity at zero. Here we formulate results about the integral representation of solutions of equations with similar singular potentials. From the potential only a majority is required by a certain function summed at infinity. In particular, q = x −2 is the singular potential, q = x −2−ε , ε > 0, is the strongly singular potential, and q = e−αx /x are the Yukawa potentials of the Bargman and Batman–Shadan types [497] and a number of others. Moreover, no additional conditions of the type q(x) impose fast oscillations at the

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Transmutations, Singular and Fractional Differential Equations

origin or constant sign, which allows one to study attractive and repulsive potentials using a single method. It should be noted that operators constructed in this book are transmutations of a special kind that differ from previously known ones in some details. Prior to this, only cases of the same limits were considered (both types [0; a] and [a; ∞]) in the basic integral equation for the kernel of the transmutation operator. Here it is shown that the case of various limits in the main integral equation can be considered. It is this arrangement of limits that made it possible to cover a wider class of potentials with singularities at zero. In addition, in comparison with the arguments on the model of the classical work of B. M. Levitan [317], we are making an improvement to this scheme. The Green function used in the proofs, as it turned out, can be expressed not only through the general Gauss hypergeometric function, but also more specifically through the Legendre function, which depends on a smaller number of parameters, which allows us to get rid of the indefinite constants in the estimates from previous works. Due to the limited volume of the book, this subsection only presents the statement of the problem, a summary of the main results, and consequences without proof (for a detailed exposition, see [233,521,528–530]).

8.2.2 The solution of the basic integral equation for the kernel of the transmutation operator Let us introduce variables and functions by formulas t +x t −x , η= , ξ ≥ η > 0, 2   2 x α K(x, t) = P (x, t), u(ξ, η) = K(ξ − η, ξ + η), t ξ=

(8.30)

where P (x, t) is a kernel from (8.29). Denote ν = α − 1. So in order to justify the submission (8.29) for solution to Eq. (8.25) it is enough to define a function u(ξ, η). It is known [546–549] that if a twice continuously differentiable solution u(ξ, η) to the integral equation 1 u(ξ, η) = − 2

∞ ∞ η Rν (s, 0; ξ, η)q(s) ds − ds q(s + τ )Rν (s, τ ; ξ, η)u(s, τ ) dτ ξ

ξ

0

under conditions 0 < τ < η < ξ < s exists, then the function P (x, t) is determined by (8.30) through this solution u(ξ, η). The function Rν = Rα−1 is the Riemann function that arises when solving a certain Goursat problem for a singular hyperbolic equation ∂ 2 u(ξ, η) 4α(α − 1)ξ η u(ξ, η) = q(ξ + η)u(ξ, η). + ∂ξ ∂η (ξ 2 − η2 )2

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343

This function is known explicitly (see [546–549]) and has a form  Rν =

s 2 − η2 ξ 2 − τ 2 · s 2 − τ 2 ξ 2 − η2

ν

  s 2 − ξ 2 η2 − τ 2 F · −ν, −ν; 1; , 2 1 s 2 − η2 ξ 2 − τ 2

(8.31)

where 2 F1 is the hypergeometric function (1.33). This expression is simplified in [233], where it is shown that the Riemann function in this case is expressed in terms of the Legendre function by the formula   η2 − τ 2 s 2 − ξ 2 1+A · . (8.32) , A= 2 Rν (s, τ, ξ, η) = Pν 1−A ξ − τ 2 s 2 − η2 The main content of this subsection is the following result. Theorem 103. Let function q(r) ∈ C 1 (0, ∞) satisfy the condition ∞ |p(t)| dt < ∞, ∀ξ > 0.

|q(s + τ )| ≤ |p(s)|, ∀s, ∀τ, 0 < τ < s,

(8.33)

ξ

Then there exists an integral representation of the form (8.29) whose kernel satisfies the estimate  2 2   α ∞ 1 y (t + r 2 ) − (t 2 − r 2 ) t Pα−1 |p(y)| dy× |P (r, t)| ≤ r 2 2try 2 ⎡

t+r 2



⎢ t −r exp ⎣ 2



1 2



∞ Pα−1

y 2 (t 2

+ r 2 ) − (t 2 2try 2

 − r 2)

⎤ ⎥ |p(y)| dy ⎦ .

t+r 2

In this case, the kernel of the transmutation operator P (x, t) and the solution to (8.25) are twice continuously differentiable on (0, ∞) functions in their arguments. We list the classes of potentials for which the conditions in (8.33) are satisfied. If |q(x)| monotonously decreases, then it is possible to accept p(x) = |q(x)|. For potentials with an arbitrary singularity at the origin and increasing at 0 < x < M (for example, Coulomb potential q = −1/x), which are trimmed by zero at infinity, q(x) = 0, x > M, we can take p(x) = |q(M)|, x < M, p(x) = 0, x ≥ M. Potentials with estimate q(x + τ ) ≤ c|q(x)| = |p(x)| also will satisfies the conditions in (8.33). The possibility of such strengthening of Theorem 8.25 was indicated by V. V. Katrakhov. In particular, these conditions are satisfied by the following potentials encountered in applications: strongly singular potential with a power feature of the form q(x) = x −2−ε , different Bargman potentials q1 (x) = −

e−ax c2 c4 , , q2 (x) = , q3 (x) = 2 (1 + βe−ax )2 (1 + c3 x)2 ch (c5 x)

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and Yukawa e−ax , q5 (x) = q4 (x) = − x

∞

e−at dc(t).

x

(see, for example, [497]). Remark 14. In fact, in the proof of the above theorem, an explicit form of the Riemann function (8.32) is not required. Only the existence of the Riemann function, its positivity, and some special property of monotonicity are used. These facts are quite general, so the results can be extended to a fairly wide class of differential equations. The estimate from Theorem 103 for potentials of a general form can be transformed into a grosser, but also more visible one. Theorem 104. Let the conditions of Theorem 103 be satisfied. Then the kernel of the transmutation operator P (x, t) satisfies the estimate     2 1 t α t + x2 |P (x, t)| ≤ Pα−1 × 2 x 2tx ⎡ ⎤ ∞    2 ∞ 2 + x 1 t − x t |p(y)| dy exp ⎣ |p(y)| dy ⎦ . Pα−1 2 2 2tx x

x

Note that for x → 0 the kernel of the integral representation can have an exponential singularity. For a class of potentials with a power singularity of the form q(x) = x −(2β+1) , β > 0,

(8.34)

obtained estimates can be simplified without reducing their accuracy. The restriction on β is caused by the condition of summability at infinity. Theorem 105. Consider the potential of the form (8.34). Then Theorem 103 performed with evaluation  α   2 t (β)4β−1 t + x2 −β |P (x, t)| ≤ · Pα−1 × x 2tx (t 2 − x 2 )β     t − x (β)4β−1 −β t 2 + x 2 P , exp x 2tx (t 2 − x 2 )β α−1 μ

where Pν (·) is a Legendre function, β is defined by (8.34), and α is defined by (8.26). Note that this estimate is obtained after quite lengthy calculations using the famous Slater–Marichev [361] theorem, which helps to calculate the necessary integrals in terms of hypergeometric functions after they are reduced to the Mellin convolution.

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345

The simplest such estimate was obtained in [233] for the potential q(x) = cx −2 , for 1 −1 which β = . As follows from [20], in this case the Legendre function Pν 2 (z) can be 2 expressed through elementary functions. Therefore, the corresponding estimate can be expressed in terms of elementary functions. Another potential for which the obtained estimate can be simplified and expressed in terms of elementary functions is a potential of the form q(x) = x −(2β+1) , when the parameters are related by the relation β = α − 1. Corollary 17. Let the relation between the parameters β = α − 1 be true. Then the estimate from Theorem 103 takes the form   β+1 β−2  2 β   β  t 2 t − x 2β−2 t 2 + x 2 t + x2 |P (x, t)| ≤ exp = x β 2tx 2 β 2tr    2 β  1 2β−2 t − x 1 t + x2 2 2 β (t + x ) exp . (8.35) 4β x 2β+1 β 2 2tx Let us note that for α = 0 in (8.25)–(8.29), Theorem 103 reduces to well-known estimates for the kernel of the integral representation of Jost solutions for the Sturm– Liouville equation. The above technique is fully transferred to the problem of constructing nonclassical operators of generalized translation. This problem is essentially equivalent to expressing solutions to the equation Bα,x u(x, y) − q(x)u(x, y) = Bβ,y u(x, y)

(8.36)

through solutions of the unperturbed Euler–Poisson–Darboux equation (or wave equation in the nonsingular case) in the presence of additional conditions ensuring correctness. Such representations are obtained already from the existence of transmutation operators and have been studied for the nonsingular case (α = β = 0) in [316,317, 319] as a consequence of the theory of generalized translation (see also [375]). An interesting original technique for obtaining such representations was also developed in the nonsingular case in the works of A. V. Borovskikh [33,34]. The presented results imply the integral representations of a certain subclass of solutions of Eq. (8.36) in the general singular case for sufficiently arbitrary potentials with singularities at the origin. Moreover, the estimates for the solutions do not contain any indefinite constants, and for the kernels of integral representations the integrals are written in explicit form equations that they satisfy.

8.2.3 Application of the method of transmutation operators to the problem of E. M. Landis In a paper of E. M. Landis [307], the following problem is posed: Prove that the solution of the stationary Schrödinger equation with bounded potential of the form u(x) − q(x)u(x) = 0, x ∈ Rn , |x| ≥ R0 > 0,

(8.37)

346

Transmutations, Singular and Fractional Differential Equations

|q(x)| ≤ λ2 , λ > 0, u(x) ∈ C 2 (|x| ≥ R0 ) , satisfying the estimate |u(x)| ≤ const · e−(λ+ε)|x| , ε > 0, is identically equal to zero. V. Z. Meshkov (see [379,380]) gave a counterexample to this problem. Also the existence of counterexamples with solutions that are complex functions was proved. Moreover, it was shown that if we strengthen the estimate in the hypothesis of E. M. Landis to |u(x)| ≤ const · e−(λ+ε)|x| , ε > 0, 4/3

then the answer will be “such nonzero solutions do not exist.” Recently, interest in these results has not disappeared. Topics related to the hypothesis of E. M. Landis and the results of V. Z. Meshkova are actively developed by leading mathematicians in the field of differential equations, such as J. Bourgein, K. Koenig, and several others (see [35,79,236,237,481]). The main question remains the study of the hypothesis of E. M. Landis for real solutions, and the answer to this question has not yet been obtained. In connection with the foregoing, it seems reasonable to name the following text the Landis–Meshkov problem. Landis–Meshkov problem. Is it true that for given domains D and positive functions r(x), s(x) only the zero classical solution of the stationary Schrödinger equation u(x) − q(x)u(x) = 0, x ∈ Rn , |q(x)| ≤ r(x),

(8.38)

satisfies the estimate |u(x)| ≤ s(x)?

(8.39)

Let D be the exterior of some circle, q(x) = λ2 , s(x) = e−(λ+ε)|x| , ε > 0. Then from the results of V. Z. Meshkov, the negative answer to this problem in the case of complex solutions follows. If D is the exterior of some circle, q(x) = λ2 , 4/3 s(x) = e−(λ+ε)|x| , ε > 0, the positive answer in this problem in the case of complex solutions has place. For real valued solutions, even in these particular cases, the answers are unknown. Further, we show that despite the general negative solution of V. Z. Meshkov for the initial statement of the problem of E. M. Landis for some classes of potentials, the problem is solved positively for real solutions. In this case, the method of transmutation operators of a special form is used [518–520]. Further, this problem is solved for the case of a potential that depends on only one variable: q(x) = q(xi ), where 1 ≤ i ≤ n. Further, for definiteness it is considered that i = 1. This case is a particular case of Eq. (8.37): u − q(x1 )u = 0.

(8.40)

Applications of transmutations to different problems

347

Here the potential q(x1 ) is bounded by an arbitrary nondecreasing function. The solution is based on the use of transmutation operators reducing Eq. (8.40) to the Laplace equation. Let the conditions of the problem (8.37) be satisfied in semispace x1 ≥ R0 and let the functions in (8.37) be invariant under the change of variables z = x1 − R0 . Therefore, we will consider the problem (8.37) in the half-space z ≥ 0 or, keeping the previous notation for the variable x1 , x1 ≥ 0. It will be proved that the solution to the problem (8.37) is zero in the half-space x1 ≥ 0, and then by virtue of Calderon’s theorem on the uniqueness of continuation (see [382], Chapter 6, p. 14), such a solution is identically zero in the space Rn . Denote by T (δ) the set of functions satisfying in part of the space Rn+ the following conditions:  u(x) ∈ C 2 Rn+ ,

(8.41)

−δ|x|

(8.42)

|u(x)| ≤ c1 e , δ > 0,    ∂u  −δ|x|   .  ∂x  ≤ c2 e

(8.43)

1

Let us construct for functions from T (λ + ε) the transmutation operator of the form (see [518–520]) ∞ Su(x) = u(x) +

K(x1 , t)u(t, x 1 ) dt,

(8.44)

x1

so that the following equality holds:  S

 ∂ 2u ∂2 − q(x1 )u = 2 Su, |q(x1 )| ≤ λ2 , 2 ∂x1 ∂x1

(8.45)

where (x1 , x 1 )=(x1 , x2 , . . . , xn ). Substitution of expression (8.44) into formula (8.45) leads to the equalities ∂ 2K ∂ 2K − = q(t)K, ∂t 2 ∂x12 ∂K(x1 , x1 ) = q(x1 ), 3 ∂x1 ∂K(x1 , t) ∂u(t, x 1 ) lim K(x1 , t) − lim u(t, x 1 ) = 0. t→∞ t→∞ ∂t ∂t

(8.46) (8.47) (8.48)

t + x1 1 , v = t−x Performing a standard variable change w = 2 , we reduce the sys2 tem (8.46)–(8.47) to a simpler (satisfying conditions (8.48) on solutions of the prob-

348

Transmutations, Singular and Fractional Differential Equations

lem (8.37) will be shown later) ∂ 2K = q(w + v)K, ∂w∂v w 1 K(w, 0) = q(s) ds. 3

(8.49) (8.50)

0

The problem (8.49)–(8.50) is a consequence of the integral equation 1 K(w, v) = 3

w

w q(s) ds +

0

|q| ≤ λ , 2

v q(α + β)K(α, β) dβ,

dα 0

(8.51)

0

w ≥ v ≥ 0.

Eq. (8.51) differs from that usually used when considering transformation operators on an infinite interval of the integral equation by changing the integration region from the semiaxis (w, ∞) to the interval (0, w), which implies an exponential kernel growth K (x1 , t). It is further proved that such a kernel exists and a transmutation operator with such a kernel (8.44) is defined on the set T (λ + ε). The possibility to reduce problem (8.46)–(8.48) to nonequivalent integral equations follows from the underdetermination of the Cauchy problem (8.49)–(8.50). It should be noted that the integral equation (8.51) can be solved in a wider region without the limitations of w ≥ v, otherwise the kernel will not be defined under the signs of the integrals. The proof of the existence of a solution in this wider field is carried out in the same way as the proof below. The nuance in proving the existence of a solution to the integral equation (8.51) usually does not pay attention to this (remark of A. V. Borovskikh). Lemma 20. There is a unique continuous solution to Eq. (8.51), satisfying the inequality  λ w  √ |K(w, v)| ≤ (8.52) I1 2λ wv , 3 v where I1 (x) is a modified Bessel function. Moreover, on the valid potential q(x1 ) ≡ λ2 in (8.52) an equal sign is reached. Remark 15. Further, the symbol c denotes an absolute positive constant, the value of which does not play a role. Proof. We introduce the notation 1 K0 (w, v) = 3

w q(s) ds, 0

Applications of transmutations to different problems

w P K(w, v) =

349

v q(α + β)K(α + β) dβ.

dα 0

0

Then Eq. (8.51) can be written as K = K0 + P K. We are looking for its solution in the form of the Neumann series K = K0 + P K0 + P 2 K0 + . . . .

(8.53)

For the terms of the series (8.53), taking into account the condition |q(x1 )| ≤ λ2 , we get  n+1 w n+1 v n  n  P K0 (w0 v) ≤ 1 λ2 , n = 0, 1, 2, . . . . 3 (n + 1)! n!

(8.54)

Application of the representation of I1 (x) as a series I1 (x) =

∞  (x/2)2k+1 k=0

k!(k + 1)!

gives inequality (8.52). The estimate (8.52) is exact, since for q(x1 ) ≡ λ2 , the inequalities in (8.54) turn into equalities for all integers n ≥ 0. The lemma is proved. Lemma 21. In terms of the variables x1 , t, the estimate |K(x1 , t)| ≤ c t eλt is valid. Proof. Let us consider the inequality    1  I1 (x) ≤ c ex , x ≥ 0.  x To verify the truth of this inequality it is necessary to parse cases (i) x ≥ 1 and (ii) 0 ≤ x ≤ 1 and use the well-known asymptotics of the function I1 (x) when x → ∞ and x → +0 (see [21]). Hence, using the obvious inequalities  √ x1 + t ≤ t, 2 wv = t 2 − x12 ≤ t 2 and estimate (8.52) follows the statement of the lemma. It follows from the lemma that expression (8.44) is defined on functions from T (λ + ε). We show that expression (8.44) gives the transmutation operator T (λ + ε).

350

Transmutations, Singular and Fractional Differential Equations

To do this, it remains to check the relation (8.48). From u(x) ∈ T (λ + ε) and from Lemma 21 it follows that lim K(x1 , t)

t→∞

∂u(t, x 1 ) = 0. ∂t

Therefore, it remains to prove that if u(x) ∈ T (λ + ε), then lim

t→∞

∂K(x1 , t) u(x1 , t) = 0. ∂t

The last relation follows from the estimate    ∂K(x1 , t)   ≤ c t eλt .    ∂t

(8.55)

To prove inequality (8.55), we need to go over to the variables w, v and, using the al∂K ∂K ready established estimates for the kernel K(x1 , t), estimate the derivatives , , ∂w ∂v differentiating Eq. (8.51). Since   ∂K 1 ∂K ∂K = + , ∂t 2 ∂w ∂v we get (8.55).

8.2.4 The solution to the E. M. Landis problem belongs to T (λ + ε) We show that any solution to the problem (8.37) belongs to T (λ + ε) and, therefore, the operator (8.44) is defined on such solutions. To do this, let us verify the condition (8.43). Lemma 22. Let the function u(x) ∈ C 2 (|x| ≥ R0 ) be a solution to the problem (8.37). Then there is a constant c > 0 such that    ∂u  −(λ+ε)|x|   .  ∂x  ≤ c e 1 Proof. By a priori Schauder estimates, in a closed ball B(x, 1) of unit radius centered at x, |x| ≥ R0 + 1, we have (see [384], Theorem 33, II)   λ1 1 u1 ≤ c u1,λ1 1+λ1 · u0 λ1 +1 + u0 , where u0 = u(x)C 0 (B(x,1)) , u1 = u(x)C 1 (B(x,1))

Applications of transmutations to different problems

351

and u1,λ1 is the sum of the Hölder coefficients of the function u(x) and its derivatives ∂u , 1 ≤ i ≤ n. It follows that of the first order ∂xi     λ1 1  ∂u(x)    ≤ c u1,λ 1+λ1 · u0 λ1 +1 + u0 . (8.56) 1  ∂x  1 Note that since all the conditions (see [384], Statement 33 V) are satisfied, the constant c in formula (8.56) does not depend on x. From Morrey’s results (see [382], Theorem 39, IV) the following estimate for u1,λ1 follows:   u1,λ1 ≤ c uL2 (B(x,1)) + q uL2 (B(x,1)) . (8.57) The constant in (8.57) does not depend on x. From the conditions |q(x1 )| ≤ λ2 , using the mean value theorem, we obtain from (8.57) ⎛ ⎜ u1,λ1 ≤ c ⎝



⎞1/2 ⎟ |u(y)|2 dy ⎠

≤ c1 e−(λ+ε)|x| .

B(x,1)

Substituting the last inequality into (8.56), we obtain      1 + λ  ∂u   ≤ c e−(λ+ε)|x| 1+λ 1+λ + e−(λ+ε)|x| ≤ c e−(λ+ε)|x| .   ∂x  1 Thus, the required inequality is established for |x1 | ≥ R0 + 1. Since the set R0 ≤ |x| ≤ R0 + 1 is compact in Rn , this inequality is true for |x| ≥ R0 . Lemma 22 is proved. Changing the coordinate again z = x1 − R0 , we obtain that Lemma 22 is valid in the half-space x1 ≥ 0 (we will redesignate z by x1 ). We apply the operator S to Eq. (8.40). From the identity (8.45) and the permutation ∂ 2u of S with the derivatives 2 , 2 ≤ i ≤ n, we obtain that in the half-space Rn+ ∂xi S (u − q(x1 )u) = Su = 0.  Denoting Su by v from (8.44), (8.51), we obtain that if u(x) ∈ C 2 Rn+ , q(x) ∈   C Rn+ , then v(x) ∈ C 2 Rn+ . Let us show that v(x) decreases exponentially in Rn+ when |x| → ∞ and therefore is equal to zero. Lemma 23. Let u(x) ∈ T (λ + ε). Then for x ∈ Rn+ |v| = |Su| ≤ c |x| e−ε|x| , ε > 0.

352

Transmutations, Singular and Fractional Differential Equations

Proof. From (8.44) and Lemma 22, we obtain ∞ |Su| ≤ |u(x)| +

$ t 2 +|x 1 |2

t eλt c e−(λ+ε)

x1 ∞

⎛ c ⎝e−(λ+ε)|x| +

te

$ −(λ+ε) t 2 +|x 1 |2

dt ≤

⎞ dt ⎠ .

x1

$ Calculating the integral by changing variables y = t 2 + |x 1 |2 with subsequent integration in parts, we obtain the required estimate. The lemma is proved. So v(x) = 0 in Rn+ . Define on T (λ + ε) the inverse to S operator P by the formula ∞ P u(x) = u(x) +

N (x1 , t)u(t, x 1 ) dt. x1

Then for the kernel N (x1 , t), statements of Lemmas 20–22 are valid. In addition, if Su ∈ T (λ + ε), then P Su(x) = u(x).

(8.58)

Since obviously 0 ∈ T (λ + ε), applying (8.58) to both parts of Su = 0 we obtain that u = 0 in Rn+ . It was shown above that this implies u ≡ 0 in all Rn . Remark 16. Consideration of the part of space Rn+ used in the proof is necessary because expression (8.44) is not defined in the area obtained by the intersection of the ball |x| ≤ R0 and the infinite half cylinder {|x 1 | ≤ R0 , |x1 | ≤ R0 }. The above reasoning leads to a theorem. Theorem 106. Any solution u(x) ∈ C 2 (|x| > R0 ) to the stationary Schrödinger equation with bounded potential u(x) − q(x1 )u = 0, x ∈ Rn , |x| ≥ R0 > 0, q(x1 ) ∈ C (|x| ≥ R0 ) , |q(x1 )| ≤ λ2 , λ > 0, satisfying the estimate |u(x)| ≤ const e−(λ+ε)|x| , ε > 0, is the identity zero. The used technique of transmutation operators allows us to strengthen the result. We denote by L2, loc (x1 ≥ R0 ) the set of functions for which for any x1 ≥ R0 the

Applications of transmutations to different problems

integral

%x1

353

ψ 2 (s) ds is finite. Suppose further that a nonnegative function g(x) is given.

R0

Let for g(x) the integral

%∞

t g(t, x 1 ) dt = p(x) be finite for any x1 ≥ R0 and for some

x1

constant α > 0  |p(x)| ≤ c · exp −α|x|δ , δ > 0. Then, according to the scheme of proof of the previous theorem, the following fact can be established. Theorem 107. Let ψ(x1 ) ∈ L2, loc (x1 ≥ R0 ), let ψ(x1 ) be a nondecreasing function, and let function g(x) satisfy the above requirements. Then any solution to the equation u(x) − q(x1 )u = 0, x ∈ Rn , |x| ≥ R0 > 0, |q(x1 )| ≤ ψ 2 (x1 ), for which ψ(x1 )|u(x)| ≤ const e−ψ(x1 )|x| g(x), g(x) ≥ 0, is the identity zero. −ε|x| Under the conditions of Theorem 106 we need to set g(x)  = e . An example of another suitable function g(x) is the function g(x) = exp −ε|x|δ , 0 < δ < 1. This case is also an example of the generalized Landis–Meshkov problem (8.38)–(8.39). In a similar way, the case of potential depending only on the radial variable can be considered. The answer in the original statement of the E. M. Landis problem after passing to spherical coordinates is also positive (see [518–520]). It is possible to consider generalizations of the E. M. Landis problem to the case of more general differential equations and the corresponding estimates of the growth of solutions. For example, it is of interest to study the questions posed for the nonlinear p-Laplacian equation [113,329].

8.3 Applications of transmutations to perturbed Bessel and one-dimensional Schrödinger equations The integral representations of solutions of one differential equation with singularities in the coefficients containing a Bessel operator perturbed by some potential are considered in this section. The existence of integral representations of a certain type for the indicated solutions is proved by the method of successive approximations using transmutation operators. In this case, potentials with strong singularities at the origin are allowed. The Riemann function is expressed not through the general hypergeometric function, but more specifically through the Legendre function, which avoids unknown constants in the estimates.

354

Transmutations, Singular and Fractional Differential Equations

8.3.1 Formulation of the problem Consider the problem of constructing an integral representation of a certain kind for solutions of a differential equation Bα u(x) − q(x)u(x) = 0,

(8.59)

where Bα is the Bessel operator (9.1) of the form Bα u = u (x) +

2α u (x), α > 0. x

(8.60)

This problem is solved by the method of transmutation operators. To do this, it suffices to construct a transmutation operator Pα of Poisson type (see (3.120)) of the form ∞ Pα u(x) = u(x) +

P (x, t)u(t) dt,

(8.61)

x

with a kernel P (x, t) which intertwines the operators Bα and Bα − q(x) by the formula Bα Pα u = Pα (Bα − q(x))u,

(8.62)

where u ∈ C 2 (0, ∞). As a result, we obtain a formula expressing solutions of Eq. (8.59) with a spectral parameter of the form Bα u(x) − q(x)u(x) = λ2 u(x) through solutions of the unperturbed equation, that is, through Bessel functions. In this case, the spectral parameter λ does not affect the form of the linear transmutation operators whose kernels are independent of it. This approach reflects one of the applications of transmutation operators, that is, the expression of solutions of more complex differential equations through similar simpler ones. The theory of transmutation operators is an important branch of modern mathematics that has numerous applications (see [51–53,234,323,376,537]). The possibility of representing the form (8.61) with a sufficiently “good” kernel P for a wide class of potentials q(x) lies at the heart of classical methods for solving inverse problems of the quantum theory of scattering [172,497]. For the Sturm–Liouville equations, transformation operators of the form (8.61) were first constructed by B. Ya. Levin (see [313,314]). The transmutation operators for the Bessel operator of the Sonine and Poisson type were introduced by Delsarte [83]. In Russian their theory was firstly presented and developed in the famous work of B. M. Levitan [317]. Then, in a number of papers, transmutation operators with the property (8.62) for variable potentials were also considered (see [284,291]). Moreover, the inverse to (8.61) Sonine type transmutation operators Sα , which satisfy the intertwining relation, is simultaneously considered,

Applications of transmutations to different problems

355

Sα (Bα − q(x))u = Bα Sα u, on suitable functions. An original method for constructing transmutation operators for the perturbed Bessel equation on the semiaxis was developed by V. V. Stashevskaya [557,558], which allowed it to include singular potentials with an estimate of zero |q(x)| ≤ cx −3/2+ε , ε > 0, for integer α. This technique was further widely developed. The case of continuous q, α > 0 was considered in detail in the works of A. S. Sokhin [546–549], as well as a number of other authors (see for more details [234,537]). Transmutation operators with “bad” potentials, such distributions from a certain class were considered in [172]. V. V. Kravchenko developed a special method for representing the kernels of transformation operators in the form of series (the Spectral Parameter Power Series method, SPPS) [276,283]. This method turned out to be well adapted for numerical solution of applied problems and computer modeling, including direct and inverse spectral problems [279,280]. Results for some special cases of the considered problem of constructing transformation operators (8.61)–(8.62) were published in [233,526]. In many mathematical and physical problems, it is necessary to consider strongly singular potentials, for example, admitting an arbitrary power singularity at zero. In this section, we formulate results on the integral representation of solutions of equations with similar singular potentials. From the potential only a majority by a certain function summed at infinity is required. In particular, the class of admissible potentials includes the singular potential q = x −2 , the strongly singular potential with a power singularity q = x −2−ε , ε > 0, the potentials of Yukawa type q = e−αx /x, Bargman and Batman–Shadan potentials [497], and a number of others. Moreover, it is not necessary to require any additional conditions for a q(x) such as fast oscillations at the origin or constant sign. This allows us to study attractive and repulsive potentials using a single method. Exemption from limiting conditions at zero is the advantage of considering Levin type transmutation operators (8.61). In this section, the main object of study is the integral equation for the kernel of the transformation operator (8.61). After reducing the problem to an integral equation, the existence and uniqueness of the solution and its necessary smoothness are proved. Estimates of the solution are obtained in terms of the parameter and potential of the original equation (8.59). Such estimates are expressed using special Legendre functions. For a particular class of potential type potentials, simpler estimates are obtained. Here a technique based on the application of the Riemann function for the Euler–Poisson–Darboux equation was used. Estimates of integrals were obtained using the Mellin transform and the Slater–Marichev theorem. It should be noted that a special kind of transmutation operators are constructed in this section. These operators differ from previously known operators by some details. Prior to this, only cases of identical limits (both types of [0; a] and [a; ∞]) in the main integral equation for the kernel of the transmutation operator were considered. Here it is shown that cases of various limits can be considered in the main integral equation. This arrangement of limits made it possible to cover a wider class of potentials with singularities at zero. In addition, in comparison with the arguments on the model of

356

Transmutations, Singular and Fractional Differential Equations

the classical work of B. M. Levitan [317], we are making some improvements to this scheme. As it turned out, the Riemann function used in the proof can be expressed not only in terms of the general Gaussian hypergeometric function with three parameters, but also more specifically in terms of the Legendre function with two parameters, which allows one to get rid of the indefinite constants in the estimates from previous papers.

8.3.2 Solution of the basic integral equation for the kernel of a transmutation operator We introduce new variables and functions using the following formulas: t +x t −x , η= , ξ ≥ η > 0, 2   2 x α K(x, t) = P (x, t), w(ξ, η) = K(ξ − η, ξ + η). t ξ=

(8.63)

Let ν = α − 1. In order to justify the representation (8.61) for solution to (8.59) it is sufficient to define the function w(ξ, η). It is known (see [233,526]) that if there exists a twice continuously differentiable solution w(ξ, η) of the integral equation 1 w(ξ, η) = − 2

∞ ∞ η Rν (s, 0; ξ, η)q(s) ds − ds q(s + τ )Rν (s, τ ; ξ, η)w(s, τ ) dτ, ξ

ξ

0

for 0 < τ < η < ξ < s, then the function P (x, t) is determined by the formulas in (8.63) using w(ξ, η). The function Rν = Rα−1 is the Riemann function that arises when solving a certain Goursat problem for a singular inhomogeneous hyperbolic equation of the form (one of the forms of the Euler–Poisson–Darboux equation) ∂ 2 w(ξ, η) 4α(α − 1)ξ η w(ξ, η) = f (ξ, η), + ∂ξ ∂η (ξ 2 − η2 )2 which in our case is converted to the form ∂ 2 w(ξ, η) 4α(α − 1)ξ η w(ξ, η) = q(ξ + η)w(ξ, η). + ∂ξ ∂η (ξ 2 − η2 )2 This Riemann function is known explicitly and has a form (see [317])  2 ν   s − η2 ξ 2 − τ 2 s 2 − ξ 2 η2 − τ 2 · · . Rν = 2 2 F1 −ν, −ν; 1; 2 s − τ 2 ξ 2 − η2 s − η2 ξ 2 − τ 2

(8.64)

This expression is simplified in [526], where it is shown that the Riemann function in this case is expressed in terms of the Legendre function by the formula   η2 − τ 2 s 2 − ξ 2 1+A · . (8.65) Rν (s, τ, ξ, η) = Pν , A= 2 1−A ξ − τ 2 s 2 − η2

Applications of transmutations to different problems

357

Theorem 108. Let function q(r) ∈ C 1 (0, ∞) satisfy the conditions ∞ |p(t)| dt < ∞, ∀ξ > 0.

|q(s + τ )| ≤ |p(s)|, ∀s, ∀τ, 0 < τ < s,

(8)

ξ

Then there exists an integral representation of the form (8.61) whose kernel satisfies the estimate |P (r, t)| ≤ ⎡

 2 2   α ∞ 1 y (t + r 2 ) − (t 2 − r 2 ) t Pα−1 |p(y)| dy· r 2 2try 2 t+r 2





⎢ t −r exp ⎣ 2

1 2



∞ Pα−1

y 2 (t 2

+ r 2 ) − (t 2

 − r 2)

2try 2

⎤ ⎥ |p(y)| dy ⎦ .

t+r 2

Moreover, the kernel of the transmutation operator P (x, t) and the solution of Eq. (8.59) are functions that are twice continuously differentiable on (0, ∞) with respect to their arguments. We break the proof of Theorem 108 into some lemmas. Let us introduce the notation 1 Iq (ξ, η) = 2

∞ Rν (y, 0; ξ, η)|p(y)| dy = ξ

1 2



∞ Pν ξ

 y 2 (ξ 2 + η2 ) − 2ξ 2 η2 |p(y)| dy, y 2 (ξ 2 − η2 )

w0 (ξ, η) = −

1 2

(8.66)

∞ Rν (s, 0; ξ, η)|p(s)| ds, ξ

∞ Aw0 (ξ, η) = −

η q(s + τ )Rν (s, τ ; ξ, η)u0 (s, τ ) dτ.

ds ξ

0

Let us prove the uniform convergence of the operator Neumann series ∞ 

Ak w0 (ξ, y)

k=0

and the possibility of its double differentiation.

(8.67)

358

Transmutations, Singular and Fractional Differential Equations

Lemma 24. The estimate |w0 (ξ, η)| ≤ Iq (ξ, η) is valid The proof follows immediately from the definition (8.66). Lemma 25. Let 0 < τ < η < ξ < s. Then the inequality Iq (s, t) ≤ Iq (ξ, η)

(8.68)

is valid. Proof. By the statement of the lemma we have 0 < τ < η < ξ < s < y. We show that then τ 2 (y 2 − s 2 ) η2 (y 2 − ξ 2 ) ≤ (≤ 1). · · s 2 (y 2 − τ 2 ) ξ 2 (y 2 − η2 ) Indeed, this inequality is equivalent to τ 2 ξ 2 (y 2 − s 2 )(y 2 − η2 ) ≤ η2 s 2 (y 2 − ξ 2 )(y 2 − τ 2 ), which is obvious, since each of the factors on the left does not exceed the corresponding factor on the right. Further, we consider for 0 < x < 1 the function f (x) =

2 1+x > 0, 0 < x < 1. ≥ 1, f (x) = 1−x (1 − x)2

Therefore, this function increases in x. Therefore, 1+

τ2 s2

·

1−

τ2 s2

·

(y 2 −s 2 ) (y 2 −τ 2 ) (y 2 −s 2 ) (y 2 −τ 2 )

≤

1+ 1−

η2 ξ2 η2 ξ2

· ·

(y 2 −ξ 2 ) (y 2 −η2 ) (y 2 −ξ 2 ) (y 2 −η2 )

.

The Legendre function Pν (x) for x ∈ (1, ∞), ν > −1, monotonously increases, and in addition, Pν (x) > 1. So ⎛ ⎞ ⎛ ⎞ 2 2 (y 2 −s 2 ) (y 2 −ξ 2 ) 1 + τs 2 · (y 1 + ηξ 2 · (y 2 −τ 2 ) 2 −η2 ) ⎠ ≤ Pν ⎝ ⎠. Pν ⎝ 2 2 (y 2 −s 2 ) (y 2 −ξ 2 ) 1 − τs 2 · (y 1 − ηξ 2 · (y 2 −τ 2 ) 2 −η2 ) The last inequality can be written differently, i.e.,  2 2  2 2   y (s + τ 2 ) − 2s 2 τ 2 y (ξ + η2 ) − 2ξ 2 η2 Pν ≤ Pν . y 2 (s 2 − τ 2 ) y 2 (ξ 2 − η2 ) Note that we have actually proved the inequality for the Riemann function

Applications of transmutations to different problems

359

Rν (y, 0; s, τ ) ≤ Rν (y, 0; ξ, η),

(12)

when 0 < τ < η < ξ < s < y. From the calculations we obtain the estimate 1 Iq (s, τ ) = 2

∞

1 Rν (y, 0; s, τ )|p(y)| dy ≤ 2

s

∞ Rν (y, 0; s, τ )|p(y)| dy. ξ

Replacing the lower limit of integration of s by ξ < s, we can only increase the value of the integral, since the Riemann function is positive, Rν > 0. As a result, we arrive at the estimate (8.68). Lemma 26. For the n-th member of the Neumann series (8.67), the estimate |wn (ξ, η)| ≤ Iq (ξ, η) ·

[ηIq (ξ, η)]n n!

(8.69)

is valid. Proof. We apply the method of mathematical induction. For n = 0, inequality (8.69) reduces to the already proved inequality from Lemma 24. Let (8.69) be valid for some n = k. Then for the next member of the Neumann series we get ∞ |wk+1 (ξ, η)| ≤ |

η

∞

0

η Rν (s, τ ; ξ, η)|q(s + τ )|Iq (s, τ )

ds ξ

Rν (s, τ ; ξ, η)wk (s, τ )q(s + τ ) dτ | ≤

ds ξ

[ηIq (s, τ )]k dτ. k!

0

Repeating the arguments of the previous lemma, we obtain Rν (s, τ ; ξ, η) ≤ Rν (s, 0; ξ, η),

(8.70)

since  Rν (s, τ ; ξ, η) = Pν

 η2 − τ 2 s 2 − ξ 2 1+A · , , A= 2 1−A ξ − τ 2 s 2 − η2

and the maximum value for A is achieved when τ = 0. Taking into account inequality (8.70) and the alleged inequality (8.69), we arrive at the estimate [τ Iq (ξ, η)]k |wk+1 (ξ, η)| ≤ Iq (ξ, η) · k!

∞

η |q(s + τ )|τ k dτ ds.

Rν (s, 0; ξ, η) ξ

0

360

Transmutations, Singular and Fractional Differential Equations

We consider potentials for which the inequality |q(s + τ )| ≤ |p(s)|, 0 < τ < s. Finally we get |wk+1 (ξ, η)| ≤ Iq (ξ, η)

[Iq (ξ, η)]k+1 ηk+1 · , k! (k + 1)

which gives estimate (8.69) for all n. Now we complete the proof of Theorem 108. Summarizing the estimates in (8.69), we obtain that the Neumann series converges uniformly in the domain 0 < η < ξ and its sum is some continuous function satisfying the inequality |w(ξ, η)| ≤ Iq (ξ, η) exp[η · Iq (ξ, η)].

(8.71)

It follows from (8.71) that we could prove the convergence of series (8.67) for the integrable potential q, which can be approximated by continuous potentials. Returning to the functions K and P , we obtain the inequalities       t −x t +x t −x t +x t −x , exp Iq , , |K(x, t)| ≤ Iq 2 2 2 2 2  α       t t −x t +x t −x t +x t −x Iq |P (x, t)| ≤ , exp Iq , . x 2 2 2 2 2 We convert the value Iq included in the estimates  Iq

t +x t −x , 2 2



1 = 2



∞ Pα−1

 y 2 (t 2 + x 2 ) − (t 2 − x 2 ) |p(y)| dy. 2txy 2

t+x 2

Thus, we arrive at estimate (8.66). To complete the proof of Theorem 108, it remains to justify the existence of second continuous derivatives of the function P (x, t) with respect to the variables x and t under the condition q ∈ C 1 (x > 0). Obviously, this is equivalent to the existence of second continuous derivatives of the function u(ξ, η) with respect to the variables ξ, η. The proof of the last statement is carried out according to the above model by the method of successive approximations and completely repeats the corresponding fragment of the proof from [526]. The theorem is proved. We list the classes of potentials for which conditions (10.18) are satisfied. If |q(x)| is monotonically decreasing, then we can take p(x) = |q(x)|. For potentials with an arbitrary singularity at the origin and increasing for 0 < x < M (for example, Coulomb potential q = − x1 ), which are cut off by zero at infinity, q(x) = 0, x > M, we can take p(x) = |q(M)|, x < M, p(x) = 0, x eqM. Potentials with the estimate q(x + τ ) ≤ c|q(x)| = |p(x)| will also satisfy condition (10.18). V. V. Katrakhov drew

Applications of transmutations to different problems

361

the attention of one of the authors to the possibility of a similar strengthening of Theorem 108. In particular, the following potentials, encountered in applications, satisfy the above conditions: a strongly singular potential with a power-law singularity of the form q(x) = x −2−ε , various Bargman potentials q1 (x) = −

e−ax , (1 + βe−ax )2

q2 (x) =

c2 , (1 + c3 x)2

q3 (x) =

c4 ch2 (c5 x)

,

and Yukawa q4 (x) = −

e−ax , x

∞ q5 (x) =

e−at dc(t)

x

(see, for example, [497]). Remark 17. In fact, in the proof of the above theorem, an explicit form of the Riemann function (8.65) is not needed. We use only the existence of the Riemann function, its positivity, and some special property of monotonicity (8.70). These facts are quite general, so the results can be extended to a fairly wide class of differential equations. The estimate from Theorem 108 for potentials of a general form can be transformed into a less accurate, but more visible one. Theorem 109. Let the conditions of Theorem 108 be satisfied. Then the kernel of the transmutation operator P (x, t) satisfies the estimate    ∞  2 t + x2 1 t α Pα−1 |p(y)| dy× |P (x, t)| ≤ 2 x 2tx x ⎤ ⎡ ∞     2  2 t +x 1 t −x |p(y)| dy ⎦ . Pα−1 exp ⎣ 2 2 2tx x

Note that for x → 0 the kernel of the integral representation can have an exponential singularity.

8.3.3 Estimates for the case of a power singular at zero potential For a class of potentials with a power singularity of the form q(x) = x −(2β+1) , β > 0,

(8.72)

obtained estimates can be simplified without reducing their accuracy. The restriction on β is caused by the condition of integrability at infinity.

362

Transmutations, Singular and Fractional Differential Equations

Theorem 110. Consider a potential of the form (8.72). Then Theorem 108 holds with the estimate  α   2 t (β)4β−1 t + x2 −β |P (x, t)| ≤ · P × α−1 x 2tx (t 2 − x 2 )β     t − x (β)4β−1 −β t 2 + x 2 P , exp x 2tx (t 2 − x 2 )β α−1 μ

where Pν (·) is the Legendre function (1.42), the value β is determined from (8.72), and the value α is determined from (8.60). Anticipating the proof, we note that this estimate is obtained after rather lengthy calculations using the famous Slater–Marichev theorem [361], which helps to calculate the necessary integrals in terms of hypergeometric functions after they are reduced to the Mellin convolution. For this class of potentials, we simplify estimate (8.71), which constitutes the content of Theorem 108, without reducing its accuracy. For this, the value Iq included in estimate (8.71) will be calculated explicitly. We divide the proof of Theorem 110 into two lemmas. Lemma 27. For a potential of the form (8.72), we have the relation 1

1 Iq (ξ, η) = 2β 4ξ

Pν (2αz + 1)(1 − z)β−1 dz,

(8.73)

0

where Pν is the Legendre function, α = η2 /(ξ 2 − η2 ). Proof. Let us consider 1 Iq (ξ, η) = 2



∞ Pν ξ

t 2 (ξ 2 + η2 ) − 2ξ 2 η2 t 2 (ξ 2 − η2 )



dt t 2β+1

.

We carry out the change of variables, denoting the argument of the Legendre function by x, x=

4ξ 2 η2 t 2 (ξ 2 + η2 ) − 2ξ 2 η2 , dx = dt. t 2 (ξ 2 − η2 ) t 3 (ξ 2 − η2 )

With such a change of variables, the numbers of integration become new limits 1, 1 +

2η2 ξ 2 + η2 = = B > 1, ξ 2 − η2 ξ 2 − η2

and the variable t will have the form 1  2 2 . t = ξη 2 2 2 2 ξ + η − x(ξ − η )

Applications of transmutations to different problems

363

This leads to the following expression for Iq : 1 Iq (ξ, η) = 2

B Pν (x)

t 3 (ξ 2 − η2 ) dt = 4ξ 2 η2 t 2β+1

1

1 2

B



  2 β−1 ξ 2 − η2 ξ + η2 − x(ξ 2 − η2 ) Pν (x) dx. · 4ξ 2 η2 2ξ 2 η2

1

In the last integral, we make another change of variables by the formula z = (x − 1)

ξ 2 − η2 , 2η2

  ξ 2 − η2 dz = dx. 2η2

Then we obtain 1 Iq (ξ, η) = 2 ⎡ ⎣



ξ 2 − η2 4ξ 2 η2

ξ 2 + η2 − (ξ 2 − η2 )

 1 Pν (2αz + 1) 

0 2η2 ξ 2 −η2

 ⎤β−1

z+1

⎦

2ξ 2 η2

1 4ξ 2β

2η2 × − η2

ξ2

dz =

1 Pν (2αz + 1)(1 − z)β−1 dz, 0

where α = η2 /(ξ 2 − η2 ). We get (8.73).

Lemma 28. Let the conditions a > 0, β > 0 be satisfied. Then the following formula is valid: 

1 Pν (2αx + 1)(1 − x)β−1 dx = (β)

1+α α

β

2

Pν−β (2α + 1).

(8.74)

0

Proof. In the proof we will use notation and a technique based on the Slater–Marichev theorem [361]. In the integral from (8.74), we change the variables t = 1/x. We get 1

∞ Pν (2αx + 1)(1 − x)

β−1

0

dx = 1

  α dt Pν 2 + 1 (t − 1)β−1 t −β = t t

364

Transmutations, Singular and Fractional Differential Equations

∞

  α dt β−1 Pν 2 + 1 (t − 1)+ t −β = I (α), t t

0 λ = x λ when x > 0, and where the notation for the truncated power function is used, x+ λ x+ = 0 when x ≤ 0. We apply the Mellin transform (see Definition 11) with respect to the variable α, (α > 0) to the function I (α). Using the Mellin convolution theorem (see [361]), we obtain

M[I (α)](s) = M[Pν (2x + 1)](s) · M[x −β (x − 1)+ ](s). β−1

Using formulas (6)(1), (4), and (2)(4) from [361], we obtain sin πν (s)(−ν − s)(1 + ν − s)(β)(1 − s) = M[I (α)](s) = − π (1 − s)(1 + β − s)   sin πν s, −ν − s, 1 + ν − s , (β)  − 1+β −s π where the Slater designation is used for the ratio of the products of gamma functions. In the notation of the Slater–Marichev theorem, we have (a) = (0), (b) = (−ν, 1 + ν), (c) = ∅, (d) = (1 + β), A = 1, B = 2, C = 0, D = 1. Using the Slater–Marichev theorem, we obtain formulas for I (α) for 0 < α < 1: I (α) = − β

sin πν (1 + ν)(−ν) 2 F1 (−ν, 1 + ν; 1 + β; −α) = π (1 + β) β

(β)α − 2 (1 + α) 2 Pν−β (1 + 2α),

(8.75)

where formula (3) from [19], p. 126, and the identity for gamma functions (see [19]) (−ν) =

π ν (ν) sin πν

are used. For α ≥ 1 we get another expression: sin πν I (α) = − (β)× π     & 1 1 + ν + ν, −ν F + −ν, 1 − 1 − β − ν; 1 − 1 − ν − ν; − αν  2 1 1+β +ν α   '  1 −ν − 1 − ν, 1 + ν F = 1+ν, 1−1−β + 1 + ν; 1 + ν; − α −1−ν  2 1 1+β −1−ν α &   (2ν + 1)(−ν) 1 sin πν F (β) · α ν + −ν, −β − ν; −2ν; − − 2 1 π (1 + β + ν) α

Applications of transmutations to different problems

α −1−ν

365

'  (−1 − 2ν)(1 + ν) 1 . 2 F1 1 + ν, 1 + ν − β; 1 + ν; − (β − ν) α

But from [19], p. 131, formula (19), it follows that equality is obtained for I (α) when 0 < α < 1 and α ≥ 1 coincide. From (8.75) it follows that we obtained the desired formula (1.101); however, its conclusion is not completely rigorous, since we did not check the legality of applying the Mellin transform and the validity conditions of the Slater–Marichev theorem (which is, in our case, rather complicated). However, now we can apply the Mellin transform to both sides of the obtained formal equality (1.101). As a result, we prove that for α > 0, β > 0, relation (1.101) is an identity. As a consequence, we now obtain the necessary estimate and we prove Theorem 110. The simplest similar estimate was obtained in [526] for the potential q(x) = cx −2 , −1

for which β = 12 . As follows from [19], in this case the Legendre function Pν 2 (z) can be expressed in terms of elementary functions. Therefore, the corresponding estimate can be expressed in terms of elementary functions. Another potential for which the obtained estimate can be simplified and expressed in terms of elementary functions is a potential of the form q(x) = x −(2β+1) , when the parameters are related by the relation β = α − 1. Corollary 18. Let the relation between the parameters β = α − 1. Then the estimate of Theorem 110 has a form  β   β   β+1 β−2  2 2 t − x 2β−2 t 2 + x 2 t + x2 t · exp = |P (x, t)| ≤ x β 2tx 2 β 2tr    2 β  1 2β−2 t − x 1 t + x2 2 2 β (t + x ) exp . (8.76) 4β x 2β+1 β 2 2tx Proof. In this case, we transform the estimate from Theorem 110 as follows:  β2  2 (β)4β−1 2−β t 2 + x2 −1 = = 2 2tx (t − x 2 )β (β + 1)  β 2β−2 (t 2 − x 2 )β (t 2 + x 2 )β 2β−2 t 2 + x 2 1 = , β (t 2 − x 2 )β (2tx)β β 2tx

(β)4β−1 −β P (t 2 − x 2 )β β



t 2 + x2 2tx



where the transformations used the formula (see [19]) Pν−ν (z) =

ν 2−ν (z2 − 1) 2 , z > 1. (ν + 1)

Therefore, the inequality for the kernel with β = α − 1 takes the form (8.76).

366

Transmutations, Singular and Fractional Differential Equations

Note that for α = 0 in formulas (8.59)–(8.60), Theorem 108 reduces to well-known estimates for the kernel of the Levin type integral representation for the Sturm– Liouville equation. The above technique is fully transferred to the problem of constructing nonclassical operators of generalized translation. This problem is essentially equivalent to expressing solutions to the equation Bα,x u(x, y) − q(x)u(x, y) = Bβ,y u(x, y)

(8.77)

through solutions of the unperturbed Euler–Poisson–Darboux equation with Bessel operators in each variable (in the nonsingular case, the wave) in the presence of additional conditions that ensure correctness. Such representations are obtained already from the fact of the existence of transmutation operators and were studied for the nonsingular case (α = β = 0) in [316,321] as a consequence of the generalized translation theory. An interesting original technique for obtaining such representations was also developed in the nonsingular case in the papers of A. V. Borovskikh [34]. From the results of this subsection, integral representations of a certain subclass of solutions of Eq. (8.77) in the general singular case for sufficiently arbitrary potentials with singularities at the origin of coordinates follow. Moreover, the estimates for the solutions do not contain any indefinite constants, and for the kernels of integral representations the integral equations are written in explicit form, which they satisfy.

8.3.4 Asymptotically exact inequalities for Legendre functions In this subsection we show how one can use formula (1.101) to establish an asymptotically exact lower bound for Legendre functions. This assessment, in our opinion, is of independent interest. Consider the obvious inequalities (α > 0) 1

− 12

Pν (1 + 2αx)(1 − x)

1 dx ≤

0

1

1

Pν (1 + 2αx)x − 2 (1 − x)− 2 dx ≤

0

1 Pν (1 + 2α)

1

1

x − 2 (1 − x)− 2 dx.

0

The second integral is known, and, as follows, for example, from [19], p. 225, it is calculated by the formula  ( )2 √ 1 1 Pν ( 1 + α) . I =π2  2 Therefore, the inequality √

( √ )2 1 1 −1 π α − 4 (α + 1)− 4 Pν 2 (2α + 1) ≤ π Pν ( 1 + α)

Applications of transmutations to different problems

367

is valid. Expressing the Legendre function in elementary functions, we finally get )2 √ Pν ( 1 + α) ≥ & 2ν+1 √ −2ν−1 ' √ √ √ 1 x + x +1 − x + x +1 . √ π(2ν + 1) x

(

The proved inequality gives the correct asymptotics for all parameters included in it, as follows from the asymptotic formula given in [448], p. 107, *

√ √ 2ν+1 +2 √ x + x+1 1 . Pν ( 1 + x) ∼ √ 2πν x

The second of the considered integral inequalities also leads to an interesting estimate ( √ )2 Pν ( 1 + α) ≤ Pν (1 + 2α). In connection with the last inequalities presented, we note that the Legendre functions are associated with complete and incomplete elliptic Legendre integrals of three kinds; these two classes of special functions are expressed through each other at certain values. On the other hand, Legendre functions are particular cases of the Gauss hypergeometric function. Various inequalities for elliptic Legendre integrals are considered in [221,222], and inequalities for hypergeometric functions are considered in [223,224].

8.4

Iterated spherical mean in the computed tomography problem

Spherical means have numerous applications in theoretical mathematics and its applications. In the classic books [75,155,170], various applications of spherical means to the theory of partial differential equations, including elliptic, hyperbolic, and ultrahyperbolic types, are given. In addition, spherical means are the object of study of integral geometry with the application of research results to tomography [163], for example photoacoustics [95] and diffraction tomography [18]. The close connection of spherical means with the Fourier transform and Riesz potentials is also known (see [162]). Representation of solutions of various partial differential equations using spherical means is also related to the theory of transmutation operators [234]. In this section we consider the weighted spherical mean (3.183) and obtain identities for iterated weighted spherical means, which are necessary to obtain explicit formulas when restoring a function from its weighted spherical means. In addition, using weighted spherical means we give the formula for representing the function through the generalized translation operators (3.144) and the Hankel transform (12). Such formulas are used in various applied problems of tomography and integral geometry.

368

Transmutations, Singular and Fractional Differential Equations

8.4.1 Iterated weighted spherical mean and its properties The iterated weighted spherical mean has the form (see [536]) γ

γ

γ

If (x; λ, μ) = Iλ,μ f (x) = Mλ Mμγ f (x) =   1 γ λζ γ μξ Tx Tx [f (x)]ζ γ ξ γ dS(ξ )dS(ζ ), |S1+ (n)|2γ S1+ (n) S1+ (n)

γ

γ

where Mλ and Mμ are weighted spherical means (3.183). Using the permutation property of the generalized translation (3.150), we obtain that the iterated weighted spherical mean is symmetric with respect to γ

γ

If (x; λ, μ) = If (x; μ, λ). It is clear that γ

γ

γ

If (x; λ, 0) = If (x; 0, λ) = Mλ f (x) and γ

If (x; 0, 0) = f (x). Following [155], p. 73, we prove the equality expressing the iterated spherical mean γ γ If (x; λ, μ) through a single integral from the weighted spherical mean Mr [f (x)]. γ

Theorem 111. Let f ∈ L1 . Then γ

γ

If (x; λ, μ) = ν Tμλ Mf (x; μ),

(8.78)

where ν=n+|γ |−1. The following formulas are also true:   | 2  n+|γ 2 1 γ   × If (x; λ, μ) = √ n+|γ |−2 π |γ |+n−1 (2λμ) 2

λ+μ 



λ−μ

 n+|γ |−3 2 γ Mr [f (x)] rdr, (λ2 − (r − μ)2 )((r + μ)2 − λ2 )

(8.79)

 β −α β +α , = 2 2  | β   n+|γ |−3  n+|γ 2 2n+|γ |−1 2 γ 2 2 2 2   − r )(r − α ) Mr [f (x)]rdr. (β √ |γ |+n−1 (β 2 − α 2 )n+|γ |−2 π γ

If



x; 

2

α

(8.80)

Applications of transmutations to different problems

369

Proof. Let g(s) be an arbitrary continuous finite function of one variable. Consider the integral ∞ J=

λn+|γ |−1 g(λ)If (x; λ, μ)dλ = γ

0

1 + |S1 (n)|2γ

∞





λn+|γ |−1 g(λ)dλ

γ

Txλζ γ Txμξ [f (x)]ζ γ ξ γ dS(ξ )dS(ζ ).

S1+ (n) S1+ (n)

0

Using the property (3.150) of generalized translation and formula (1.104), we can write ⎡ ⎤  ∞  ⎢ ⎥ γ 1 γ μξ ⎢ γ λζ J= + Tx ⎣ λn+|γ |−1g(λ)dλ Tx [f (x)]ζ γ dS(ζ )⎥ ⎦ξ dS(ξ )= 2 |S1 (n)|γ S1+ (n)

1 |S1+ (n)|2γ

 γ

⎢ Txμξ ⎢ ⎣ lim

1

1 |S1+ (n)|2γ

R

R→+∞

S1+ (n)

λn+|γ |−1 g(λ)dλ

⎡

 γ

γ

R→+∞ BR+ (n)

S1+ (n)

⎡

 γ

⎢ Txμξ ⎣

S1+ (n)

 γ

γ

⎥ Txλζ [f (x)]ζ γ dS(ζ )⎥ ⎦× ⎤



⎢ Txμξ ⎢ ⎣ lim

⎤



S1+ (n)

0

ξ γ dS(ξ ) =

|S1+ (n)|2γ

S1+ (n)

0

⎡

⎥ γ Txz [f (x)]g(|z|)zγ dz⎥ ⎦ ξ dS(ξ ) = ⎤

⎥ Txz [f (x)]g(|z|)zγ dz⎦ ξ γ dS(ξ ).

Rn+

Now, applying the properties of associativity (3.151) and self-adjointness (3.157), we obtain    γ μξ x 1 Tz Tz [f (z)] g(|z|)zγ dz ξ γ dS(ξ ) = J= + 2 |S1 (n)|γ S1+ (n) R+ n

1 |S1+ (n)|2γ C(γ ) |S1+ (n)|2γ





γ

Tzx [f (z)] γ Tzμξ [g(|z|)] zγ dz ξ γ dS(ξ ) =

γ

Txz [f (x)]

S1+ (n) R+ n





S1+ (n) R+ n

π ... 0

π , n 0 i=1

sinγi −1 αi ×

370

Transmutations, Singular and Fractional Differential Equations

 g( μ2 ξ12 + ... + μ2 ξn2 + z12 + ... + zn2 − 2μξ1 z1 cos α1 − ... − 2μξn zn cos αn )× dα1 ...dαn zγ dz ξ γ dS(ξ ). Passing to the spherical coordinates z = rη, |η| = 1, r ≥ 0 in the integral by z and taking into account that |ξ | = 1, we obtain C(γ ) J= + |S1 (n)|2γ



∞ r

n+|γ |−1

 γ

dr

S1+ (n) 0

Txrη [f (x)]×

S1+ (n)

π π ,  n ... sinγi −1 αi g( μ2 + r 2 −2rμξ, η cos α)dα1 ...dαn ηγ dS(η) ξ γ dS(ξ ), 0 i=1

0

where ξ, η cos α = ξ1 η1 cos α1 + ... + ξn ηn cos αn . Using the multi-dimensional Poisson operator (3.137) we can write the integral J in the form 1 J= + |S1 (n)|2γ 

+∞  n+|γ |−1 r dr

γ

Txrη [f (x)]ηγ dS(η)×

S1+ (n)

0

 Pηγ g( r 2 + μ2 − 2rμξ, η) ξ γ dS(ξ ).

S1+ (n)

Applying to the integral

% S1+ (n)

$ γ Pη g( r 2 + μ2 − 2rμξ, η) ξ γ dS(ξ ) the formula (see

[247]) 

γ Pξ g(ξ, x)x γ dS(x) = C(γ )

S1+ (n)

1 g(|ξ |p)(1 − p 2 )

n+|γ |−3 2

dp

−1

and noting that |η| = 1, we obtain n -

 

γi +1 2



1   + J=√ |γ |+n−1 |S1 (n)|2γ π2n−1  i=1

2

1 (1 − p 2 ) −1

n+|γ |−3 2

∞ r

n+|γ |−1

0

 g( r 2 + μ2 − 2rμp) dp =

 dr S1+ (n)

γ

Txrη [f (x)]ηγ dS(η)×

Applications of transmutations to different problems n -

 



γi +1 2

1 √ n−1  |γ |+n−1  |S + (n)| γ π2  1 i=1

2

1

∞

371

r n+|γ |−1 Mr [f (x)] dr× γ

0

 n+|γ |−3 (1 − p 2 ) 2 g( r 2 + μ2 − 2rμp) dp.

−1

Now, instead of the variable p, we introduce the following formula associating the variable λ associated with p: r 2 + μ2 − 2rμp = λ2 . We get p=

r 2 + μ 2 − λ2 (λ2 − (r − μ)2 )((r + μ)2 − λ2 ) λ , , dp = − dλ, 1 − p 2 = 2rμ rμ (2rμ)2

and for p = −1, λ = |r + μ|, and for p = 1, λ = |r − μ|. Then using (1.107) we obtain   | 2  n+|γ 2  × J = (2μ)2−n−|γ | √ |γ |+n−1 π 2 |λ+μ| 

∞



g(λ) λ dλ

 n+|γ |−3 2 γ Mr [f (x)] rdr. (λ2 − (r − μ)2 )((r + μ)2 − λ2 )

|λ−μ|

0

Since g(λ) is an arbitrary function, from the equality ∞

 λn+|γ |−1 g(λ)If (x; λ, μ)dλ = (2μ)2−n−|γ | √ γ

0

∞

2 π



n+|γ | 2



|γ |+n−1 2

×

λ+μ 

 n+|γ |−3  2 γ Mr [f (x)] rdr (λ2 − (r − μ)2 )((r + μ)2 − λ2 )

g(λ) λ dλ λ−μ

0

it follows that γ



If (x; λ, μ) = √ 1 (2λμ)n+|γ |−2

2 π

λ+μ 





n+|γ | 2



|γ |+n−1 2

×

 n+|γ |−3 2 γ Mr [f (x)] rdr. (λ2 − (r − μ)2 )((r + μ)2 − λ2 )

λ−μ

(8.81)

372

Transmutations, Singular and Fractional Differential Equations

We removed the modules within the integration due to the fact that the integrand is odd by r. Now taking into account (3.154) for ν=n+|γ |−1, we get γ

γ

If (x; λ, μ) = ν Tμt Mf (x; μ). If in (8.81) we put α = λ − μ, β = λ + μ, (β > α), then we obtain   β −α β +α γ If x; , = 2 2   | β  n+|γ |−3  n+|γ 2 2n+|γ |−1 2 γ 2 2 2 2   Mr [f (x)]rdr. (β − r )(r − α ) √ |γ |+n−1 (β 2 − α 2 )n+|γ |−2 π 2

α

The theorem is proved. We give a corollary of the proved theorem expressing the action of iterated weighted averages on Bessel functions. Corollary 19. For the function jγ (x, ξ ), the following equality is true:  γ

I β−α

β+α 2 , 2

β 

jγ (x, ξ ) = √

 π

(β 2 − r 2 )(r 2 − α 2 )

n+|γ | 2





|γ |+n−1 2



2n+|γ |−1 jγ (x, ξ ) × (β 2 − α 2 )n+|γ |−2

 n+|γ |−3 2

j n+|γ |−2 (r)rdr.

(8.82)

2

α

Proof. In Theorem 111, let us choose f (x) = jγ (x, ξ ). Using formula (3.190) of the form Mμγ jγ (x, ξ ) = jγ (x, ξ ) j n+|γ |−2 (μ), 2

we get γ

Iλ,μ jγ (x, ξ ) = jγ (x, ξ ) j n+|γ |−2 (μ) j n+|γ |−2 (λ). 2

2

Applying (8.80), we obtain (8.82). We can rewrite (8.82) in the form     β −α β +α j n+|γ |−2 j n+|γ |−2 = 2 2 2 2   | β  n+|γ |−3  n+|γ 2 2n+|γ |−1 2 2 2 2 2   − r )(r − α ) j n+|γ |−2 (r)rdr. (β √ |γ |+n−1 (β 2 − α 2 )n+|γ |−2 2 π 2

α

Applications of transmutations to different problems

373

γ

If for Iλ,μ jγ (x, ξ ) we write equality (8.78), we get the known formula (3.152): ν

Tμλ j ν−1 (μ) = j ν−1 (μ) j ν−1 (λ), 2

2

ν = n + |γ | − 1.

2

8.4.2 Application of identity for an iterated spherical mean to the task of computed tomography Consider one application of formula (8.79) from Theorem 111 to computed tomography. In problems of diffraction tomography and backscattering, the Hankel transform is the measured data (see, for example, [168], [302], p. 126, and [3], p. 90). Similar formulas are used to restore function. Let us prove a formula expressing a function through its Hankel transform and a generalized translation. This is a generalization of the well-known formula for a simpler problem, which uses the representation of a function through its Fourier transform and the usual shift. In this form, similar representations are used to restore functions in the indicated problems of tomography and integral geometry. In the theory of scattering, the surface of a ball |x| < 2λ, where λ is the given wavelength and x is the space vector called the Ewald sphere (see [129]). The Ewald sphere can be used to find the maximum resolution available for a given X-ray wavelength and the unit cell dimensions.  be a function with support inside part of the ball Theorem 112. Let F + (n) = {x ∈ Rn+ : |x| < 2λ}. B2λ

The equality 



F (y) = C(n, γ )

 γ λζ Tλξ

S1+ (n) S1+ (n)

(λξ )jγ (λξ, y) |λξ |F (4λ2 − |tξ |2 )

n+|γ |−3 2

 ζ γ ξ γ dS(ξ )dS(ζ ), (8.83)

where

  π22n−3 λ2n+2|γ |−4  |γ |+n−1 2 C(n, γ ) = ,    n γ +1 n+|γ | j + 2 2   |S1 (n)|γ 2 2 √

j =1

is true. Proof. Let us put μ = λ in (8.79). Then γ

If (x; λ, λ) =   | 2λ* + n+|γ |−3 2  n+|γ 2 1 2 γ 2 2   −r r n+|γ |−2 Mr [f (x)]dr. 4λ √ |γ |+n−1 (2λ2 )n+|γ |−2 π 2 0 (8.84)

374

Transmutations, Singular and Fractional Differential Equations

Let us consider the function fy (x) =

|x| (4λ2 − |x|2 )

(x)jγ (x, y), F

n+|γ |−3 2

x, y ∈ Rn+ .

We now find the weighted spherical mean of fy (x) when x = 0: γ Mr fy (x)|x=0

1 + |S1 (n)|γ

1 = + |S1 (n)|γ 



γ

Txrz

S1+ (n)



1 |S1+ (n)|γ

 [ γ Txrz fy (x)]x=0 zγ dS(z) = S1+ (n)



|x| − |x|2 )

(4λ2

n+|γ |−3 2

r|z|

S1+ (n)

(4λ2 − r|z|2 )

r 1 |S1+ (n)|γ (4λ2 − |r|2 ) n+|γ2 |−3

n+|γ |−3 2



(x)jγ (x, y) F

zγ dS(z) = x=0

(rz)jγ (rz, y)zγ dS(z) = F

(rz)jγ (rz, y)zγ dS(z). F

S1+ (n)

Using (8.84) we get γ

If (0; λ, λ) =   | 2λ* + n+|γ |−3 2  n+|γ 2 1 2 γ 2 2   − r r n+|γ |−2 Mf (0, r)dr = 4λ √ |γ |+n−1 (2λ2 )n+|γ |−2 π 2 0   2λ n+|γ |  n+|γ |−3 2 2 (2λ2 )2−n−|γ | * 2 2 + 2 r   −r r n+|γ |−2 dr× 4λ n+|γ |−3 √ + |γ |+n−1 2 2) 2 |S (n)| γ π (4λ − r 1 2 0  (rz)jγ (rz, y)zγ dS(z) = F S1+ (n)



n+|γ | 2



(2λ2 )2−n−|γ |   √ |S1+ (n)|γ π |γ |+n−1 2

2

2λ r

n+|γ |−1

 dr

0

(rz)jγ (rz, y)zγ dS(z). F

S1+ (n)

Applying (1.104) we can write  γ

If (0; λ, λ) = √

2 π



n+|γ | 2



|γ |+n−1 2



(2λ2 )2−n−|γ | |S1+ (n)|γ

 + B2λ

(z)jγ (z, y)zγ dz = F

Applications of transmutations to different problems

 2 √



π  2

√



π 2

√

n+|γ | 2



|γ |+n−1 2 n+|γ | 2



|γ |+n−1 2





n+|γ | 2



(2λ2 )2−n−|γ | |S1+ (n)|γ



375

(z)jγ (z, y)zγ dz = F

Rn+

  n (2λ2 )2−n−|γ | |γ |−n , 2 γj + 1 2  F (y) = 2 |S1+ (n)|γ j =1

n





j =1

π22n−3 λ2n+2|γ |−4 

γj +1 2





|γ |+n−1 2

 F (y).

Therefore, √ F (y) =

  π22n−3 λ2n+2|γ |−4  |γ |+n−1 2 γ If (0; λ, λ).    n γ +1 |  2 n+|γ  j2 2

(8.85)

j =1

From the other side, γ If (0; λ, λ) =

1 + |S1 (n)|2γ



 γ

tζ

Ttξ [f (tξ )]ζ γ ξ γ dS(ξ )dS(ζ ).

S1+ (n) S1+ (n)

From (8.85) and (8.86) we get   π22n−3 λ2n+2|γ |−4  |γ |+n−1 2 × F (y) =    n γ +1 n+|γ | j + 2 2   |S1 (n)|γ 2 2 √





S1+ (n) S1+ (n)

 λζ Tλξ

j =1

|λξ | (4λ2 − |tξ |2 )

  F (λξ )jγ (λξ, y) ζ γ ξ γ dS(ξ )dS(ζ ). n+|γ |−3 2

(8.86)

Fractional powers of Bessel operators

9

In this chapter we study the fractional powers (Bγ )α , α ∈ R, of the differential Bessel operator in the form Bγ = D 2 +

γ D, x

γ ≥ 0, D :=

d . dx

(9.1)

Of course fractional powers of the Bessel operator (9.1) were studied in many papers. But in most of them fractional powers were defined implicitly as a power function multiplication under Hankel transform. This definition via integral transforms leads to many restrictions. Just imagine that for the classical Riemann–Liouville fractional integrals we have to work only with its definitions via Laplace or Mellin transforms and nothing more without explicit integral representations. If it would be true, then 99% of the classical “Bible” [494] and other books on fractional calculus would be empty as they mostly use explicit integral definitions! But for fractional powers of the Bessel operator in most papers implicit definitions via Hankel transform are still used. Such situation is not natural and in some papers different approaches to step closer to explicit formulas were studied. Let us mention that in [367] explicit formulas were derived as compositions of Erdélyi–Kober fractional integrals [494] on distribution spaces; in this monograph results on fractional powers of Bessel and related operators are gathered from McBride’s and earlier papers. An important step was made in [555] in which explicit definitions were derived in terms of the Gauss hypergeometric functions with different applications to partial differential equations; we also use basic formulas from [555] in this chapter. The most general study was carried out by I. Dimovski and V. Kiryakova [90–92,252] for the more general class of hyper-Bessel differential operators related to the Obrechkoff integral transform. They constructed explicit integral representations of the fractional powers of these operators by using Meijer G-functions as kernels, and also intensively and successfully used for this the theory of transmutations. Note that in this and other fields of theoretical and applied mathematics, the methods of transmutation theory are very useful and productive and for some problems are even irreplaceable (see, e.g., [89]). In [527,531] simplified representations for fractional powers of the Bessel operator were derived with Legendre functions as kernels, and based on them general definitions were simplified and unified with standard fractional calculus notation as in [494], and also important generalized Taylor formulas were proved which mix integer powers of Bessel operators (instead of derivatives in the classical Taylor formula) with fractional powers of the Bessel operator as integral remainder term (cf. also [268,532]). In this chapter we study fractional powers of the Bessel differential operator. The fractional powers are defined explicitly in the integral form without the use of integral transforms in the definitions. Some general properties of the fractional powers of the Transmutations, Singular and Fractional Differential Equations With Applications to Mathematical Physics https://doi.org/10.1016/B978-0-12-819781-3.00015-X Copyright © 2020 Elsevier Inc. All rights reserved.

378

Transmutations, Singular and Fractional Differential Equations

Bessel differential operator are proved and others are listed. Among them are different variations of definitions, relations with the Mellin and Hankel transforms, group properties, the generalized Taylor formula with Bessel operators, and an evaluation of the resolvent integral operator in terms of the Wright or generalized Mittag-Leffler functions. At the end, some topics are suggested for further study and possible generalizations.

9.1

Fractional Bessel integrals and derivatives on a segment

In this section we give the definitions of the fractional Bessel integrals and derivatives which correspond of the Riemann–Liouville fractional integrals and fractional derivatives on a finite interval, and we consider their properties. We also consider fractional Bessel derivatives on a finite interval of Gerasimov–Caputo type.

9.1.1 Definitions Let [a, b] (0 ≤ a < b < ∞) be a finite interval on the real semiaxis [0, ∞). Definition 44. Let α > 0. The right-sided fractional Bessel integral Bγ−α ,b− on a segment [a, b] for f ∈L1 (a, b), a, b ∈ [0, ∞), is defined by the formula α (Bγ−α ,b− f )(x) = (I Bγ ,b− f )(x) =

1 (2α)

b 

y 2 −x 2 2y

2α−1

  γ −1 x2 α+ f (y)dy. , α; 2α; 1− F 2 1 2 y2

(9.2)

x

The left-sided fractional Bessel integral Bγ−α ,a+ on a segment [a, b] for f ∈L1 (a, b), a, b ∈ (0, ∞), is defined by the formula α (Bγ−α ,a+ f )(x) = (I Bγ ,a+ f )(x) =   x  γ  2 2 2α−1 1 γ −1 x −y y2 y α+ f (y)dy. (9.3) , α; 2α; 1− F 2 1 (2α) x 2x 2 x2 a n−α 2n Definition 45. Let α > 0, n = [α] + 1, f ∈L1 (a, b), I Bγn−α ,b− f, I Bγ ,a+ f ∈Cev (a, b). The right-sided and left-sided fractional Bessel derivatives on a segment of the Riemann–Liouville type for α = 0, 1, 1, ... are defined, respectively, by the equalities

(Bγα,b− f )(x) = (DBγα,b− f )(x) = Bγn (I Bγn−α ,b− f )(x),

n = [α] + 1,

(9.4)

(Bγα,a+ f )(x) = (DBγα,a+ f )(x) = Bγn (I Bγn−α ,a+ f )(x),

n = [α] + 1.

(9.5)

and

Fractional powers of Bessel operators

379

When α = n ∈ N ∪ {0}, then (Bγ0,b− f )(x) = (Bγ0,a+ f )(x) = f (x), (Bγn,b− f )(x) = (Bγn,a+ f )(x) = Bγn f (x), where Bγn is an iterated Bessel operator (9.1). Remark 18. In some cases for fractional Bessel integrals on a segment [a, b] it is con−α α α venient to use the notations Bγ−α ,b− and Bγ ,a+ and in other cases I Bγ ,b− and I Bγ ,a+ . Similarly, for fractional Bessel integrals on a segment [a, b] sometimes we will use the notations Bγα,b− and Bγα,a+ and sometimes DBγα,b− and DBγα,a+ . Definition 44 is based on integral representations introduced for special cases a = 1, b = 1 in [555]. It was noted in [515,527,531] that Definition 44 may be simplified, as the kernels are expressed in a more simple way via Legendre functions (the Legendre functions are a two-parameter family but the Gauss hypergeometric functions are in general a three-parameter family). These simplifications are based on the formula [457] 2 F1 (a, b; 2b; z) =

      1 1 1 1 z 1 −b b−a− 12 , 22b−1  b + P2 z 2 −b (1 − z) 2 1 − √ a−b− 12 2 2 1−z

(9.6)

and have forms (Bγ−α ,b− f )(x)=

√

 b  γ 1   π y 2 2 α− 12 y 2 2 −α 1 x (y − x ) P + f (y)dy, γ x y x 22α−1 (α) 2 −1 2 x

√

 x  γ 1   π y 2 2 α− 12 y 2 2 −α 1 x f )(x)= (x − y ) P + f (y)dy. (Bγ−α γ ,a+ x y x 22α−1 (α) 2 −1 2 a

Now we would like to have another explicit formula for Bγα when α > 0. For applications it is better to use the generalization of the Gerasimov–Caputo fractional derivative (2.30). n−α 2n Definition 46. Let n = [α] + 1, f ∈L[0, ∞), I Bγn−α ,b− f, I Bγ ,a+ f ∈Cev (0, ∞). The right-sided fractional Bessel derivatives on a segment [a, b] of Gerasimov– Caputo type for α > 0, α = 0, 1, 2, ..., is defined by the equality n (Bγα,b− f )(x) = (I Bγn−α ,b− Bγ f )(x).

The left-sided fractional Bessel derivatives on a segment [a, b] of Gerasimov–Caputo type for α > 0, α = 0, 1, 2, ..., is defined by the equality n (Bγα,a+ f )(x) = (I Bγn−α ,a+ Bγ f )(x).

380

Transmutations, Singular and Fractional Differential Equations

Here I Bγn−α ,b− is the right-sided fractional Bessel integral (9.2) on a segment [a, b] n−α and I Bγ ,a+ is the left-sided fractional Bessel integral (9.3) on a segment [a, b]. When α = n ∈ N ∪ {0}, then (Bγ0 ,b− f )(x) = (Bγ0 ,a+ f )(x) = f (x), (Bγn,b− f )(x) = (Bγn,a+ f )(x) = Bγn f (x), where Bγn is an iterated Bessel operator (9.1).

9.1.2 Basic properties of fractional Bessel integrals on a segment Lemma 29. For γ = 0, f (x)∈L1 (a, b), a ≥ 0, fractional Bessel integrals on a segment [a, b] are −α f )(x) = (B0,b−

1 (2α)

b 2α (y − x)2α−1 f (y)dy = (Ib− f )(x) x

and −α (B0,a+ f )(x) =

1 (2α)

x 2α (x − y)2α−1 f (y)dy = (Ia+ f )(x), a

2α and I 2α are Riemann–Liouville fractional integrals and derivatives on a where Ib− a+ segment [a, b] defined by (2.11) and (2.12), respectively.

Proof. Indeed, we have −α f )(x) = (B0,b−

1 (2α)

b 

y 2 −x 2 2y

2α−1

  x2 1 F , α; 2α; 1− α− f (y)dy 2 1 2 y2

x

and −α f )(x) = (B0,a+

1 (2α)

  x  γ  2 2 2α−1 y2 y x −y 1 F , α; 2α; 1− α− f (y)dy. 2 1 x 2x 2 x2 a

Using the formula that is obtained from the integral representation of the Gauss hypergeometric function (1.34) of the form

Fractional powers of Bessel operators

 2 F1

381

1 x2 α − , α; 2α; 1 − 2 2 y





2y = x+y

2α−1 ,

we obtain provable formulas. Consider now the case when α = 1. Lemma 30. The following equalities hold: (Bγ−1 ,b− f )(x) =

1 γ −1

b  1−γ x y − 1 f (y)dy, y

1 γ −1

x   y γ −1  f (y)dy, y 1− x

f (x)∈L1 (a, b),

a ≥ 0,

f (x)∈L1 (a, b),

a ≥ 0.

x

and (Bγ−1 ,a+ f )(x) =

a

Proof. Applying the formula  2 F1

γ +1 x2 , 1; 2; 1 − 2 2 y



y2 2 = 1 − γ x2 − y2

  x 1−γ −1 , y

which is valid for the Gauss hypergeometric function, we obtain provable statements. In the following lemma we indicate the conditions under which the operators Bγ−1 ,b−

and Bγ−1 ,a+ will be left inverse to the Bessel differential operator on the segment.

2 (a, b), f (x) = B g(x), f (x)∈L (a, b), a ≥ 0. The equality Lemma 31. Let g ∈ Cev γ 1

(Bγ−1 ,b− Bγ g)(x) = g(x) is true if g(b − 0) = lim g(y) = 0, y→b−0

g (b − 0) = lim g (y) = 0. y→b−0

The equality (Bγ−1 ,a+ Bγ g)(x) = g(x) is true if g(a + 0) = lim g(x)=0, x→a+0

g (a + 0) = lim g (x)=0. x→a+0

382

Transmutations, Singular and Fractional Differential Equations

γ



Proof. Let us consider Bγ−1 ,b− . Putting f (x) = Bγ g(x) = g (x) + x g (x), we obtain −1 (Bγ−1 ,b− f )(x) = (Bγ ,b− Bγ g)(x) =

  b  1−γ 1 x γ



y − 1 g (y) + g (y) dy = γ −1 y y x ⎤ ⎡ b



 1−γ  b  1−γ 1 ⎣ x x y − 1 g

(y)dy + γ − 1 g (y)dy ⎦ . γ −1 y y x

x

(9.7) Twice integrating by parts the first term in (9.7) leads to

b  1−γ x y −1 g

(y)dy = y x

 

y=b b  x 1−γ

y −1 g (y) − (γ x 1−γ y γ −1 −1)g (y)dy = y y=x x y=b     x 1−γ

1−γ γ −1 −1 g (b − 0) − (γ x y − 1)g(y) + b b y=x b γ (γ − 1)x

1−γ

y γ −2 g(y)dy =

x

   x 1−γ −1 g (b − 0) − (γ x 1−γ bγ −1 − 1)g(b − 0)+ b b b 1−γ (γ − 1)g(x) + γ (γ − 1)x y γ −2 g(y)dy. x

Integrating in parts the second term in (9.7) we get  

y=b b  1−γ

b  x 1−γ γ −1 x

−1 g (y)dy = −1 g(y) − γ −1 y γ −2 g(y)dy = y y x y=x x

x

   b x 1−γ 1−γ − 1 g(b − 0) − (γ − 1)x y γ −2 g(y)dy. b x

Then (Bγ−1 ,b− Bγ g)(x) =

Fractional powers of Bessel operators

383

   x 1−γ b 1 (γ x 1−γ bγ −1 − 1)g(b − 0) + g(x)+ −1 g (b − 0) − γ −1 b γ −1 b 1−γ γx y γ −2 g(y)dy+ x

   b x 1−γ γ 1−γ − 1 g(b − 0) − γ x y γ −2 g(y)dy = γ −1 b x    x 1−γ b g(x) + −1 g (b − 0) − g(b − 0). γ −1 b From the last equality it is obvious that in order to have (Bγ−1 ,b− Bγ g)(x) = g(x) it is

necessary that g(b − 0) = lim g(y) = 0, g (b − 0) = lim g (y) = 0. Similarly, it shows lim

x→a+0

y→b−0 that (Bγ−1 ,a+ Bγ g)(x)

y→b−0

= g(x) is true when

g (x)=0.

lim g(x)=0 and

x→a+0

9.1.3 Fractional Bessel integrals and derivatives on a segment of elementary and special functions Statement 21. For μ > −1 the following formulas hold: 2 2 μ Bγ−α ,b− (b − x ) = 2α+μ  α  x2 x −2α b4α+2μ x2 2− 2 1− 2 × 22α (2α) b b   γ −1 x2 + μ + 1; 2α + μ + 1; 1 − 2 2 F1 α, α+ 2 b

and 2 2 μ Bγ−α ,a+ (x − a ) = 2α+μ    x 1−γ a2 x 2α+2μ (μ + 1) 1− 2 × 2α a 2 (2α + μ + 1) x   1−γ a2 γ −1 x2 F3 ,α + , μ + 1, α, 2α + μ + 1; 1 − 2 ; 1 − 2 . 2 2 a x 2 2 μ Proof. Let μ > −1. Find the fractional Bessel integral Bγ−α ,b− from (b − x ) : 2 2 μ Bγ−α ,b− (b − x ) =

384

Transmutations, Singular and Fractional Differential Equations

1 (2α)



b (b − y ) 2

2 μ

y 2 −x 2 2y

2α−1

 2 F1

 x2 γ −1 , α; 2α; 1− 2 dy. α+ 2 y

x 1

Replacing variable y by the formula 1 − xy 2 = t, we get y = x(1 − t)− 2 , dy = 12 x(1 − 2

3

t)− 2 dt, y = b, t = 1 −

x2 , b2

y = x, t = 0, and

2 2 μ Bγ−α ,b− (b − x ) = 2

x 2α b2μ

1− x 2



b

t

22α (2α)

2α−1

(1−t)

−α−1−μ

   μ  x2 γ −1 ; 2α; t dt. 1− 2 −t 2 F1 α, α+ 2 b

0

Using formula (2.21.1.21) from [457] of the form y x c−1 (1 − ωx)a−c−β (y − x)β−1 2 F1 (a, b; c; ωx)dx = 0

y c+β−1 (1 + ωy)α (1 − ωy)−c B(c, β) 2 F1 (a, b + β; c + β; ωy), y, Re c, Re β > 0, | arg(1 − ωy)| < π,

(9.8)

we obtain a = α, c = 2α, β = μ + 1, a − c − β = α − 2α − μ − 1 = −α − μ − 1 and 2 2 μ Bγ−α ,b− (b − x ) = 2α+μ  α  2 −2α  x 2α b2μ x2 x x2 × 2− 2 1− 2 22α (2α) b b b2   x2 γ −1 + μ + 1; 2α + μ + 1; 1 − 2 2 F1 α, α+ 2 b 2α+μ  α  −2α 4α+2μ 2 2 b x x x × 2− 2 1− 2 22α (2α) b b   x2 γ −1 + μ + 1; 2α + μ + 1; 1 − 2 . 2 F1 α, α+ 2 b 2 2 μ Now let us find the fractional Bessel integral Bγ−α ,a+ from (x − a ) , μ > −1: 2 2 μ Bγ−α ,a+ (x − a ) =   x  γ  2 2 2α−1 y2 y x −y γ −1 1 , α; 2α; 1− 2 (y 2 − a 2 )μ dy 2 F1 α+ (2α) x 2x 2 x a

Fractional powers of Bessel operators

385

Replacing the variable y by the formula 1 − 1

dy = − 12 x(1 − t)− 2 dt, y = a, t = 1 −

a2 , x2

y2 x2

1

= t, we obtain y = x(1 − t) 2 ,

y = x, t = 0, and

2 2 μ Bγ−α ,a+ (x − a ) = 2

x 2α+2μ 22α (2α)

1− a 2



x

t 2α−1 (1 − t)

γ −1 2

 1−

  μ  a2 γ −1 F , α; 2α; t dt. −t α+ 2 1 2 x2

0

Using formula (2.21.1.20) from [457] of the form y

x c−1 (1 − zx)−ρ (y − x)β−1 2 F1 (a, b; c; ωx)dx =

0

  y c+β−1 yz B(c, β) F3 ρ, a, β, b, c + β; ; ωy , (1 − yz)ρ yz − 1 y, Re c, Re β > 0, |arg(1 − ωy)|, |arg(1 − z)| < π, we get 2 2 μ Bγ−α ,a+ (x − a ) = 2α+μ    x 2α+2μ (μ + 1) x 1−γ a2 × 1 − 2α 2 a 2 (2α + μ + 1) x   a2 γ −1 x2 1−γ ,α + , μ + 1, α, 2α + μ + 1; 1 − 2 ; 1 − 2 . F3 2 2 a x

9.1.4 Fractional Bessel derivatives on a segment as inverse to integrals Theorem 113. If f (x) ∈ L1 (a, b). Then DBγα,b− is a left inverse operator to I Bγα,b− and DBγα,a+ is a left inverse operator to I Bγα,a+ : (DBγα,b− (I Bγα,b− f )(y))(x) = f (x), (DBγα,a+ (I Bγα,a+ f )(y))(x) = f (x). α Proof. Let us find first (I Bγn−α ,b− (I Bγ ,b− f )(y))(x). We obtain α (I Bγn−α ,b− (I Bγ ,b− f )(y))(x)= ⎛ ⎞   b  2 2 2α−1 2 1 ⎝ n−α y t −y γ −1 I Bγ ,b− , α; 2α; 1− 2 f (t)dt ⎠(x) = 2 F1 α+ (2α) 2t 2 t y

386

Transmutations, Singular and Fractional Differential Equations

1 × (2α)(2n − 2α)   b  2 2 2n−2α−1 x2 y −x γ −1 , n − α; 2n − 2α; 1− 2 dy× 2 F1 n − α+ 2y 2 y x

b 

t 2 −y 2 2t

2α−1

 2 F1

α+

 y2 γ −1 , α; 2α; 1− 2 f (t)dt = 2 t

y

b  t  2 2 2n−2α−1  2 2 2α−1 1 y −x t −y f (t)dt × (2α)(2n − 2α) 2y 2t x x     x2 y2 γ −1 γ −1 , n−α; 2n−2α; 1− 2 2 F1 α+ , α; 2α; 1− 2 dy = 2 F1 n−α+ 2 2 y t b t 1 1−2α t f (t)dt y 2α−2n+1 (y 2 −x 2 )2n−2α−1(t 2 −y 2 )2α−1× 22n−2 (2α)(2n−2α) x x     x2 y2 γ −1 γ −1 F F , n−α; 2n−2α; 1− , α; 2α; 1− n−α+ α+ dy. 2 1 2 1 2 2 y2 t2 (9.9) Let us denote the internal integral in (9.9) by I . Replacing variables by the formulas y 2 = η, x 2 = ξ, t 2 = τ in the internal integral in (9.9) we get 1 I= 2 

τ ηα−n (η−ξ )2n−2α−1 (τ −η)2α−1 × ξ

2 F1 n − α+

ξ γ −1 , n − α; 2n − 2α; 1− 2 η



  η γ −1 F , α; 2α; 1− dη. α+ 2 1 2 τ

Now, introducing a new variable substitution η = τ − w(τ − ξ ), we can write η = τ − w(τ − ξ ),

η = τ ⇒ w = 0, η = ξ ⇒ w = 1, dη = −(τ − ξ )dw,  α−n   ξ α−n α−n α−n = (τ − w(τ − ξ )) =τ , w 1− 1− η τ (η−ξ )2n−2α−1 = (τ − ξ )2n−2α−1 (1 − w)2n−2α−1 , (τ −η)2α−1 = (τ − ξ )2α−1 w 2α−1 ,     1 − τξ (1 − w) η ξ ξ   , 1− = 1 − 1− = w, η τ τ 1− 1− ξ w τ

Fractional powers of Bessel operators

387

and  α−n   1 1 α−n ξ 2n−1 I= τ (τ − ξ ) w 2α−1 (1 − w)2n−2α−1 1 − 1 − × w 2 τ 0     ξ γ −1 , α; 2α; 1 − w × 2 F1 α+ 2 τ   ⎛ ⎞ 1 − τξ (1 − w) γ − 1   ⎠ dw. , n − α; 2n − 2α; 2 F1 ⎝n − α+ ξ 2 1− 1− w τ

For the product of two Gauss hypergeometric functions (see [19]), we have



2 F1 (a, b; c; x) 2 F1 (a , b ; c ; y) =

∞  (a)m (a )k (b)m (b )k x m y k . (c)m (c )k m! k!

m,k=0

Therefore, ∞ 1  α−n−m−k τ (τ − ξ )2n−1+k+m × I= 2 m,k=0 γ −1 (α+ 2 )m (n − α+ γ −1 2 )k (α)m (n − α)k

1 1 × m! k!  α−n−k   1 ξ w 2α−1+m (1 − w)2n−2α+k−1 1 − 1 − dw = w τ (2α)m (2n − 2α)k

0 ∞ 1  α−n−m−k τ (τ − ξ )2n−1+k+m × 2

m,k=0 γ −1 (α+ γ −1 2 )m (n − α+ 2 )k (α)m (n − α)k

1 1 × m! k!

(2α)m (2n − 2α)k   ξ (2α + m)(2n + k − 2α) = 2 F1 n + k − α, 2α + m; 2n + k + m; 1 − (2n + k + m) τ   ∞ ξ 2n−1+k+m 1  n+α−1 τ × 1− 2 τ m,k=0 γ −1 (α+ γ −1 2 )m (n − α+ 2 )k (α)m (n − α)k

1 1 × (2α)m (2n − 2α)k m! k!   (2α + m)(2n + k − 2α) ξ . 2 F1 n + k − α, 2α + m; 2n + k + m; 1 − (2n + k + m) τ

388

Transmutations, Singular and Fractional Differential Equations

Consider the expression   ∞  (n − α+ γ −1 ξ 2n−1+k+m 2 )k (n − α)k (2n + k − 2α) 1 × 1− S= (2n − 2α)k (2n + k + m) k! τ k=0   ξ . 2 F1 n + k − α, 2α + m; 2n + k + m; 1 − τ Noting that (2n + k − 2α) (2n + k − 2α)(2n − 2α) = = (2n − 2α), (2n − 2α)k (2n − 2α + k) (2n + k + m) = (2n + m)k (2n + m), we obtain     ∞ γ −1 (2n − 2α) ξ 2n−1+m  (n − α)k (n − α+ 2 )k 1 ξ k × S= 1− 1− (2n + m) τ (2n + m)k k! τ k=0   ξ . 2 F1 n − α + k, 2α + m; 2n + m + k; 1 − τ Using formula (6.7.1.7) from [457] of the form  (a)k (b )k k=0

k!(c)k

x k 2 F1 (a + k, b; c + k; x) = 2 F1 (a, b + b ; c; x),

we obtain   (2n − 2α) ξ 2n−1+m × 1− (2n + m) τ   ξ γ −1 F ; 2n + m; 1 − . n − α, α + m + n+ 2 1 2 τ S=

Returning to variables x and t by the formulas ξ = x 2 , τ = t 2 , we can write ∞ γ −1 (2n − 2α) n+α−1  (2α + m)(α+ 2 )m (α)m 1 τ × 2 (2α)m (2n + m) m! m=0     ξ γ −1 ξ 2n−1+m ; 2n + m; 1 − = 1− 2 F1 n − α, α + m + n+ τ 2 τ ∞ γ −1 (2n − 2α) 2(n+α−1)  (2α + m)(α+ 2 )m (α)m 1 t × 2 (2α)m (2n + m) m! m=0 2n−1+m    2 x2 γ −1 x F ; 2n + m; 1 − n − α, α + m + n+ . 1− 2 2 1 2 t t2

I=

Fractional powers of Bessel operators

389

Taking into account the form of I , let us write α (I Bγn−α ,b− (I Bγ ,b− f )(y))(x)= ∞ γ −1 1 (2n − 2α)  (2α + m)(α+ 2 )m (α)m 1 × 2 (2α)m (2n + m) m! 22n−2 (2α)(2n − 2α) m=0



b

t 1−2α t 2(n+α−1) 1 −

2 2n−1+m

x t2

×

x

  x2 γ −1 F ; 2n + m; 1 − n − α, α + m + n+ f (t)dt = 2 1 2 t2  2n−1+m b ∞  (2α + m)(α+ γ −1 1 x2 2n−1 2 )m (α)m 1 t × 1 − (2α)m (2n + m) m! 22n−1 (2α) t2 m=0 x   x2 γ −1 ; 2n + m; 1 − 2 f (t)dt. 2 F1 n − α, α + m + n+ 2 t Since (2α + m) = (2α), (2α)m we have 2n−1+m   ∞  (α+ γ −1 x2 2 )m (α)m 1 × 1− 2 (2n + m) m! t m=0 x   x2 γ −1 2n−1 ; 2n + m; 1 − 2 f (t)dt. t 2 F1 n − α, α + m + n+ 2 t b

1 α (I Bγn−α ,b− (I Bγ ,b− f )(y))(x)= 2n−1 2

Now we show that α (Bγn )x (I Bγn−α ,b− (I Bγ ,b− f )(y))(x) = f (x).

Let  2n−1+m   x2 x2 γ −1 F Mn (x, t)= 1− 2 ; 2n + m; 1− n − α, α + m + n+ . 2 1 2 t t2 It is obvious that Mn (x, x) = 0,

n = 1, 2, ... .

Let us find (Bγ )x Mn applying formula (15.2.4) from [2] of the form d c−1 c−2 [z 2 F1 (a, b; c; z)] = (c − 1)z 2 F1 (a, b; c − 1; z). dz

390

Transmutations, Singular and Fractional Differential Equations

We obtain  2n−1+m 1 ∂ γ ∂ x2 x 1− 2 (Bγ )x Mn = γ x ∂x ∂x t   x2 γ −1 ; 2n + m; 1− 2 = 2 F1 n − α, α + m + n+ 2 t  2n+m−2 x2 2 1 ∂ γ +1 × x 1− 2 −(2n + m − 1) 2 γ t x ∂x t   x2 γ −1 ; 2n + m − 1; 1 − 2 = 2 F1 n − α, α + m + n+ 2 t γ +1    2n+m−2 x2 2t γ +1 1 ∂ x 2 2 × 1 − −(2n + m − 1) 2 x γ ∂x t 2 t t2   x2 γ −1 ; 2n + m − 1; 1 − 2 . 2 F1 n − α, α + m + n+ 2 t Using formula (15.2.9) from [2] of the form d c−1 [z (1 − z)a+b−c 2 F1 (a, b; c; z)] = dz (c − 1) zc−2 (1 − z)a+b−c−1 2 F1 (a − 1, b − 1; c − 1; z), we can find (Bγ )x Mn =  2n+m−3  2  γ −1 2 2t γ +1 1 2x x2 x (2n + m − 1)(2n + m − 2) 2 × 1− 2 γ 2 2 x t t t t   x2 γ −1 ; 2n + m − 2; 1 − 2 = 2 F1 n − 1 − α, α + m + n − 1+ 2 t   2  2n+m−3 x2 2 × 1− 2 (2n + m − 1)(2n + m − 2) t t   x2 γ −1 ; 2n + m − 2; 1 − 2 = 2 F1 n − 1 − α, α + m + n − 1+ 2 t  2 2 Mn−1 . (2n + m − 1)(2n + m − 2) t That gives (Bγ )x Mn = (2n + m − 1)(2n + m − 2)

 2 2 Mn−1 . t

(9.10)

Fractional powers of Bessel operators

391

Applying (9.10) (n − 1) times, we obtain (Bγ )n−1 x Mn = (2n + m − 1)(2n + m − 2)...(2 + m)

 2(n−1) 2 M1 = t

 2(n−1) 2 (2n + m − 1)(2n + m − 2)...(2 + m) × t  1+m  x2 γ +1 1− 2 ; 2 + m; 1 − 2 F1 1 − α, α + m+ 2 t   (2n + m) 2 2(n−1) × (m + 2) t  1+m  x2 γ +1 1− 2 ; 2 + m; 1 − 2 F1 1 − α, α + m+ 2 t

x2 t2

 =

 x2 . t2

So α (Bγn )x (I Bγn−α ,b− (I Bγ ,b− f )(y))(x)=

1 22n−1

∞  (α+ γ −1 2 )m (α)m 1 (2n + m) × (2n + m) m! (m + 2)

m=0

 1+m b  2(n−1) x2 2 2n−1 t × 1− 2 (Bγ )x t t x   x2 γ +1 F ; 2 + m; 1 − 1 − α, α + m+ f (t)dt = 2 1 2 t2 ∞ γ −1 1  (α+ 2 )m (α)m 1 × 2 (2 + m) m! m=0

1+m   b  x2 x2 γ +1 ; 2 + m; 1 − 2 f (t)dt. (Bγ )x t 1 − 2 2 F1 1 − α, α + m+ 2 t t x

Applying formula (15.3.3) from [2] of the form F (a, b; c; z) = (1 − z)c−a−b F (c − a, c − b; c; z), we obtain 

 x2 γ +1 ; 2 + m; 1 − 2 = 2 F1 1 − α, α + m+ 2 t 1−γ    2 2 x2 1−γ x F ; 2 + m; 1 − α + m + 1, 1 − α + . 2 1 2 t2 t2

392

Transmutations, Singular and Fractional Differential Equations

Now  m+1   ∂ x2 x2 γ +1 1− 2 ; 2 + m; 1 − 2 = 2 F1 1 − α, α + m+ ∂x 2 t t  2  1−γ     m+1 2 ∂ x x2 x2 1−γ F ; 2 + m; 1 − 1− α + m + 1, 1− α + . 2 1 ∂x t 2 2 t2 t2 Using formula (15.2.9) from [2] of the form d c−1 [z (1 − z)a+b−c 2 F1 (a, b; c; z)] = dz (c − 1) zc−2 (1 − z)a+b−c−1 2 F1 (a − 1, b − 1; c − 1; z), we obtain  m+1   x2 x2 ∂ γ +1 1− 2 ; 2 + m; 1 − 2 = 2 F1 1 − α, α + m+ ∂x 2 t t  2  1−γ   m+1 2 x2 ∂ x × 1− 2 2 ∂x t t   x2 1−γ ; 2 + m; 1 − 2 = 2 F1 α + m + 1, 1 − α + 2 t 1−γ   2  2 −1  m  x2 x 2x × 1 − (m + 1) − 2 t t2 t2   x2 1−γ F ; m + 1; 1 − α + m, −α + . 2 1 2 t2 Returning to the series, we write α (Bγn )x (I Bγn−α ,b− (I Bγ ,b− f )(y))(x)= ∞ γ −1 1  (α+ 2 )m (α)m 1 × 2 (2 + m) m! m=0

1+m   b  x2 x2 γ +1 ; 2 + m; 1 − 2 f (t)dt = (Bγ )x t 1 − 2 2 F1 1 − α, α + m+ 2 t t x

1+m b  x2 t 1− 2 × t m=0 x   x2 γ +1 F ; 2 + m; 1 − 1 − α, α + m+ f (t)dt = 2 1 2 t2 1 2

∞ 

(α+ γ −1 2 )m (α)m 1 1 ∂ γ ∂ x (2 + m) m! x γ ∂x ∂x

Fractional powers of Bessel operators

393

∞ γ −1 1  (α+ 2 )m (α)m 1 × 2 (m + 1)! m! m=0

   2  1−γ  m b 2 −1 2x x x2 1 ∂ γ t (m + 1) − 2 × x 1− 2 x γ ∂x t t2 t x   x2 1−γ ; m + 1; 1 − 2 f (t)dt = 2 F1 α + m, −α + 2 t ∞ γ −1  (α+ 2 )m (α)m 1 × − m! m! m=0

1 ∂ x γ ∂x

b

 m   x2 x2 1−γ tγ 1 − 2 F ; m + 1; 1 − α + m, −α + f (t)dt. 2 1 2 t t2

x

Applying formula (6.7.1.7) from [457] of the form  (a)k (b )k k=0

k!(c)k

x k 2 F1 (a + k, b; c + k; x) = 2 F1 (a, b + b ; c; x),

we sum up the row:  m   ∞  (α+ γ −1 x2 x2 1−γ 2 )m (α)m 1 F 1− 2 ; m + 1; 1− α+ m, −α+ = 2 1 m! m! 2 t t2 m=0   x2 = 1. 2 F1 α, 0; 1; 1 − 2 t So α (Bγn )x (I Bγn−α ,b− (I Bγ ,b− f )(y))(x)= −

1 ∂ x γ ∂x

b t γ f (t)dt = x

1 γ x f (x) = f (x). xγ It is similarly proved that (DBγα,a+ (I Bγα,a+ f )(y))(x) = f (x).

9.2

Fractional Bessel integral and derivatives on a semiaxis

The fractional powers (Bγ )α , α ∈ R, in the case of a semiaxis were not studied in [555] as they require more delicate considerations and estimates when applied. But

394

Transmutations, Singular and Fractional Differential Equations

they seem to be very important as in most applications boundary conditions for differential equations are given exactly at zero or infinity. So we introduce fractional Bessel integrals and derivatives for these special values.

9.2.1 Definitions Definition 47. Let α > 0. The right-sided fractional Bessel integral on a semiaxis [2α]+1 (0, +∞) is defined by the formula Bγ−α ,− for f (x)∈C α (Bγ−α ,− f )(x) = (I Bγ ,− f )(x) =   ∞ 2 2 2α−1 x2 γ −1 y −x 1 , α; 2α; 1− 2 f (y)dy. 2 F1 α+ (2α) 2y 2 y

(9.11)

x [2α]+1[0,+∞) The left-sided fractional Bessel integral on a semiaxis Bγ−α ,0+ for f (x)∈C is defined by the formula α (Bγ−α ,0+ f )(x) = (I Bγ ,0+ f )(x) =

1 (2α)

  x  γ  2 2 2α−1 y2 y γ −1 x −y , α; 2α; 1− 2 f (y)dy. 2 F1 α+ x 2x 2 x 0

(9.12) Using (9.6), we can write (Bγ−α ,− f )(x) =

√

π 2α−1 2 (α)

∞  y  γ 1 −α  1  x y  1 2 (y 2 −x 2 )α− 2 P γ2 −1 + f (y)dy x 2 y x 2 x

and (Bγ−α ,0+ f )(x) =

√

x   γ 1 −α  1  x y  π 2 2 α− 12 y 2 (x −y ) P γ2 −1 + f (y)dy. x 2 y x 22α−1 (α) 2 0

The expression of the fractional Bessel integrals through the Legendre functions is useful and is a simplification of the original definition, since the Gauss hypergeometric function depends on three parameters, and the Legendre function depends only on two parameters. n−α 2n Definition 48. Let α > 0, n = [α] + 1, f ∈L1 (a, b), I Bγn−α ,− f, I Bγ ,0+ f ∈Cev (0, ∞). The right-sided and left-sided fractional Bessel derivatives on a semiaxis of the Riemann–Liouville type for α = 0, 1, 1, ... are defined, respectively, by the equalities

(Bγα,− f )(x) = (DBγα,− f )(x) = Bγn (I Bγn−α ,− f )(x),

n = [α] + 1,

(9.13)

Fractional powers of Bessel operators

395

and (Bγα,0+ f )(x) = (DBγα,0+ f )(x) = Bγn (I Bγn−α ,0+ f )(x),

n = [α] + 1.

(9.14)

When α = n ∈ N ∪ {0}, then (Bγ0,− f )(x) = (Bγ0,0+ f )(x) = f (x), (Bγn,− f )(x) = (Bγn,0+ f )(x) = Bγn f (x), where Bγn is an iterated Bessel operator (9.1). In [367] spaces adapted to work with operators of the form Bγα,0+ and Bγα,− , α ∈ R, were introduced:   k ∞ kd ϕ p Fp = ϕ ∈C (0, ∞) : x ∈ L (0, ∞) for k = 0, 1, 2, ... , 1 ≤ p < ∞, dx k   dkϕ F∞ = ϕ ∈C ∞ (0, ∞) : x k k →0 as x →0+ and as x →∞ for k = 0, 1, 2, ... , dx and   Fp,μ = ϕ : x −μ ϕ(x) ∈ Fp ,

1 ≤ p ≤ ∞,

μ ∈ C.

We present here two theorems that are special cases of theorems from [367]. Theorem 114. Let α ∈ R. For all p, μ, and γ > 0 such that μ= p1 −2m, γ = p1 −μ−2m+1, m=1, 2..., the operator Bγα,0+ is a continuous linear mapping from Fp , μ into Fp,μ−2α . If also 2α = μ − p1 + 2m and γ − 2α = p1 − μ − 2m + 1, m = 1, 2..., then Bγα,0+ is a homeomorphism from Fp , μ onto Fp,μ−2α with inverse Bγ−α ,0+ . Theorem 115. Let α ∈ R. For all p, μ, and γ > 0 such that μ= p1 −2m + 1, γ = p1 −μ−2m, m=1, 2..., the operator Bγα,− is a continuous linear mapping from Fq,−μ+2α into Fq,μ , where

1 q

= 1 − p1 . If also 2α = μ −

μ − + 2m, m = 1, 2..., then with inverse Bγ−α ,− . 1 p

Bγα,−

1 p

+ 2m − 1 and γ + 2α =

is a homeomorphism from Fq,−μ+2α onto Fq,−μ

n−α 2n Definition 49. Let n = [α] + 1, f ∈L[0, ∞), I Bγn−α ,− f, I Bγ ,0+ f ∈Cev (0, ∞). The right-sided fractional Bessel derivatives on a semiaxis of Gerasimov–Caputo type for α > 0, α = 0, 1, 2, ... is defined by the equality n (Bγα,− f )(x) = (I Bγn−α ,− Bγ f )(x).

The left-sided fractional Bessel derivatives on a semiaxis of Gerasimov–Caputo type for α > 0, α = 0, 1, 2, ... is defined by the equality n (Bγα,0+ f )(x) = (I Bγn−α ,0+ Bγ f )(x).

396

Transmutations, Singular and Fractional Differential Equations

Here I Bγn−α ,− is the right-sided fractional Bessel integral (9.11) on a semiaxis, and is the left-sided fractional Bessel integral (9.12) on a semiaxis. When α = n ∈ I Bγn−α ,0+ N ∪ {0}, then (Bγ0 ,− f )(x) = (Bγ0 ,0+ f )(x) = f (x), (Bγn,− f )(x) = (Bγn,0+ f )(x) = Bγn f (x), where Bγn is an iterated Bessel operator (9.1).

9.2.2 Basic properties of fractional Bessel integrals on a semiaxis Lemma 32. For γ = 0, f ∈ L1 (0, ∞) the following formulas are valid: −α (B0,− f )(x) =

1 (2α)

∞ (y − x)2α−1 f (y)dy = (I−2α f )(x) x

and −α f )(x) = (B0,0+

1 (2α)

x 2α (x − y)2α−1 f (y)dy = (I0+ f )(x), 0

2α are right-sided (2.25) and left-sided (2.26) fractional Bessel intewhere I−2α and I0+ grals on a semiaxis, respectively.

Proof. Indeed, we have −α f )(x) = (B0,− 2α−1   ∞ 2 1 x2 1 y − x2 2 F1 α − , α; 2α; 1 − 2 f (y)dy (2α) 2y 2 y x

and −α f )(x) = (B0,0+   x  2 2 2α−1 1 y2 x −y 1 F , α; 2α; 1− α− f (y)dy. 2 1 (2α) 2x 2 x2 0

Using the formula that is obtained from the integral representation of the Gauss hypergeometric function (1.34)     x2 1 2y 2α−1 , 2 F1 α − , α; 2α; 1 − 2 = 2 x +y y we obtain provable formulas.

Fractional powers of Bessel operators

397

−α Let us now consider the connection of fractional Bessel integrals Bγ−α ,− and Bγ ,0+ with corresponding fractional Saigo integrals (2.41) and (2.42).

Lemma 33. Let f ∈L1 (0, ∞). The following equalities hold:  γ −1 √  1 2α, γ −1 2 −α,−α J x 2 f ( x) , 22α x 2 1−γ  γ −1 √  x 2 −α 2α, γ −1 −α,−α f )(x) = Ix 2 2 x 2 f ( x) , (Bγ−α ,0+ 2α 2

(Bγ−α ,− f )(x) =

γ ,β,η

where Jx γ ,β,η is the fractional Saigo integral (2.41) and Ix integral (2.42). Here γ > 0, β, θ are real numbers.

(9.15) (9.16) is the fractional Saigo

Proof. Replacing the variable y 2 = t in the Bessel fractional integral on the semiaxis (9.11), we get (Bγ−α ,− f )(x) =   ∞ √ x2 1 γ −1 2 2α−1 −α (t − x ) t F , α; 2α; 1 − f ( t)dt. α + 2 1 2α 2 t 2 (2α) x2

Comparing the resulting expression with (2.41), we obtain γ = 2α,

β=

γ −1 − α, 2

−γ − β = −α −

γ −1 , 2

η = −α,

Which gives (9.15). Similarly, we obtain (9.16). We consider now the case when α = 1. Lemma 34. The following equalities hold: (Bγ−1 ,− f )(x) =

1 γ −1

(Bγ−1 ,0+ f )(x) =

∞  1−γ x y − 1 f (y)dy, y

f (x)∈L1 (0, ∞),

x

1 γ −1

x   y γ −1  y 1− f (y)dy, x

f (x)∈L1 (0, ∞).

0

Proof. Applying the formula  2 F1

x2 γ +1 , 1; 2; 1 − 2 2 y



y2 2 = 1 − γ x2 − y2

  x 1−γ −1 , y

which is valid for the Gauss hypergeometric function, we obtain provable statements.

398

Transmutations, Singular and Fractional Differential Equations

2 (0, ∞), f (x) = B g(x), f (x)∈L (0, ∞). When Lemma 35. Let g ∈ Cev γ 1

lim g(x) = 0,

x→+∞

lim g (x) = 0,

x→+∞

we have (Bγ−1 ,− Bγ g)(x) = g(x). When lim g (x)=0,

lim g(x)=0,

x→0+

x→0+

we have (Bγ−1 ,0+ Bγ g)(x) = g(x). Proof. Let f (x) = Bγ g(x) = g

(x) + γx g (x). Then −1 (Bγ−1 ,− f )(x) = (Bγ ,− Bγ g)(x) =

  ∞  1−γ x γ 1 y − 1 g

(y) + g (y) dy = γ −1 y y x ⎞ ⎛∞



 1−γ  ∞  1−γ x x 1 ⎝ y −1 g

(y)dy + γ −1 g (y)dy ⎠ . γ −1 y y x

x

(9.17) Twice integrating by parts the first term in (9.17), we obtain

∞  1−γ x y −1 g

(y)dy= y x

  y=∞ ∞  x 1−γ −1 g (y) − (γ x 1−γ y γ −1 −1)g (y)dy = y y y=x x

   y=∞ 1−γ y=∞   x

1−γ γ −1  −1 g (y) − (γ x y − 1)g(y) + y y y=x y=x ∞ γ (γ − 1)x

1−γ

y γ −2 g(y)dy.

x

Integrating by parts the second term in (9.17), we obtain

∞  1−γ x − 1 g (y)dy = y x

Fractional powers of Bessel operators

399

  y=∞ ∞  γ −1 x 1−γ  − 1 g(y) − γ −1 y γ −2 g(y)dy. y x y=x x

Then, obviously, when lim g(x)= 0, lim g (x) = 0 leads to (Bγ−1 ,− Bγ g)(x)=g(x). x→+∞

x→+∞

Similarly, when lim g(x)=0 and lim g (x)=0, we have (Bγ−1 ,0+ Bγ g)(x) = g(x). x→0+

x→0+

Lemma 36. Let f (x)∈L1 (0, ∞). Fractional Bessel integrals on the semiaxis are related by the equality ∞

γ f (x)(Bγ−α ,0+ g)(x)x dx

∞ =

0

γ g(x)(Bγ−α ,− f )(x)x dx.

(9.18)

0

Proof. Let us consider (Bγ−α ,0+ f )(x) using its kernel representation as a Legendre function: ∞

γ f (x)(Bγ−α ,0+ g)(x)x dx =

0

  ∞  α + 12  (2α) 0   1 ∞  α+ 2 (2α) 0   1 ∞  α+ 2 (2α) 0   1 ∞  α+ 2 (2α)

x f (x)x dx

y γ

2

x

1

−α

P γ2 −1 2

   1 x y + g(y)dy = 2 y x

0

∞  y  γ 1 −α  1  x y  1 2 g(y)dy (x 2 −y 2 )α− 2 P γ2 −1 + f (x)x γ dx = x 2 y x 2 y

  γ 1   ∞ 2 1 x y 2 2 α− 12 x 2 −α g(y)y dy (x −y ) P γ −1 + f (x)dx = y 2 y x 2 γ

y

∞   γ 1 −α  1  x y  2 2 α− 12 y 2 g(x)x dx (y −x ) P γ2 −1 + f (y)dy = x 2 y x 2

0

∞

1

(x 2 −y 2 )α− 2

γ

γ

x

γ g(x)(Bγ−α ,− f )(x)x dx.

0

This proves the lemma.

9.2.3 Factorization Following [555] and [367] we present the following results.

400

Transmutations, Singular and Fractional Differential Equations η,α

Let Re (2η + μ) + 2 > 1/p and f ∈ Fp,μ . For Re α > 0, we define I2 f by the formula 2 x −2η−2α (α)

η,α I2 f (x) =

x (x 2 − u2 )α−1 u2η+1 f (u)du.

(9.19)

0 η,α

Let Re (2η − μ) > −1/p and f ∈ Fp,μ . For Re α > 0, we define K2 f by the formula η,α K2 f (x) =

2 x 2η (α)

∞ (u2 − x 2 )α−1 u1−2(η+α) f (u)du.

(9.20)

x

The definitions are extended to Re α ≤ 0 by means of the formulas η,α

1 η,α+1 df f + I2 x 2 dx

(9.21)

1 η,α+1 df f − K2 x . 2 dx

(9.22)

η,α+1

I2 f = (η + α + 1)I2 and η,α

η,α+1

K2 f = (η + α)K2

Theorem 116. The following factorizations of (9.11) and (9.12) are valid: 1−γ

−2α K2 2 (Bγ−α ,− f )(x) = 2

,α

K20,α x 2α f (x)

(9.23)

and (Bγ−α ,0+ f )(x) =

 x 2α 2

γ −1

I2 2

,α 0,α I2 f (x),

where K20,α f (x) =

2 (α)

∞ (u2 − x 2 )α−1 u1−2α f (u)du, x

1−γ 2

K2

,α

2 f (x) = x 1−γ (α)

∞ (u2 − x 2 )α−1 uγ −2α f (u)du, x

I20,α f (x) =

2 x −2α (α)

x (x 2 − u2 )α−1 uf (u)du, 0

γ −1 2 ,α

I2

2 f (x) = x 1−γ −2α (α)

x (x 2 − u2 )α−1 uγ f (u)du. 0

(9.24)

Fractional powers of Bessel operators

401

Proof. We have 1−γ

,α

−2α K2 2 K20,α x 2α f = Bγ−α ,− f = 2 ∞ 21−2α 1−γ 2 ,α (u2 − y 2 )α−1 uf (u)du = K (α) 2

22−2α 1−γ x  2 (α)

y ∞ 

2 α−1 γ −2α

(y − x ) 2

y

∞ dy (u2 − y 2 )α−1 uf (u)du =

x

22−2α 1−γ x  2 (α)

y

∞

u uf (u)du

x

(u2 − y 2 )α−1 (y 2 − x 2 )α−1 y γ −2α dy.

x

For the inner integral we have u

(y 2 − x 2 )α−1 (u2 − y 2 )α−1 y γ −2α dy =

x

√   2α−1 x2 1 21−2α π(α)  2 γ −1 2 γ −1 −2α   u −x x u ; 2α; 1 − 2 2 F1 α, α + 2  α+ 1 2 u 2

and Bγ−α ,− f

√ 21−2α 21−2α π(α) 1−γ   x = 2 ×  (α)  α + 1 2

∞

u −x 2

2

2α−1

x

γ −1 −2α

u

  x2 γ −1 ; 2α; 1 − 2 uf (u)du = 2 F1 α, α + 2 u

x

21−2α (2α)

∞

u −x 2

2

2α−1

 1−2α

u

2 F1

 x2 γ −1 ; 2α; 1 − 2 f (u)du = α, α + 2 u

x

1 (2α)

∞

u2 − x 2 2u

2α−1

  x2 γ −1 , α; 2α; 1 − 2 f (u)du. 2 F1 α + 2 u

x

This coincides with formula (9.23). Now we proof (9.24). We have (Bγ−α ,0+ f )(x) = 1 (2α)

2α−1   x  γ  2 u2 u x − u2 γ −1 , α; 2α; 1 − 2 f (u)du = 2 F1 α + x 2x 2 x 0

402

Transmutations, Singular and Fractional Differential Equations γ −1

2−2α x 2α I2 2

,α 0,α I2 f

21−2α x 2α γ −1 ,α I 2 y −2α (α) 2

= y (y 2 − u2 )α−1 uf (u)du = 0

22−2α x 2α −γ +1−2α x  2 (α)

x

2 α−1 γ −2α

(x − y ) 2

y

y

0

22−2α 1−γ x  2 (α)

(y 2 − u2 )α−1 uf (u)du =

dy 0

x

x uf (u)du

(y 2 − u2 )α−1 (x 2 − y 2 )α−1 y γ −2α dy.

u

0

Let us find x

(y 2 − u2 )α−1 (x 2 − y 2 )α−1 y γ −2α dy = {y 2 = t} =

u

1 2

x 2 γ −1 (t − u2 )α−1 (x 2 − t)α−1 t 2 −α dt = u2

√

  2α−1 1−γ x2 π(α)  2  x − u2  , α; 2α; 1 − 2 . u−2α+γ −1 2 F1 α + 2 u 22α  α + 12 Using the formula  −a F (a, b; c; z) = (1 − z) F 2 1 a, c − b; c; 2 1 we obtain

 z , z−1



   1−γ 1−γ x2 x2 , α; 2α; 1 − 2 = 2 F1 α, α + ; 2α; 1 − 2 = 2 F1 α + 2 2 u u  2 −α   2 x γ −1 u ; 2α; 1 − 2 = 2 F1 α, α + 2 u2 x  2 −α   x γ −1 u2 α + , α; 2α; 1 − F 1 2 2 u2 x2

and x (y − u ) 2

2 α−1

2 α−1 γ −2α

(x − y ) 2

u



x 2 − u2

2α−1

u−2α+γ −1

y



x2 u2

−α

√

π(α) ×  22α  α + 12   γ −1 u2 α, α + = ; 2α; 1 − F 2 1 2 x2

dy =

Fractional powers of Bessel operators

403

√

  2α−1 π(α)  2 u2 γ −1   x − u2 uγ −1 x −2α 2 F1 α, α + ; 2α; 1 − 2 . 2 x 22α  α + 12 Finally, (Bγ−α ,0+ f )(x) =

√ 22(1−2α) π   x 1−γ −2α × 1 (α) α + 2

  x  2α−1 u2 γ −1 2 2 γ u 2 F1 α + , α; 2α; 1 − 2 f (u)du. x −u 2 x 0

Applying the duplication formula (1.7), we obtain 21−2α 1−γ −2α × x (2α)   x  2α−1 u2 γ −1 2 2 γ u 2 F1 α + , α; 2α; 1 − 2 f (u)du = x −u 2 x (Bγ−α ,0+ f )(x) =

0

1 (2α)

x 

x 2 − u2 2x

2α−1     u2 u γ γ −1 , α; 2α; 1 − 2 f (u)du, 2 F1 α + x 2 x

0

which gives (9.24). The proof is complete.

9.2.4 Fractional Bessel integrals on semiaxes of elementary and special functions −α Statement 22. Let f (x) = x m , x > 0, m ∈ R. Then the integrals Bγ−α ,− and Bγ ,0+ of the power function are defined by the formulas

m −α − m2 , − γ −1 − α − −α m 2α+m −2α 2 2 2  , m + 2α + γ < 1, Bγ ,− x = x 1−γ −m , − m2 2

m+γ +1 m , + 1 m 2α+m −2α 2 2 Bγ−α 2  . ,0+ x = x α + m2 + 1, α + m+γ2 +1

Proof. We have 1 (2α)

∞

2α−1

  x2 γ −1 F , α; 2α; 1 − α + y m dy = 2 1 2 y2 x   2 1 −3 x − 12 2 dt, y = x, t = 1, y = +∞, t = 0 = = t, y = xt , dy = − xt 2 y2 m Bγ−α ,− x =

y2 − x2 2y

404

Transmutations, Singular and Fractional Differential Equations

1 1 2 (2α)

1 

x 2 t −1 −x 2

2α−1

1

2xt − 2

0

x 2α+m 22α (2α)

1 t

−α− m 2 −1

  1 3 γ −1 , α; 2α; 1− t (xt − 2 )m xt − 2 dt = 2 F1 α+ 2 

(1 − t)

2α−1

2 F1

 γ −1 , α; 2α; 1 − t dt. α+ 2

0

Using the following formula (2.21.1.11) from [457], p. 265, 

z x

μ−1

(z − x)

c−1

0

z

c+μ−1

 

2 F1

 x dx = a, b; c; 1 − z

c, μ, c − a − b + μ c − a + μ, c−b+μ

 ,

z > 0, Re c > 0, Re (c − a − b + μ) > 0, we have γ −1 m ,a = α + , b = α, c = 2α ⇒ 2 2 m γ −1 −α− >0 c−a−b+μ=− 2 2 z = 1, μ = −α −

and x 2α+m 2α, −α − m2 , − γ −1 2 −α−  1−γ −m = 2α m 2 (2α) , −2 2

m −α − m2 , − γ −1 2 −α− 2 x 2α+m 2−2α  . 1−γ −m m , −2 2

m Bγ−α ,− x

m 2

=

m Now let us consider Bγ−α ,0+ x : m Bγ−α ,0+ x =

1 (2α)

2α−1   x  γ  2 y2 y x − y2 γ −1 F , α; 2α; 1 − α + y m dy = 2 1 x 2x 2 x2 0

1 1 y2 1 { 2 = t, y = xt 2 , dy = xt − 2 dt, y = 0, t = 0, y = x, t = 1} = 2 x     1 γ −1 m x x(1 − t) 2α−1 γ −1 2 t , α; 2α; 1 − t x m t 2 dt = 2 F1 α + 2(2α) 2 2

0

Fractional powers of Bessel operators

x 2α+m 22α (2α)

1 t

m+γ +1 −1 2

405

(1 − t)

2α−1

  γ −1 , α; 2α; 1 − t dt. 2 F1 α + 2

0

Using the following formula (2.21.1.11) from [457], p. 265, 

z x

μ−1

(z − x)

0

= zc+μ−1 

c−1



2 F1

 x dx = a, b; c; 1 − z

c, μ, c − a − b + μ c − a + μ, c−b+μ

 ,

z > 0, Re c > 0, Re (c − a − b + μ) > 0, we have γ −1 m+γ +1 ,a = α + , b = α, 2 2 m c = 2α ⇒ c − a − b + μ = + 1 > 0, 2

m+γ +1 2α+m m x 2α, , + 1 −α m 2 2  Bγ ,0+ x = 2α = 2 (2α) α + m+γ2 +1 α + m2 + 1,

m+γ +1 m , + 1 2 2 x 2α+m 2−2α  . α + m2 + 1, α + m+γ2 +1 z = 1, μ =

Corollary 20. The operator x12α Bγ−α ,− is of the so-called Dzhrbashyan–Gelfond– Leontiev type (see [252,494]) when m + 2α + γ < 1. This means that it acts on power series by the rule ∞  ∞   1 −α  k k B ak z = c(α, k)ak z , z2α γ ,− k=0 k=0 −α − k2 , − γ −1 −2α 2 −α−  c(α, k) = 2 1−γ −k , − k2 2 Corollary 21. The operator type:

1 B −α x 2α γ ,0+

k 2

,

ak ∈ R.

is also of Dzhrbashyan–Gelfond–Leontiev

∞  ∞   1 −α  B ak z k = d(α, k)ak zk , z2α γ ,0+ k=0 k=0

m+γ +1 m , + 1 −2α 2 2  d(α, k) = 2 , α + m2 + 1, α + m+γ2 +1

ak ∈ R.

406

Transmutations, Singular and Fractional Differential Equations

Statement 23. Let f (x) = k γ −1 (xξ ))(x), defined by (1.21), x > 0, γ > 0. Then the 2

integral (Bγ−α ,− f )(x) is defined by the formula −2α (Bγ−α k γ −1 (xξ ). ,− k γ −1 (xξ ))(x) = ξ 2

2

Proof. Using the factorization (9.23), we get

1 (2α)

(Bγ−α ,− f )(x) =

  ∞ 2 2 2α−1 x2 y −x γ −1 F , α; 2α;1− α+ f (y)dy = 2 1 2y 2 y2 x

1−γ 2

2−2α K2 22(2−α)  2 (α)

,α

K20,α x 2α f = ∞ ∞ 1−γ 2 2 α−1 γ −2α x (u − x ) u du (t 2 − u2 )α−1 tf (t)dt. x

u

Applying formula (2.16.3.7) from [456] of the form ∞

x 1±ρ (x 2 − a 2 )β−1 Kρ (cx)dx = 2β−1 a β±ρ c−β (β)Kρ±β (ac),

a

where a, c, β > 0 and Kρ is the modified Bessel function of the second kind (1.17), we obtain (Bγ−α ,− k γ −1 (xξ ))(x) = 2

22(1−α)  2 (α)

x

1−γ

∞ ∞ 2 2 α−1 γ −2α (u − x ) u du (t 2 − u2 )α−1 t k γ −1 (tξ )dt = 2

x

22(1−α) 1−γ 2 x (α)

2

2(1−α) 1−γ

2

ξ

γ −1 2



ξ γ −1 2

γ −1 2

 

x

−2α

2

γ −1 2

 

(xξ )

γ +1 2

γ −1 2



γ +1 2



ξ 

u

2

  ∞ 1+ γ −1 2 −α (u2 −x 2 )α−1 u K γ −1 −α (uξ )du=

α−1 −α

ξ

2

γ +1 2

γ −1 2



x

22(α−1) ξ −2α x

γ −1 2

K γ −1 (xξ ) = 2

K γ −1 (xξ ) = ξ −2α k γ −1 (xξ ). 2

2

Fractional powers of Bessel operators

407

9.3 Integral transforms of fractional powers of Bessel operators An integral transform maps the original space into or onto the image space. Usually difficult operations in the original space are converted into simple operations in the image space. For example, the Fourier transform converts a derivative of order n into multiplication by the power n of the variable with some constant. This is the reason why the Fourier transform is beneficial to use for solution to differential equations. Since the Hankel transform applied to a Bessel operator of order n gives multiplication of a Hankel image of a function by the power 2n of the variable with some constant, this transform is used instead of the Fourier transform when a differential equation with Bessel operator is solved. The action of the Hankel transform on fractional Bessel derivatives of order α on semiaxes has the next property: it does not involve multiplication by some power in the dual variable under the Hankel transform (see Theorem 119). In this section we collect some integral transforms which can be used to solve differential equations with fractional Bessel derivatives on semiaxes.

9.3.1 The Mellin transform Using formula (2.21.1.11) from [457], p. 265, of the form   x x α−1 (z − x)c−1 2 F1 a, b; c; 1 − dx = z 0   c, α, c − a − b + α c+α−1  , z c − a + α, c−b+α z

(9.25)

z > 0, Re c > 0, Re (c − a − b + α) > 0, we prove the following theorems. Theorem 117. Let α > 0. Mellin transforms of I Bγα,− and I Bγα,0+ are MI Bγα,− f (s) =

1  22α

α+

s 2

s 2,

−

γ −1 2 ,

s 2

− γ −1 2 α + 2s

f ∗ (2α + s),

(9.26)

f ∗ (2α + s),

(9.27)

where s > γ − 1, I Bγα,− f ∈ Pab , 1 MI Bγα,0+ f (s) = 2α  2



γ −s+1 2

where I Bγα,0+ f ∈ Pab , 2α + s < 2.

− α, 1 − 2s − α γ −s+1 1 − 2s , 2



408

Transmutations, Singular and Fractional Differential Equations

Proof. Let us start from the definitions ((I Bγα,− f )(x))∗ (s) =

∞ x s−1 (I Bγα,− f )(x)dx = 0

1 (2α)

+∞

∞

y2 − x2 2y

x s−1 dx

2α−1

  x2 γ −1 F , α; 2α; 1 − α + f (y)dy 2 1 2 y2

x

0

  ∞ y 1 x 2 s−1 γ −1 1−2α 2 2 2α−1 f (y)(2y) dy (y −x ) , α; 2α; 1− 2 x dx. 2 F1 α+ (2α) 2 y 0

0

Using (9.25), let us find the inner integral for s > γ − 1:   x2 γ −1 (y − x ) , α; 2α; 1 − 2 x s−1 dx = 2 F1 α + 2 y 0

γ −1 s s y 4α+s−2 2α, 2, 2 − 2  . γ −1 s s 2 α+ 2 − 2 , α+ 2

y

2

2 2α−1

We obtain ((I Bγα,− f )(x))∗ (s) = 1  22α

α+

s 2

s 2,

−

1  22α

γ −1 2 ,

s 2

α+

− γ −1 2 α + 2s

s 2,

−

s 2

γ −1 2 ,

s 2

− γ −1 2 α + 2s

f ∗ (2α + s).

Similarly, we have ((I Bγα,0+ f )(x))∗ (s) =

∞

x s−1 (Bγ−α ,0+ f )(x)dx =

0

1 (2α)

∞



0



0

x  γ  2 2 2α−1 y x −y s−1 x dx × x 2x 0

 y2 γ −1 , α; 2α; 1− 2 f (y)dy = 2 F1 α+ 2 x ∞ ∞    γ  2 2 2α−1 1 1 x −y f (y)y γ dy × (2α) x 2x 2 F1 α+

y

 y2 γ −1 , α; 2α; 1− 2 x s−1 dx. 2 x

∞ f (y)y 2α+s−1 dy = 0

Fractional powers of Bessel operators

409

Let us find the inner integral:   ∞ γ  2 2 2α−1 y2 1 x −y γ −1 , α; 2α; 1− 2 x s−1 dx = 2 F1 α+ x 2x 2 x y

  ∞ 2α−s+γ  2α−1 y2 1 γ −1 2 2 , α; 2α; 1− 2 dx = x −y 2 2 F1 α+ x 2 x y   1 =t = x   1/y  2α−1 γ −1 1−2α γ −2α−s 2 2 2 2 2 t , α; 2α; 1−t y dt = {ty = z} = 1−t y 2 F1 α+ 2 1−2α

0

21−2α y 2α+s−γ −1

1

   2α−1 γ −1 2 zγ −2α−s 1−z2 F , α; 2α; 1−z α+ dz = 2 1 2

0

{z = s} = 2

1 2α+s−γ −1 y 22α

1 s

γ −s−1 −α 2

  γ −1 , α; 2α; 1−s ds. (1−s)2α−1 2 F1 α+ 2

0

Using (9.25), for 2α + s < 2 we get   γ −1 , α; 2α; 1−s ds = (1−s)2α−1 2 F1 α+ 2 0

γ −s+1 s 1 2α+s−γ −1 2α, − α, 1 − − α 2 2 y  γ −s+1 22α 1 − 2s , 2

1

s

γ −s−1 −α 2

and ∗ ((Bγ−α ,0+ f )(x)) (s) =

1  22α



γ −s+1 2

− α, 1 − 2s ,

1  22α



γ −s+1 2

1 − 2s − α γ −s+1 2

− α, 1 − 2s ,

1−

s 2 −α γ −s+1 2

∞ f (y)y 2α+s−1 dy = 0

f ∗ (2α + s).

This completes the proof. In order to obtain formulas for Mellin transform of fractional Bessel derivatives on semiaxes we should proof the next statement.

410

Transmutations, Singular and Fractional Differential Equations

Lemma 37. Let Bγn f ∈ Pab . Then for n ∈ N MBγn f (s) = 22n 

n+1− 1 − 2s

s 2

1−s+γ +n 2 1−s+γ 2

f ∗ (s − 2n).

(9.28)

Proof. Using formulas for Mellin transform from [94], we obtain Mf (s) = (1 − s)Mf (s − 1),

1 M f (s) = Mf (s − 1), x

1 M f (s) = (Mf (t − 1))(s) = (2 − s)Mf (s − 2), x Mf

(s) = (2 − s)(1 − s)Mf (s − 2), MBγ f (s) = (2 − s)(1 − s)f ∗ (s − 2) + γ (2 − s)f ∗ (s − 2) = (2 − s)(1 − s + γ )f ∗ (s − 2). So MBγ f )(s) = (2 − s)(1 − s + γ )f ∗ (s − 2). Applying formula (9.29) n times, we obtain MBγn f (s) = (2 − s)(4 − s)...(2n − s)(1 − s + γ )(3 − s + γ )...× (2n − 1 − s + γ )f ∗ (s − 2n). Since

 s  s s  (2 − s)(4 − s)...(2n − s) = 2n 1 − 2− ... n − = 2 2 2    2n  n + 1 − 2s s   2n 1 − = 2 n  1 − 2s

and (1 − s + γ )(3 − s + γ )...(2n − 1 − s + γ ) =      1−s +γ 1−s +γ n 1−s +γ 2 + 1 ... +n−1 = 2 2 2     +n 2n  1−s+γ 2 n 1−s +γ   , 2 = 2 n  1−s+γ 2

we have

     n + 1 − 2s  1−s+γ + n 2 f ∗ (s − 2n) = MBγn f (s) = 22n   1−s+γ   s  1− 2  2

(9.29)

Fractional powers of Bessel operators

2n

2 

n+1− 1 − 2s

s 2

411

1−s+γ +n 2 1−s+γ 2

f ∗ (s − 2n).

This completes the proof. Theorem 118. Let α > 0, n = [α] + 1. Mellin transforms of DBγα,− and DBγα,0+ are MDBγα,− f (s) = 22α 

s 2

s 2,

−α−

γ −1 2 ,

s 2

− γ −1 2 s 2 −α

f ∗ (s − 2α),

(9.30)

f ∗ (s − 2α),

(9.31)

b where s − 2n > γ − 1, I Bγn−α ,− f ∈ Pa , and

MDBγα,0+ f (s) = 22α 

1 − 2s + α, 1 − 2s ,

γ −s+1 +α 2 γ −s+1 2



b where 2α − 2n + s < 2, I Bγn−α ,0+ f ∈ Pa .

Proof. Applying (9.26) and (9.28), we obtain ∗ ((DBγα,− f )(x))∗ (s) = ((Bγn (I Bγn−α ,− f (x)) (s) =

n + 1 − 2s 1−s+γ +n ∗ 2 ((I Bγn−α 22n  ,− f (x)) (s − 2n) = 1−s+γ 1 − 2s 2



1−s+γ s s s n+1− 2 +n − n, − n − γ −1 2α 2 2 2 2 2  f ∗ (s − 2α).  s γ −1 1−s+γ s 1 − 2s 2 2 −α− 2 , 2 −α (9.32)

Using the formula (1 − z)(z) =

π , sin (πz)

z∈ / Z,

in the numerator, we obtain   s  s π (−1)n π  1+n−  −n = = , s 2 2 sin( 2 − n)π sin( 2s )π     1−s +γ s −γ +1  +n  −n = 2 2     1−s +γ 1−s +γ −n  +n =  1− 2 2 n (−1) π π = . 1−s+γ sin( 2 + n)π sin( 1−s+γ 2 )π

412

Transmutations, Singular and Fractional Differential Equations

So   (−1)n π 1+s −γ n   , = (−1)  2  1−s+γ sin( 1−s+γ 2 2 )π s  (−1)n π   = (−1)n  . s s 2  1 − 2 sin( 2 )π Substituting the obtained expressions in (9.32), we obtain (9.30). Similarly, using (9.27) and (9.28) we have ∗ ((DBγα,0+ f )(x))∗ (s) = ((Bγn (I Bγn−α ,0+ f (x)) (s) =

+n n + 1 − 2s 1−s+γ ∗ 2 22n  ((I Bγn−α ,0+ f (x)) (s − 2n) = 1−s+γ s 1− 2 2

1−s+γ s +n +α n+1− 2 1 − 2s + α, γ −s+1 2α 2 2 2  f ∗ (s − 2α)=  1−s+γ γ −s+1 s 1 − 2s 1 − + n, + n 2 2 2

γ −s s + α, + α 1 − 2 2 22α  f ∗ (s − 2α). γ −s+1 1 − 2s , 2

9.3.2 The Hankel transform −α Theorem 119. Let Bγ−α ,0+ f, Bγ ,− f ∈L1 (R+ ). Then γ

−2α Fγ [(Bγ−α ,0+ f )(x)](ξ ) = ξ

∞   f (t) cos(απ)j γ −1 (ξ t)−sin(απ)y γ −1 (ξ t) t γ dt, 2

2

0

(9.33) where 4α − 2 < γ < 4 − 2α and −2α Fγ [(Bγ−α ,− f )](ξ ) = ξ

∞ j 1γ −1 (tξ )f (t)t γ dt, 2

(9.34)

,α

0

where j 1γ −1 (tξ ) = 2

,α

2

γ −1 2

 

(tξ ) ∞ 

+1 2

γ −1 2

 J γ1−1 (tξ ), 2

,α

(−1)n   J γ −1 (tξ ) = γ +1 2 ,α + α + n n=0 (α + n + 1) 2 1



tξ 2

2n+ γ −1 +2α 2

.

Fractional powers of Bessel operators

413

Proof. Using the factorization formula (9.24) and denoting g(x) = I20,α f (x), we obtain Fγ [(Bγ−α ,0+ f )(x)](ξ ) =

∞

γ j γ −1 (xξ ) (Bγ−α ,0+ f )(x)x dx = 2

0

1 22α

∞

γ −1

j γ −1 (xξ ) I2 2 2

,α 0,α I2 f (x)x 2α+γ

dx =

0

1 22α

∞

γ −1

j γ −1 (xξ ) I2 2 2

,α

g(x)x 2α+γ dx =

0

1 2α−1 2 (α)

∞

x 2

0

1 22α−1 (α)

(x 2 − u2 )α−1 uγ g(u)du =

j γ −1 (xξ ) x dx 0

∞

∞ uγ g(u)du (x 2 − u2 )α−1 j γ −1 (xξ ) x dx. 2

u

0

Let us consider the inner integral: ∞ (x 2 − u2 )α−1 j γ −1 (xξ ) x dx = 2

u

2

γ −1 2

  ξ

γ +1 2

γ −1 2



∞ γ −1 (x 2 − u2 )α−1 J γ −1 (xξ ) x 1− 2 dx. 2

u

Using formula (2.12.4.17) from [456] of the form ∞

x 1−ρ (x 2 − a 2 )β−1 Jρ (cx)dx = 2β−1 a β−ρ c−β (β)Jρ−β (ac),

a

a, c, β > 0,

(2β − ρ) < 3/2,

we obtain for 4α − γ < 2 ∞ γ −1 γ −1 (x 2 − u2 )α−1 J γ −1 (xξ ) x 1− 2 dx = 2α−1 uα− 2 ξ −α (α)J γ −1 −α (uξ ) 2

u

2

414

Transmutations, Singular and Fractional Differential Equations

and Fγ [(Bγ−α ,0+ f )(x)](ξ ) = 2

γ +1 2 −α

 

(α)ξ 2

γ +1 2 −α



γ +1 2

γ −1 2 +α

∞ u

γ −1 2 +α



(α)ξ

γ +1 2



2

γ −1 2 −α

ξ

γ +1 2 −α

 

γ +1 2



γ −1 2 +α

∞ uα+

2

u (u2 − y 2 )α−1 yf (y)dy =

J γ −1 −α (uξ )du

0

∞

J γ −1 −α (uξ )g(u)du =

0

2



γ +1 2

0

∞ γ +1 yf (y)dy (u2 − y 2 )α−1 u 2 −α J γ −1 −α (uξ )du. 2

y

0

Let us calculate the inner integral using formula (2.12.4.17) from [456] of the form ∞ x 1+ρ (x 2 − a 2 )β−1 Jρ (cx)dx = a

2β−1 a β+ρ c−β (β)[cos(βπ)Jρ+β (ac) − sin(βπ)Yρ+β (ac)], a, c, β > 0, (2β + ρ) < 3/2. We obtain ∞ γ +1 (u2 − y 2 )α−1 u 2 −α J γ −1 −α (uξ )du = 2

y

2α−1 y

γ −1 2

ξ −α (α)[cos(απ)J γ −1 (ξy) − sin(απ)Y γ −1 (ξy)] 2

2

for 2α + γ < 4 and Fγ [(Bγ−α ,0+ f )(x)](ξ ) =   γ −1 ∞ 2 2  γ +1 γ +1 2 y 2 f (y)[cos(απ)J γ −1 (ξy) − sin(απ)Y γ −1 (ξy)]dy = γ −1 2 2 ξ 2 +2α 0 ξ

−2α

∞

  f (t) cos(απ)j γ −1 (ξ t) − sin(απ)y γ −1 (ξ t) t γ dt. 2

2

0

So (9.33) is proved. Now let us consider (9.34). Let g(x) = K20,α x 2α f (x). Using the factorization (9.23), we obtain −2α Fγ [(Bγ−α ,− f )](ξ ) = 2

∞

1−γ

j γ −1 (xξ ) x γ K2 2 2

0

,α

K20,α x 2α f (x)dx =

Fractional powers of Bessel operators

2

−2α

∞

415

1−γ

j γ −1 (xξ ) x γ K2 2

,α

2

g(x)dx =

0

∞

21−2α (α)

∞ j γ −1 (xξ ) xdx (u2 − x 2 )α−1 uγ −2α g(u)du = 2

x

0 ∞

21−2α (α)

γ −2α

g(u)u

u j γ −1 (xξ )(u2 − x 2 )α−1 xdx.

du

2

0

0

Using formula (2.12.4.7) from [456] of the form a x 1−ρ (a 2 − x 2 )β−1 Jρ (cx)dx =

21−ρ a β−ρ sρ+β−1,β−ρ (ac), cβ (ρ)

0

a > 0,

Re β > 0,

we obtain for the inner integral u (u2 − x 2 )α−1 j γ −1 (xξ ) x dx = 2

0

2

γ −1 2

  ξ

γ +1 2



u (u2 − x 2 )α−1 J γ −1 (xξ ) x 1−

γ −1 2

γ −1 2

2

dx =

0

  (α) u2 ξ 2 γ +1 u2α 1 F2 1; α + 1, ;− . 2(α + 1) 2 4 So Fγ [(Bγ−α ,− f )](ξ ) = 1 22α (α + 1)

1 22α (α + 1) 

∞ 1 F2

  u2 ξ 2 γ +1 F ; − g(u)uγ du= 1; α + 1, 1 2 2 4

∞ 0

 u2 ξ 2 γ +1 ;− uγ K20,α u2α f (u)du = 1; α + 1, 2 4

0

21−2α (α)(α + 1) 21−2α (α)(α + 1)

∞

  ∞ u2 ξ 2 γ +1 γ ;− u du (t 2 − u2 )α−1 tf (t)dt = 1 F2 1; α + 1, 2 4 u

0

  u2 ξ 2 γ +1 tf (t)dt (t 2 − u2 )α−1 1 F2 1; α + 1, ;− uγ du. 2 4

∞

t

0

0

416

Transmutations, Singular and Fractional Differential Equations

Using Wolfram Mathematica, we obtain   u2 ξ 2 γ +1 (t 2 − u2 )α−1 1 F2 1; α + 1, ;− uγ du = 2 4 0     (α) γ +1 2 t 2ξ 2 γ +1 2α+γ −1  t ;− 1 F2 1; α + 1, α + 2 4 2 α + γ +1 2

t

and Fγ [(Bγ−α ,− f )](ξ ) =    γ +1 2  22α (α + 1) α +

  ∞ t 2ξ 2 γ +1  f (t) t 2α+γ 1 F2 1; α + 1, α + ;− dt. γ +1 2 4 2

0

Since  t 2ξ 2 γ +1 ;− = 1 F2 1; α + 1, α + 2 4  ∞  (−1)n γ +1   (α + 1) α + 2 (α + n + 1) α + 

n=0

 γ +1 2

+n



tξ 2

2n

and the Wright function through which the Hankel transform of Bγ−α ,− f is expressed in [515] is given by J γ1−1 (tξ ) = 2

,α

∞ 

(−1)n   γ +1 + α + n n=0 (α + n + 1) 2

we obtain Fγ [(Bγ−α ,− f )](ξ ) =

ξ

−2α

2

γ −1 2

  ξ

γ +1 2

γ −1 2

 ξ

−2α

∞ f (t) t



γ +1 2

tξ 2

2n+ γ −1 +2α 2

,

J γ1−1 (tξ )dt = 2

,α

0

∞ j 1γ −1 (tξ )f (t)t γ dt. 2

,α

0

Thus, (9.34) is proved. γ

Since Fγ [(Bγn f )](ξ ) = (−1)n ξ 2n Fγ [f ](ξ ), we obtain for Bγα,0+ f, Bγα,− f ∈L1 (R+ ) −(n−α)

Fγ [(Bγα,0+ f )(x)](ξ ) = Fγ [(Bγn Bγ ,0+

f )(x)](ξ ) =

Fractional powers of Bessel operators

417

(−1)n ξ 2n Fγ [Bγ−(n−α) ,0+ f (x)](ξ ) = ∞ n 2α

(−1) ξ

  f (t) cos((n − α)π)j γ −1 (ξ t) − sin((n − α)π)y γ −1 (ξ t) t γ dt, 2

2

0

n = [α] + 1,

4(n − α) − 2 < γ < 4 − 2(n − α)

and Fγ [(Bγα,− f )(x)](ξ ) = Fγ [(Bγn Bγ−(n−α) f )(x)](ξ ) = ,− −(n−α)

(−1)n ξ 2n Fγ [Bγ ,− ∞ (−1)n ξ 2α j 1γ −1 2

,n−α

f (x)](ξ ) = n = [α] + 1.

(tξ )f (t)t γ dt,

0

9.3.3 The Meijer transform The integral Meijer transform (1.58) plays the same role for the left-sided Bessel fractional derivative at a semiaxis as the Laplace transform (1.54) plays for the left-sided Riemann–Liouville fractional derivative at a semiaxis (compare (2.43) and (9.35)). Theorem 120. Let α > 0. The Meijer transform of Bγ−α ,0+ for proper functions is −2α Kγ f (ξ ). Kγ [(Bγ−α ,0+ f )(x)](ξ ) = ξ

(9.35)

Proof. Let g(x) = I20,α f (x). Then using the factorization (9.24), we obtain Kγ [(Bγ−α ,0+ f )(x)](ξ ) =

∞

γ k γ −1 (xξ ) (Bγ−α ,0+ f )(x)x dx = 2

0

1 22α 1 22α

∞

γ −1

k γ −1 (xξ ) I2 2 2

0 ∞

γ −1

k γ −1 (xξ ) I2 2 2

,α 0,α I2 f (x)x 2α+γ

,α

dx =

g(x)x 2α+γ dx =

0

1 2α−1 2 (α)

∞

x 2

0

1 22α−1 (α)

(x 2 − u2 )α−1 uγ g(u)du =

k γ −1 (xξ ) x dx ∞

0

∞ γ u g(u)du (x 2 − u2 )α−1 k γ −1 (xξ ) x dx. 2

0

u

418

Transmutations, Singular and Fractional Differential Equations

Let us consider the inner integral. Using formula (2.16.3.7) from [456] of the form ∞ x 1±ρ (x 2 −a 2 )β−1 Kρ (cx)dx = 2β−1 a β±ρ c−β (β)Kρ±β (ac),

a, c, β > 0,

a

(9.36) we get ∞ (x 2 − u2 )α−1 k γ −1 (xξ ) x dx = 2

u

   

2

γ +1 2

2

∞ γ −1 (x 2 − u2 )α−1 K γ −1 (xξ ) x 1− 2 dx =

1−γ 2

 ξ

γ −1 2

ξ

γ −1 2

2

u

1−γ 2

γ +1 2



· 2α−1 uα−

γ −1 2

ξ −α (α)K γ −1 −α (uξ ) 2

and Kγ [(Bγ−α ,0+ f )(x)](ξ ) = 2  (α) 2  (α)

3−γ 2

3−γ 2



ξ

γ −1 2 +α

−α

γ +1 2



ξ



∞

−α

γ +1 2



u

2

1−γ 2

γ +1 2



γ +1 2 −α

∞

−α

ξ

γ −1 2 +α

uα+

γ +1 2

2

0

u (u2 − t 2 )α−1 tf (t)dt =

K γ −1 −α (uξ )du 2

0 ∞

γ −1 2 +α

K γ −1 −α (uξ )g(u)du =

0

∞ (u2 − t 2 )α−1 u

tf (t)dt

γ +1 2 −α

K γ −1 −α (uξ )du. 2

t

0

Using again (9.36), we can write ∞ γ +1 γ −1 (u2 − t 2 )α−1 u 2 −α K γ −1 −α (uξ )du = 2α−1 t 2 ξ −α (α)K γ −1 (tξ ) 2

2

t

and Kγ [(Bγ−α ,0+ f )(x)](ξ ) = 2  (α)

3−γ 2

−α

γ +1 2



ξ

γ −1 2 +α

· 2α−1 ξ −α (α)

∞ f (t)K γ −1 (tξ )t 2

0

γ +1 2

dt =

Fractional powers of Bessel operators

−2α

ξ

∞

419

f (t)k γ −1 (tξ )t γ dt = ξ −2α Kγ f. 2

0

Lemma 38. Let n ∈ N and let the Meijer transform of Bγn f exist. Then for 0 ≤ γ < 1 Kγ [Bγn f ](ξ ) = ξ 2n Kγ [f ](ξ )− n 

ξ 2k−1−γ Bγn−k f (0+) −

k=1





1−γ 2



2γ 



γ +1 2

 lim

x→0+

n 

ξ 2k−2 x γ

k=1

d [B n−k f (x)], dx γ (9.37)

for γ = 1 Kγ [Bγn f ](ξ ) = ξ 2n Kγ [f ](ξ ) −

n 

ξ 2k−1−γ Bγn−k f (0+)+

k=1

lim

n 

x→0+

ξ 2k−2 ln xξ

k=1

d [B n−k f (x)], dx γ

(9.38)

and for 1 < γ Kγ [Bγn f ](ξ ) = ξ 2n Kγ [f ](ξ )− n 

 1 d lim ξ 2k−1−γ x [Bγn−k f (x)], (9.39) γ − 1 x→0+ dx n

ξ 2k−1−γ Bγn−k f (0+) −

k=1

k=1

where Bγn−k f (0+) = lim Bγn−k f (x). x→+0

Proof. Let us find Kγ [Bγn f ](ξ ): ∞ Kγ [Bγn f ](ξ ) =

k γ −1 (xξ ) [Bγn f (x)] x γ dx = 2

0

∞ k γ −1 (xξ ) 2

0

d γ d x [B n−1 f (x)] dx = dx dx γ

∞ ∞  d d d [Bγn−1 f (x)] k γ −1 (xξ ) [Bγn−1 f (x)] dx = − xγ 2 2 dx dx dx x=0 0       d d −k γ −1 (xξ ) x γ + xγ [Bγn−1 f (x)] k γ −1 (xξ ) [Bγn−1 f (x)] + 2 dx dx 2 x=0 x=0

k γ −1 (xξ ) x γ

420

Transmutations, Singular and Fractional Differential Equations

 ∞  n−1 γ γ d n−1 [Bγ k γ −1 (xξ )] [Bγ f (x)]x dx = −k γ −1 (xξ ) x [Bγ f (x)] + 2 2 dx x=0 0



  ∞  d n−1 2  +ξ k γ −1 (xξ ) [Bγn−1 f (x)]x γ dx = ... x k γ −1 (xξ ) [Bγ f (x)] 2 dx 2 x=0 γ

0

∞ ... = ξ 2n

k γ −1 (xξ ) f (x)x γ dx+ 2

0

n−1  k=0

    γ d n−1−k γ d n−1−k ξ f (x)]−k γ −1 (xξ ) x f (x)]  . k γ −1 (xξ ) [Bγ [Bγ x 2 dx 2 dx x=0 2k

Let 0 ≤ γ < 1. Then using (1.27), we obtain lim k γ −1 (xξ ) x γ

x→0+

2

 

1−γ 2



d d   lim x γ [Bγn−1−k f (x)] = [B n−1−k f (x)]. γ +1 x→0+ dx dx γ 2γ  2

For γ = 1 using (1.28) we obtain lim k0 (xξ )

x→0+

d d [B n−1−k f (x)] = − lim ln xξ [Bγn−1−k f (x)]. x→0+ dx γ dx

When 1 < γ using (1.26) we obtain lim k γ −1 (xξ ) x γ

x→0+

2

d d 1 [Bγn−1−k f (x)] = lim xξ 1−γ [B n−1−k f (x)]. dx γ − 1 x→0+ dx γ

Next we have 1−γ

3−γ

1−γ

d 2 2 ξ 2 x 2   K γ +1 (xξ ), k γ −1 (xξ ) = − 2 dx 2  γ +1 2 and using (1.18) for small x 1−γ

γ +1 3−γ d 2 2  x 2 ξ 2 K γ +1 (xξ ) ∼ k γ −1 (xξ ) = −  x γ +1 2 dx 2  2   γ +1 1−γ γ +1 3−γ  γ +1 2 2 2 2 2 x ξ (ξ x)− 2 = −ξ 1−γ , −  γ +1 γ +1 1− 2  2 2

γ

Therefore,

x →0+.

  d k γ −1 (xξ ) [Bγn−1−k f (x)] = −ξ 1−γ Bγn−1−k f (0+), xγ x→0+ dx 2 lim

Fractional powers of Bessel operators

421

and for 0 ≤ γ < 1 Kγ [Bγn f ](ξ ) = ξ 2n Kγ [f ](ξ ) −   2γ 

1−γ 2





γ +1 2



 2γ 

1−γ 2



n−1 

ξ 2k lim x γ x→0+

k=0 n 

d [B n−1−k f (x)] = dx γ

ξ 2k−1−γ Bγn−k f (0+)−

k=1



γ +1 2

ξ 2k+1−γ Bγn−1−k f (0+)−

k=0

ξ 2n Kγ [f ](ξ ) − 

n−1 

 lim

n 

x→0+

ξ 2k−2 x γ

k=1

d [B n−k f (x)], dx γ

for γ = 1 Kγ [Bγn f ](ξ ) = ξ 2n Kγ [f ](ξ ) −

n 

ξ 2k−1−γ Bγn−k f (0+)+

k=1

lim

x→0+

n 

ξ 2k−2 ln xξ

k=1

d [B n−k f (x)], dx γ

and for 1 < γ Kγ [Bγn f ](ξ ) = ξ 2n Kγ [f ](ξ ) −

n 

ξ 2k−1−γ Bγn−k f (0+)−

k=1

1 lim γ − 1 x→0+

n 

ξ 2k−1−γ x

k=1

d [B n−k f (x)]. dx γ

d Remark 19. Let n ∈ N, let dx [Bγn−k f (x)] be bounded, let the Meijer transform of n Bγ f exist, and let γ = 1. Then

Kγ [Bγn f ](ξ ) = ξ 2n Kγ [f ](ξ ) −

n 

ξ 2k−1−γ Bγn−k f (0+).

k=1

If

d n−k β dx [Bγ f (x)] ∼ x ,

β > 0, when x → 0+, then (9.40) holds for γ = 1.

Remark 20. Since k− 1 (x) = e−x . we have 2

K0 [f ](ξ ) = L[f ](ξ ),

(9.40)

422

Transmutations, Singular and Fractional Differential Equations

where L[f ] is a Laplace transform of f . It is well known that L[f

](ξ ) = ξ 2 L[f ](ξ ) − ξf (0) − f (0). From the other side,     1−γ  2    = 1, γ =0 2γ  γ +1 2

n 

xγ

k=1

  d [Bγn−k f (x)] = f (x), dx γ =0,n=1

and K0 [B0 f ](ξ ) = Lf

(ξ ) = ξ 2 K0 [f ](ξ ) − ξf (0) − f (0) = L[f

](ξ ). The same situation holds for K0 [B0n f ](ξ ). Theorem 121. Let n = [α] + 1 for fractional α and n = α for α ∈ N and let the Meijer transform of the left-sided fractional Bessel derivatives on a semiaxis of the Riemann–Liouville type Bγα,0+ f exist. Then for 0 ≤ γ < 1 Kγ [Bγα,0+ f ](ξ ) = ξ 2α Kγ [f ](ξ ) −   2γ 

1−γ 2



ξ 2k−1−γ Bγα−k ,0+ f (0+)−

k=1



γ +1 2

n 

 lim

x→0+

n  k=1

xγ

d α−k B f (x), dx γ ,0+

(9.41)

for γ = 1 Kγ [Bγα,0+ f ](ξ ) = ξ 2n Kγ [f ](ξ ) −

n 

ξ 2k−1−γ Bγα−k ,0+ f (0+) + lim

x→0+

k=1

n  k=1

ln x

d α−k B f (x), dx γ ,0+ (9.42)

and for 1 < γ Kγ [Bγα,0+ f ](ξ ) = ξ 2n Kγ [f ](ξ ) −

n 

 d 1 lim x Bγα−k f (x), γ − 1 x→0+ dx ,0+ n

ξ 2k−1−γ Bγα−k ,0+ f (0+) −

k=1

k=1

(9.43) where α−k Bγα−k ,0+ f (0+) = lim Bγ ,0+ f (x). x→+0

Fractional powers of Bessel operators

423

Proof. Using (9.40) and (9.35) for 0 ≤ γ < 1, we obtain Kγ [Bγα,0+ f ](ξ ) = Kγ [Bγn (I Bγn−α ,0+ f )(x)](ξ ) = ξ

2n

Kγ [(I Bγn−α ,0+ f )(x)](ξ ) − 

 2γ 

1−γ 2



 2γ 

1−γ 2



x→0+

xγ

k=1

n 

d [B n−k (I Bγn−α ,0+ f )(x)] = dx γ

ξ 2k−1−γ Bγα−k ,0+ f (0+)−

k=1



γ +1 2

n 

 lim

ξ 2α Kγ [f ](ξ ) − 

ξ 2k−1−γ Bγn−k (I Bγn−α ,0+ f )(x)|x=0 −

k=1



γ +1 2

n 

 lim

n 

x→0

xγ

k=1

d (B α−k f )(x), dx γ ,0+

where we put α−k lim Bγα−k ,0+ f (x) = Bγ ,0+ f (0+).

x→+0

Similarly, we get (9.45) and (9.43). Theorem 122. Let n = [α] + 1 for fractional α and n = α for α ∈ N and let the Meijer transform of the left-sided fractional Bessel derivatives on a semiaxis of Gerasimov– Caputo type Bγα,0+ f exist. Then for 0 ≤ γ < 1 Kγ [Bγα,0+ f ](ξ ) = ξ 2α Kγ [f ](ξ ) −   2γ 

1−γ 2



ξ 2α−2k−1−γ Bγk f (0+)−

k=0



γ +1 2

n−1 

 lim

x→0+

n−1 

ξ 2α−2k−2 x γ

k=0

d [B k f (x)], dx γ

(9.44)

for γ = 1 Kγ [Bγα,0+ f ](ξ ) = ξ 2α Kγ [f ](ξ ) −

n−1 

ξ 2α−2k−1−γ Bγk f (0+)+

k=0

lim

x→0+

n−1  k=0

ξ 2α−2k−2 ln xξ

d [B k f (x)], dx γ

(9.45)

424

Transmutations, Singular and Fractional Differential Equations

and for 1 < γ Kγ [Bγα,0+ f ](ξ ) = ξ 2α Kγ [f ](ξ ) −

n−1 

ξ 2α−2k−1−γ Bγk f (0+)−

k=0

1 lim γ − 1 x→0+

n−1 

ξ 2α−2k−1−γ x

k=0

d [B k f (x)], dx γ

where α−k Bγα−k ,0+ f (0+) = lim Bγ ,0+ f (x). x→+0

Proof. Using (9.35) and (9.40) for 0 ≤ γ < 1, we obtain n 2α−2n Kγ [Bγn f ](ξ ) = Kγ [Bγα,0+ f ](ξ ) = Kγ [(I Bγn−α ,0+ Bγ f )(x)](ξ ) = ξ

ξ 2α Kγ [f ](ξ ) −   2γ 

1−γ 2



 lim

x→0+

ξ 2α Kγ [f ](ξ ) −   2γ 

1−γ 2



n 

ξ 2α−2n+2k−2 x γ

k=1

n−1 

d [B n−k f (x)] = dx γ

ξ 2α−2k−1−γ Bγk f (0+)−

k=0



γ +1 2

ξ 2α−2n+2k−1−γ Bγn−k f (0+)−

k=1



γ +1 2

n 

 lim

x→0+

n−1 

ξ 2α−2k−2 x γ

k=0

d [B k f (x)], dx γ

where we put lim Bγk ,0+ f (x) = Bγk ,0+ f (0+).

x→+0

Similarly, for γ = 1 we have n 2α−2n Kγ [Bγn f ](ξ ) = Kγ [Bγα,0+ f ](ξ ) = Kγ [(I Bγn−α ,0+ Bγ f )(x)](ξ ) = ξ

ξ

2α

Kγ [f ](ξ ) −

n 

ξ 2α−2n+2k−1−γ Bγn−k f (0+)+

k=1

lim

x→0+

n 

ξ 2α−2n+2k−2 ln xξ

k=1

ξ 2α Kγ [f ](ξ ) −

n−1  k=0

d [B n−k f (x)] = dx γ

ξ 2α−2k−1−γ Bγk f (0+)+

(9.46)

Fractional powers of Bessel operators n−1 

lim

x→0+

ξ 2α−2k−2 ln xξ

k=0

425

d [B k f (x)] dx γ

and for γ > 1 n 2α−2n Kγ [Bγn f ](ξ ) = Kγ [Bγα,0+ f ](ξ ) = Kγ [(I Bγn−α ,0+ Bγ f )(x)](ξ ) = ξ n−1 

ξ 2α Kγ [f ](ξ ) −

ξ 2α−2k−1−γ Bγk f (0+)−

k=0

1 lim γ − 1 x→0+

n−1 

ξ 2α−2k−1−γ x

k=0

d [B k f (x)]. dx γ

d Remark 21. Let k ∈ N, let dx [Bγk f (x)] be bounded, let the Meijer transform of Bγα,0+ f exist, and let γ = 1. Then

Kγ [Bγα,0+ f ](ξ ) = ξ 2α Kγ [f ](ξ ) −

n−1 

ξ 2α−2k−1−γ Bγk f (0+).

(9.47)

k=0

If

d k β dx [Bγ f (x)] ∼ x ,

β > 0, when x → 0+, then (9.47) holds for γ = 1.

9.3.4 Generalized Whittaker transform Theorem 123. The generalized Whittaker transform of Bγ−α ,0+ for proper functions is  γ −1    γ −1 −α −2α 2 2 W γ −1 Bγ ,0+ f (x) = C(γ , α, ρ)x (x), W γ −1 f ρ,

ρ+α,

4

where

 C(γ , α, ρ) =



4

   − α − ρ  3−γ − α − ρ 4     . γ +1 22α  4 − ρ  3−γ 4 −ρ γ +1 4

Proof. We have  W

γ −1 2 ρ, γ −1 4

Bγ−α ,0+ f



1 (x) = (2α)

∞ γ −1 x 2 t 2 (xt) 2 e 2 Wρ, γ −1 (x 2 t 2 )dt× 4

0

2α−1    t  γ  2 y2 y t − y2 γ −1 F , α; 2α; 1 − α + f (y)dy = 2 1 t 2x 2 t2 0

x

γ −1 2

∞

∞ γ

f (y)y dy

22α−1 (2α) 0

t y

γ −1 2 −γ −2α+1

e

x2 t 2 2

(t 2 − y 2 )2α−1 ×

426

Transmutations, Singular and Fractional Differential Equations

  y2 γ −1 Wρ, γ −1 (x 2 t 2 ) 2 F1 α + , α; 2α; 1 − 2 dt. 4 2 t Using the formula −a

2 F1 (a, b; c; z) = (1 − z)

 2 F1 a, c − b; c;

 z , z−1

we obtain       y2 t2 γ −1 y 1−γ −2α γ −1 F F , α; 2α; 1− , α; 2α; 1− α + = α + 2 1 2 1 2 t 2 t2 y2 and 

γ −1 2 ρ, γ −1 4

W

x

 Bγ−α f (x) = ,0+ ∞

γ −1 2

∞ f (y)y

22α−1 (2α)

1−2α

dy

t

γ −1 2

e

x2 t 2 2

(t 2 − y 2 )2α−1 ×

y

0

  t2 γ −1 2 2 , α; 2α; 1 − 2 dt. Wρ, γ −1 (x t ) 2 F1 α + 4 2 y Let us consider the inner integral. We have ∞ t

γ −1 2

e

x2 t 2 2

 (t − y ) 2

2 2α−1

2 2

Wρ, γ −1 (x t ) 2 F1 4

 t2 γ −1 , α; 2α; 1 − 2 dt = α+ 2 y

y

{t 2 → t, y 2 = p} =   ∞ γ −1 1 x 2 t t γ −1 1 t 4 − 2 e 2 (t − p)2α−1 Wρ, γ −1 (x 2 t) 2 F1 α + , α; 2α; 1 − dt. 4 2 2 p p

Using formula (2.21.8.2) from [457] of the form ∞ t p

p

  σt t (t − p)c−1 e 2 Wρ, a+b−c (σ t) 2 F1 a, b; c; 1 − dt = 2 p      (c)  a−b−c+1 − ρ  b−a−c+1 − ρ σp 2 2     e 2 Wρ+ c , a−b (σp), 2 2 c−a−b+1  a+b−c+1 − ρ  − ρ 2 2

a+b−c−1 2

a+b−1 2 c

σ2

p, Re c > 0, Re (c + 2ρ) < 1 − |Re(a − b)|; |arg σ |
0, (c) F (a, b; c; z) = (b)(c − b)

1

t b−1 (1 − t)c−b−1 (1 − tz)−a dt,

0

we obtain 1 1 Rλ f = − f − λ λ

+∞ 1  ∞ f (y)dy 0 k=1

x

1 k 2 λ  (αk)



y2 − x2 2y

2αk−1 ×



  −αk− γ −1 2 x2 (1 − t) dt = {k = p + 1} = 1− 1− 2 t t y 2α(p+1)−1  2 +∞ 1  ∞ 1 1 y − x2 1 f (y)dy × − f− λ λ 2y λp+1  2 (α(p + 1)) αk−1

αk−1

0 p=0

x

 t

α(p+1)−1

(1 − t)

α(p+1)−1

  −α(p+1)− γ −1 2 1 1 x2 dt = − f − × 1− 1− 2 t λ λ y

430

Transmutations, Singular and Fractional Differential Equations

 2 2α−1 1    −α− γ −1 +∞ 2 y − x2 x2 α−1 α−1 f (y) dy t (1 − t) × 1− 1− 2 t 2y y x ∞  p=0

1 λp+1  2 (α(p + 1))



0

y2

− x2 2y

2αp

 t

αp

(1 − t)

αp

  −αp x2 dt = 1− 1− 2 t y

 2 2α−1 1    −α− γ −1 +∞ 2 y − x2 x2 f (y) dy × 1− 1− 2 t 2y y x 0

p  ∞ 2 − x 2 )2 α  1 1 t (1 − t)(y 1 t α−1 (1 − t)α−1 dt. (9.51)  2 (α + αp) λ 4 y 2 − (y 2 − x 2 )t 1 1 − f− 2 λ λ

p=0

The function in (9.51) is a special case of the Wright generalized hypergeometric function defined above as (9.50). So it follows that   α p ∞  1 1 1 t (1 − t)(y 2 − x 2 )2 =  2 (α + αp) λ 4 y 2 − (y 2 − x 2 )t p=0   α  1 1 t (1 − t)(y 2 − x 2 )2 , E(α,α),(α,α) λ 4 y 2 − (y 2 − x 2 )t and we finally derive 1 1 Rλ f = − f − 2 λ λ

 2 2α−1 1 +∞ y − x2 f (y) dy t α−1 (1 − t)α−1 × 2y x



0

    −α− γ −1 α  2 x2 1 1 t (1 − t)(y 2 − x 2 )2 E(α,α),(α,α) 1− 1− 2 t dt. λ 4 y 2 − (y 2 − x 2 )t y

9.4.2 The generalized Taylor formula with powers of Bessel operators Many applications of the Riemann–Liouville fractional integrals are based on the fact that they are remainder terms in the Taylor formula. Such formulas exist also with powers of Bessel operators – they are the so-called Taylor–Delsarte series (cf. [83, 317] and especially [139]). But in the Taylor–Delsarte series not a function itself is expanded but its generalized translation; these series are in fact just operator versions of Bessel function series. But for application to numerical partial differential equation solutions, we need the classical form of the Taylor formula f (x + t) = f (x)+. . ., as only with this formula we may calculate partial differential equation solutions layer by layer. Such formulas are much harder to prove. With the abovementioned motivation as a tool for solving singular partial differential equations numerically, a first

Fractional powers of Bessel operators

431

attempt to construct the generalized Taylor formula with Bessel operators was made in [225,227]. But these results were rather vague as neither coefficients nor the integral remainder term were found explicitly: For coefficients the recurrent system of equations was found and the remainder term was evaluated as a multi-term composition of simple integral operators. The solution to the problem of finding the generalized Taylor formula with Bessel operators in the explicit form was found in [268] (cf. also [234,527,531]). Of course it is based on explicit forms for fractional powers of Bessel operators. Theorem 125. The following generalized Taylor formula is valid for proper functions:    k  b2 −x 2 2i−2  x2 γ −1 1 , i −1; 2i −1; 1− 2 × f (x) = 2 F1 i + (2i − 1) 2b 2 b i=1     1 b2 − x 2 2i−1 x2 γ −1 (B i−1 f )|b − , i; 2i; 1 − 2 × 2 F1 i + (2i) 2b 2 b k (DB i−1 f )|b + Bγ−k ,b− (B f ).

Theorem 126. The following generalized Taylor formula is valid for proper functions:    k  x 2 − a 2 2i−2  a   a2 γ −1 1 f (x) = F , i; 2i −1; 1− i + × 2 1 (2i − 1) 2x x 2 x2 i=1

1  x 2 − a 2 2i−1 × (2i) 2x   a2 γ −1 k F , i; 2i; 1 − i + a γ (Dx −γ Cγi−1 f )|a + Bγ−k 2 1 ,a+ (Cγ f ). 2 x2 (Cγi−1 f )|a +

B-potentials theory 10.1

10

Definitions of hyperbolic B-potentials, absolute convergence, and boundedness

The theory of fractional powers of elliptic operators with Bessel operator ν Bν =D 2 + D, x

D=

d dx

acting instead of all or some second derivatives in  is well developed (see [174,206, 207,343–347,351–353,501–504,506,507,511]). Fractional powers of hyperbolic operators, with Bessel operators instead of all or some second derivatives, are much less studied. Such operators have wide areas of application, such as singular differential equations, differential geometry, and random walks. In this chapter we study real powers of γ = Bγ1 − Bγ2 − ... − Bγn ,

Bγi =

∂2 γi ∂ + , 2 xi ∂xi ∂xi

i = 1, ..., n.

The composition method (see [229,230,234,523]) was used for the construction of α (γ )− 2 , α > 0.

10.1.1 Negative fractional powers of the hyperbolic expression with Bessel operators We consider fractional powers of the hyperbolic expression with Bessel operators γ = Bγ1 − Bγ2 − ... − Bγn ,

Bγi =

∂2 γi ∂ + , ∂xi2 xi ∂xi

i = 1, ..., n,

γ

in Sev and Lp . We will call negative real powers of γ hyperbolic B-potentials. Definition 50. Hyperbolic B-potentials IPα ±i0,γ for α > n + |γ | − 2 are defined by formulas n−1+|γ  |

(IPα ±i0,γ f )(x) =

e± 2 iπ Hn,γ (α)



α−n−|γ | 2

(P ±i0)γ

y

(γ Tx f )(x)y γ dy,

Rn+

yγ =

n  γ yi i, i=1

(10.1) where γ  = (γ2 , ..., γn ), |γ  | = γ2 + ... + γn , Transmutations, Singular and Fractional Differential Equations With Applications to Mathematical Physics https://doi.org/10.1016/B978-0-12-819781-3.00016-1 Copyright © 2020 Elsevier Inc. All rights reserved.

434

Transmutations, Singular and Fractional Differential Equations



n 

Hn,γ (α) =



i=1

γi +1 2

2n−α 



    α2 .

n+|γ |−α 2

For 0 ≤ α ≤ n + |γ | − 2 hyperbolic B-potentials IPα ±i0,γ are defined as (IPα ±i0,γ f )(x) = (γ )k (IPα+2k ±i0,γ f )(x) = n−1+|γ  |

e± 2 iπ (γ )k Hn,γ (α + 2k) where k =



n+|γ |−α 2

.



α+2k−n−|γ | 2

(P ± i0)γ

y

(γ Tx f )(x)y γ dy,

(10.2)

Rn+

It is well known (see for example [242]) that generalized convolution of weighted generalized functions and a regular function is a regular function. Using the property of weighted generalized functions (P ± i0)λγ (see [505]) we can rewrite (10.1) as ⎡ |  ± n−1+|γ iπ 2 e y α ⎣ (IP ±i0,γ f )(x) = r α−n−|γ | (y)(γ Tx f )(x)y γ dy+ Hn,γ (α) K+ ⎤  α−n−|γ | y e± 2 πi |r(y)|α−n−|γ | (γ Tx f )(x)y γ dy ⎦ , (10.3) K−

where K − = {x : x ∈ Rn+ : P (x) ≤ 0}, K + = {x : x ∈ Rn+ : P (x) ≥ 0},   r(y) = P (y) = y12 − y22 − ... − yn2 . Function r(y) is a Lorentz distance and K + is a part of a light cone. Introducing the notations  y (IPα+ ,γ f )(x) = r α−n−|γ | (y)(γ Tx f )(x)y γ dy,

(10.4)

K+



(IPα− ,γ f )(x) =

|r(y)|α−n−|γ | (γ Tx f )(x)y γ dy, y

(10.5)

K−

we can write 

(IPα ±i0,γ f )(x) =

n−1+|γ |

α−n−|γ | e± 2 iπ α (IP+ ,γ f )(x) + e± 2 πi (IPα− ,γ f )(x) . Hn,γ (α)

(10.6)

B-potentials theory

435

Remark 22. Let y  = (y2 , ..., yn ), |y  | = 3, we have ∞ (IPα+ ,γ f )(x) =



γ y1 1 dy1





y22 + ... + yn2 , (y  )γ = y2 2 ...ynn . For n ≥

(y12 − |y  |2 )

α−n−|γ | 2

γ

γ



(γ Tx f )(x)(y  )γ dy  , y

{|y  |y1 }+

0

(10.8) where {|y  | < y1 }+ = {y ∈ Rn+ : |y  | < y1 }, {|y  | > y1 }+ = {y ∈ Rn+ : |y  | > y1 }. For n = 2, we have ∞ (IPα+ ,γ f )(x) =

γ y1 1 dy1

0

∞ (IPα− ,γ f )(x) =

y1 α−2−|γ | γ y (y12 − y22 ) 2 (γ Tx f )(x)y2 2 dy2 , 0

∞ α−2−|γ | γ1 γ y y1 dy1 (y22 − y12 ) 2 (γ Tx f )(x)y2 2 dy2 .

0

y1

Passing to the spherical coordinates y  = ρσ in (10.7) and (10.8), we obtain (IPα+ ,γ f )(x) = |S1+ (n − 1)|γ × ∞

γ y1 1 dy1

0

y1 α−n−|γ |   y (y12 − ρ 2 ) 2 ρ n+|γ |−2 ( γ1 Tx11 )(Mργ )x  [f (x1 , x  )]dρ,

(10.9)

0

(IPα− ,γ f )(x) = |S1+ (n − 1)|γ × ∞

γ y1 1 dy1

∞ α−n−|γ |   y (ρ 2 − y12 ) 2 ρ n+|γ |−2 ( γ1 Tx11 )(Mργ )x  [f (x1 , x  )]dρ,

y1

0

where 

(Mργ )x  [f (x1 , x  )] =

1 |S1+ (n − 1)|γ



γ



Tx  f (x1 , x  )σ γ dS ρσ

S1+ (n−1)

is the weighted spherical mean (3.183). If f (x) = ϕ(x1 )G(x  ), then (10.9) and (10.10) have forms

(10.10)

436

Transmutations, Singular and Fractional Differential Equations

(IPα+ ,γ f )(x)=|S1+ (n−1)|γ × ∞ y1 α−n−|γ |   γ1 γ1 y1 ( Tx1 )[ϕ(x1 )]y1 dy1 (Mργ )x  [G(x  )](y12 − ρ 2 ) 2 ρ n+|γ |−2 dρ, 0

0

(10.11) (IPα− ,γ f )(x)=|S1+ (n−1)|γ × ∞ ∞  γ1 γ1 y1 ( Tx1 )[ϕ(x1 )]y1 dy1 (Mργ )x  [G(x  )](ρ 2 y1 0

− y12 )

α−n−|γ | 2

ρ n+|γ

 |−2

dρ. (10.12)

10.1.2 Absolute convergence and boundedness Theorem 127. Let f ∈ Sev and α > n + |γ | − 2. Then integrals (IPα ±i0,γ f )(x) converge absolutely for x ∈ Rn+ . Proof. Let us prove absolute convergence of each term in (10.3). Passing in (10.3) to spherical coordinates y=ρσ , ρ=|y|, σ  =(σ2 , ..., σn ), we obtain  y r α−n−|γ | (y)(γ Tx f )(x)y γ dy = K+ ∞



(σ12 − |σ  |2 )

ρ α−1 dρ

α−n−|γ | 2

(γ Tρσ f )(x)σ γ dS,

{S1+ (n),|σ  |n+|γ |−2. So for α > n + |γ | − 2 integrals (IPα ±i0,γ f )(x) converge absolutely. Corollary 22. The integral in (10.2) also converges absolutely.

B-potentials theory

437

We present here the Marcinkiewicz interpolation theorem in the following form (see [25]). Theorem 128. Let 1 ≤ pi ≤ qi < ∞ (i = 1, 2), q1 = q2 , 0 < τ < 1, 1 1−τ τ q = q1 + q2 . If a linear operator A has simultaneously weak (p2 , q2 )γ , then an operator A has a strong type (p, q)γ and

1 1−τ p = p1

+

τ p2 ,

types (p1 , q1 )γ and

||Af ||q,γ ≤ M||f ||p,γ ,

(10.13)

where a constant M = M(γ , τ, κ, p1 , p2 , q1 , q2 ) and does not depend on f and A. Theorem 129. Let n + |γ | − 2 < α < n + |γ |, 1 ≤ p < ||IPα ±i0,γ f ||q,γ ≤ Cn,γ ,p ||f ||p,γ ,

n+|γ | α .

For the estimate

f (x) ∈ Sev

to be valid, it is necessary and sufficient that q = depend on f .

(10.14)

(n+|γ |)p n+|γ |−αp .

Constant Cn,γ ,p does not

Proof. Necessity. Let n + |γ | − 2 < α < n + |γ |, let 1 < p < q the inequality ||IPα ±i0,γ f ||q,γ ≤ Cn,γ ,p ||f ||p,γ ,

n+|γ | α ,

and let for some

f (x) ∈ Sev

(10.15)

hold. (n+|γ |)p We show that inequality (10.15) is valid only for q= n+|γ |−αp . Let us obtain the required inequality for each term in the representation (10.6). Let us consider the extension operator τδ : (τδ f )(x) = f (δx), δ > 0. We have ⎛ ||τδ f ||p,γ



⎜ =⎝

⎞1

p

⎛

⎟ ⎜ f (δx)x dx ⎠ = ⎝δ −n−|γ | p



γ

Rn+

⎞1

p

⎟ f (y)y dy ⎠ . p

γ

Rn+

Therefore, ||τδ f ||p,γ = δ

| − n+|γ p

||f ||p,γ .

(10.16)

For (IPα+ ,γ f )(x), we obtain  α−n−|γ | y (IPα+ ,γ f )(x) = [y12 − y22 − ... − yn2 ] 2 (γ Tx τδ f )(y)y γ dy = 22n−|γ | C(γ )



K+ x n +yn

x 1 +y1

... |x1 −y1 |

K+

|xn −yn |

[y12 − y22 − ... − yn2 ] (xy)γ −1

f (δz)

n  i=1

α−n−|γ | 2

y γ dy

×

γi

zi [(zi2 − (xi − yi )2 )((xi + yi )2 − zi2 )] 2 −1 dz =

438

Transmutations, Singular and Fractional Differential Equations

{δz = s} = 22n−|γ | C(γ ) n  si

δ

i=1

δ





[y12 − y22 − ... − yn2 ] (xy)γ −1

K+

si2 − (xi − yi )2 δ2

2n−2|γ | 2n−|γ |

2

 C(γ ) K+

δ(x1 +y1 )

δ(xn +yn )

... δ|x1 −y1 |

f (s)

{δy = t} = δ

2

 C(γ ) K+

δx1 +t1

δxn +tn

... |δx1 −t1 |

δ

f (s)

2



si2 δ2

[y12 − y22 − ... − yn2 ] (xy)γ −1

δ|xn −yn |

 γi −1 2

ds = α−n−|γ | 2

y γ dy

×

γi

si [(si2 − δ 2 (xi − yi )2 )(δ 2 (xi + yi )2 − si2 )] 2 −1 ds =

α−n−|γ |

δ n+|γ |−α [t12 − t22 − ... − tn2 ] 2 δ n−|γ | (xt)γ −1

n 

δ −n−|γ | t γ dt

×

γi

si [(si2 − (δxi − ti )2 )((δxi + ti )2 − si2 )] 2 −1 ds = α−n−|γ |

C(γ ) K+

n 

(xi + yi )2 −

f (s)δ −n ×

...

i=1

|δxn −tn |

−α 2n−|γ |

δ(xn +yn )

δ|x1 −y1 |



n 

δ(x1 +y1 )

y γ dy

i=1

δ|xn −yn |

2n−2|γ | 2n−|γ |

α−n−|γ | 2

[t12 − t22 − ... − tn2 ] 2 δ |γ |−n (xt)γ −1

δx1 +t1

t γ dt

δxn +tn

... |δx1 −t1 |

f (s)×

|δxn −tn |

γi

si [(si2 − (δxi − ti )2 )((δxi + ti )2 − si2 )] 2 −1 ds =

i=1

δ −α

 2 2 2 (γ Tδx t f (t))[t1 − t2 − ... − tn ]

α−n−|γ | 2

t γ dt = δ −α τδ (IPα+ ,γ f )(x).

K+

Then (IPα+ ,γ f )(x) = δ α τδ−1 (IPα+ ,γ τδ f )(x).

(10.17)

Next, we have ⎛ γ ||τδ−1 IPα+ ,γ f ||q

⎜ =⎝

 Rn+

⎞1 q

⎟ (τδ−1 (IPα+ ,γ f )(x))q x γ dx ⎠ =

B-potentials theory

⎛ ⎜ ⎝



⎛

Rn+

δ

⎞1

⎞q



⎝

n+|γ | q

439

[y12

− y22

− ... − yn2 ]

α−n−|γ | 2

K+

q

x  ⎟ ( Ty f )(y)y dy⎠ x dx ⎠ = =t = δ x δ

γ

γ

γ

γ

||IPα+ ,γ f ||q .

Hence, ||τδ−1 IPα+ ,γ f ||q = δ γ

n+|γ | q

γ

||IPα+ ,γ f ||q .

(10.18)

Using (10.16)–(10.18), we get ||IPα+ ,γ f ||q,γ = δ α ||τδ−1 IPα+ ,γ τδ f ||q,γ = δ

n+|γ | q +α

Cn,γ ,p δ

||IPα+ ,γ τδ f ||q,γ ≤ Cn,γ ,p δ

n+|γ | n+|γ | q − p +α

n+|γ | q +α

||τδ f ||p,γ =

||f ||p,γ

or ||IPα+ ,γ f (x)||q,γ ≤ Cn,γ ,p δ

n+|γ | n+|γ | q − p +α

||f (x)||p,γ .

(10.19)

| n+|γ | n+|γ | n+|γ | If n+|γ q − p + α > 0 or q − p + α < 0, then passing to the limit at δ → 0 or γ at δ → ∞ in (10.19) accordingly we obtain that for all functions f ∈ Lp the equality

||IPα+ ,γ f ||q,γ = 0 holds, which is wrong. That means that inequality (10.19) is possible only if n+|γ | n+|γ | (n+|γ |)p Necessity is proved. q − p +α=0, i.e., for q= n+|γ |−αp .   γ γ   Sufficiency. Let x = (x2 , ..., xn ), |x | = x22 + ... + xn2 , (x  )γ = x2 2 ...xnn . Without loss of generality, we will assume that f (x)≥0 and ||f ||p,γ =1. Let 0 < δ < 1. We consider the operators  y (IPα+ ,γ ,δ f )(x) = r α−n−|γ | (y)(γ Tx f )(y)y γ dy δy12 ≥|y  |2

and

 (IPα− ,γ ,δ f )(x) =

r α−n−|γ | (y)(γ Tx f )(y)y γ dy. y

y12 ≤δ|y  |2

Let μ be some fixed real number. We introduce the notations G0δ,μ = {y ∈ Rn+ : δy12 ≥ |y  |2 , 0 ≤ y1 ≤ μ},

440

Transmutations, Singular and Fractional Differential Equations n 2  2 G∞ δ,μ = {y ∈ R+ : δy1 ≥ |y | , μ < y1 },  r α−n−|γ | (y) y ∈ G0δ,μ , + (y) = K0,δ 0 y ∈ Rn+ \ G0δ,μ ,  α−n−|γ | r (y) y ∈ G∞ + δ,μ , K∞,δ (y) = 0 y ∈ Rn+ \ G∞ δ,μ , 0 Hδ,μ = {y ∈ Rn+ : y12 ≤ δ|y  |2 , |y  | ≤ μ}, ∞ Hδ,μ = {y ∈ Rn+ : y12 ≤ δ|y  |2 , μ < |y  |},  0 , r α−n−|γ | (y) y ∈ Hδ,μ + M0,δ (y) = 0 , 0 y ∈ Rn+ \ Hδ,μ  α−n−|γ | ∞, r (y) y ∈ Hδ,μ + (y) = M∞,δ ∞. 0 y ∈ Rn+ \ Hδ,μ

In these notations we have + + ∗ f )γ + (K∞,δ ∗ f )γ , (IPα+ ,γ ,δ f )(x) = (K0,δ

(10.20)

∗ f )γ .

(10.21)

+ (IPα− ,γ ,δ f )(x) = (M0,δ

+ ∗ f )γ + (M∞,δ

To apply Marcinkiewicz’s theorem, we should prove that the operators IPα± ,γ ,δ have weak types (p1 , q1 )γ and (p2 , q2 )γ , where p1 , q1 , p2 , q2 are such that 1 q

=

1−τ q1

+

τ q2 ,

1 p

=

1−τ p1

+ pτ2 ,

0 < τ < 1. In order to do this we will be interested in the estimate of

sup λ(μγ (IPα± ,γ ,δ f, λ))1/p =   sup λ mesγ {x ∈ Rn+ : |(IPα± ,γ ,δ f )(x)| > λ} .

0 λ} mesγ {x ∈ Rn+ : |(M∞,δ

and then to apply the inequality mesγ {x ∈ Rn+ : |A + B| > λ} ≤ mesγ {x ∈ Rn+ : |A| > λ} + mesγ {x ∈ Rn+ : |B| > λ}. To estimate the generalized convolution, we will use Young’s inequality (3.178).

B-potentials theory

441

We have 

+ ||1,γ = ||K0,δ

Rn+

μ

(y12 − y22 − ... − yn2 )

α−n−|γ | 2

y γ dy =

G0δ,μ



γ



+ K0,δ (y)y γ dy =

(y12 − |y  |2 )

y1 1 dy1

α−n−|γ | 2



(y  )γ dy  = {y  = y1 z , z ∈ Rn−1 + }=

|y  |2 ≤δy12

0

μ

 y1α−1 dy1

(1 − |z |2 )

α−n−|γ | 2

(z )γ dz ≤

(1 − |z |2 )

α−n−|γ | 2

(z )γ dz =

α−n−|γ | 2





|z |2 ≤δ

0

μ

 y1α−1 dy1

0

α

|z |2 ≤1



μα



(1 − |z |2 )

1 (z )γ dz = Cα,n,γ μα ,

|z |≤1  

1 where Cα,n,γ = 21−n

α−n−|γ |+2 2



α

 n

  γ +1  i2 i=2 

α−γ1 +1 2

does not depend on δ. Therefore,

+ 1 ||K0,δ ||1,γ ≤ Cα,n,γ μα

(10.22)

+ and K0,δ ∈ L1 . + . We have Now let us consider M0,δ γ



+ ||M0,δ ||1,γ



= Rn+ 

(y  )γ dy 

|y  |≤μ



+ M0,δ (y)y γ dy



 =

(|y  |2 − y12 ) 

|y  |α−n−|γ |+γ1 +1 (y  )γ dy  

1 Dα,n,γ |y  |≤μ

α−n−|γ | 2

y γ dy =

0 Hδ,μ

y12 ≤δ|y  |2

|y  |≤μ

(y12 − y22 − ... − yn2 ) α−n−|γ | 2

y1 1 dy1 = {y1 = |y  |z1 , z1 ∈ R1+ } = γ

 (1 − z12 )

z12 ≤δ 

|y  |α−n−|γ |+γ1 +1 (y  )γ dy  ,

α−n−|γ | 2

γ

z11 dz1 ≤

442

Transmutations, Singular and Fractional Differential Equations



1 where Dα,n,γ =

z12 ≤1

(1 − z12 )

α−n−|γ | 2

γ

z11 dz1 does not depend on δ. Going over to spher-

ical coordinates y  = ρσ , we obtain + ||M0,δ ||1,γ

μ 2 ≤ Dα,n,γ

3 ρ α−1 dρ = Dα,n,γ μα , 0

3 = where Dα,n,γ

1 α





σ γ dS.

S1+ (n−1)

+ . Let us take p  such that Now we estimate the norm K∞,δ + consider ||K∞,δ ||p ,γ . Let p = 1 (i.e., p  = ∞). Then

⎛ ⎜ + ||K∞,δ ||p ,γ = ⎝ 

⎜ ⎝

⎜ ⎜ ⎝

(y12 − |y  |2 )



γ

y1 1 dy1 μ

⎛ ⎜ ⎝

 ⎟ + |K0,δ (y)|p y γ dy ⎠

∞

α−n−|γ | 2

⎟ y dy ⎠

p γ

= ⎞1/p

(y12 − |y  |2 )

α−n−|γ | 2

p

⎟  (y  )γ dy ⎟ ⎠

= {y  = y1 z , z ∈ Rn−1 + }=

|y  |2 ≤δy12 (α−n−|γ |)p  +n+|γ |−1

⎞1/p



(1 − |z |2 )

dy1

α−n−|γ | 2

p

 ⎟ (z )γ dz ⎠

≤

|z |2 ≤δ

μ



= 1. First we

⎞1/p

y1 n 

1 p

=

G∞ δ,μ

∞

+

⎞1/p

Rn+

⎛

⎛



1 p



⎛∞ ⎞1/p  γi 2+1   α−n−|γ | i=2  (1 − δ) 2 ⎝ y1(α−n−|γ |)p +n+|γ |−1 dy1 ⎠  = n+|γ  |+1 n 2  2 μ 2 (1 − δ) Cα,n,γ

α−n−|γ | 2

μ

| − n+|γ q

2−n 2 Cα,n,γ ,p =

, n 

i=2

 

γi +1 2 



((n+|γ |−α)p  −n−|γ |)1/p 



n+|γ  |+1 2

.

p Here we take into account that α−n−|γ | λ} = 0.

and mesγ {x ∈ Rn+ : (1 − δ)

Considering (10.20) and (10.21) and applying Young’s inequality (3.178), we obtain mesγ {x ∈ Rn+ : (1 − δ) mesγ {x ∈ Rn+ : (1 − δ) mesγ {x ∈ Rn+ : (1 − δ) mesγ {x ∈ Rn+ : (1 − δ) (1 − δ)

n+|γ |−α 2

n+|γ |−α 2

|(IPα+ ,δ f )(x)| > 2λ} ≤

n+|γ |−α 2

+ |(K0,δ ∗ f )γ | > λ}+

n+|γ |−α 2

+ |(K∞,δ ∗ f )γ | > λ} =

n+|γ |−α 2

+ |(K0,δ ∗ f )γ | > λ} ≤

+ ||(K0,δ ∗ f )γ ||p,γ p

λp

≤

B-potentials theory

(1 − δ)

445

n+|γ |−α 2

p

+ ||K0,δ ||1,γ ||f ||p,γ p

p

λp 1 (Cα,n,γ )p (1 − δ)

n+|γ |−α 2

p

μpα

λp C 7 (1 − δ)

n+|γ |−α 2

≤

=

1 . λq

Similarly, mesγ {x ∈ Rn+ : (1 − δ)

n+|γ |−α 2

|(IPα− ,δ f )(x)| > 2λ} ≤ C 7 (1 − δ)

n+|γ |−α 2

1 . λq

It was shown that the operators IPα± ,γ ,δ have a weak type (p, q)γ , where p and q are  (n+|γ |)p n+|γ | p(1−τ ) . Let 0 < τ < 1, p = , p ∈ 1, . The related by the equality q = n+|γ 1 1 |−αp 1−τp α     n+|γ | (n+|γ |)p1 and a weak type p , . operators IPα± ,γ ,δ have a weak type 1, n+|γ 1 |−α n+|γ |−αp1 γ

γ

Then by Marcinkiewicz’s theorem, Theorem 128, the operators IPα± ,γ ,δ have a strong   (n+|γ |)p and the inequality type p, n+|γ |−αp γ

||(1 − δ)

n+|γ |−α 2

(IPα± ,γ ,δ f )(x)||q,γ ≤ M(1 − δ)

n+|γ |−α 2

||f ||p,γ

is true. So ||(IPα± ,γ ,δ f )(x)||q,γ ≤M||f ||p,γ ,

1≤p
1



s 2, s 2

+ α,

s−γ +1 2 s−γ +1 +α 2

x −s ds.

496

Transmutations, Singular and Fractional Differential Equations

s = −2n, n = 0, 1, 2, ..., s = γ − 2k − 1, γ ∈ N,

k = 0, 1, 2, ...

Let all poles be simple. If we choose  #  1 γ > max (γ − 1 − 4α + (γ − 1)2 − 4A0 ), 0 , 2 γ = 2k + 1, then evaluation of the residues at the above poles yields    s−γ +1 s 1 1 −s 2, 2 Gα,γ ,1 (x) = 2α res x +  s s−γ +1 s=s1 (s − s1 )(s − s2 ) 2 +α 2 + α, 2   s−γ +1 s 1 −s 2, 2  s res + x s−γ +1 s=s2 (s − s1 )(s − s2 ) +α 2 + α, 2    ∞ s−γ +1 s  1 −s 2, 2  s res x = s−γ +1 s=−2n (s − s1 )(s − s2 ) +α n=0 2 + α, 2  s  s1 −γ +1 1 , 1 1 2  s 2 x −s1 + s1 −γ +1 1 (s1 − s2 ) 22α + α, + α 2   s 2 s2 −γ +1 2 , 1 −s2 2 2 + x  s s2 −γ +1 2 (s2 − s1 ) +α 2 + α, 2   ⎡ ⎤⎞ 1−γ ∞ n − n   2 (−1) ⎣   x 2n ⎦⎠ = n!(−2n − s1 )(−2n − s2 )  (α − n)  α + 1−γ − n n=0 2  s  s −γ +1 1 1 1 1 −s1 2, 2  s + x s1 −γ +1 1 (s1 − s2 ) 22α +α 2 + α, 2  s  s2 −γ +1 2 1 −s2 2, 2  s + x s2 −γ +1 2 (s2 − s1 ) +α 2 + α, 2   ⎡ ⎤⎞ 1−γ ∞ n − n   2 (−1) ⎣   x 2n ⎦⎠ = n!(2n + s1 )(2n + s2 )  (α − n)  α + 1−γ − n n=0 2  s  s1 −γ +1 1 1 1 −s1 2, 2  s x + s1 −γ +1 1 (s1 − s2 ) 22α +α 2 + α, 2  s  s2 −γ +1 2 1 −s2 2, 2  s + x s2 −γ +1 2 (s2 − s1 ) + α, + α 2 2   ⎡ ⎤⎞ 1−γ ∞ n (s  (−1) 1 + 2n)(s2 + 2n) 2 −n ⎣   x 2n⎦⎠, n! (s + 2n + 1)(s + 2n + 1) (α − n)  α + 1−γ − n n=0

1

2

2

Fractional differential equations with singular coefficients

497

where  1 s1 = (γ − 1 − 4α − (γ − 1)2 − 4A0 ), 2  1 s2 = (γ − 1 − 4α + (γ − 1)2 − 4A0 ). 2 Let us consider the last term in Gα,γ ,1 (x) as a Wright function (1.38):   ⎡ ∞ n (s1 + 2n)(s2 + 2n) 1−γ  2 −n (−1) ⎣  n! (s + 2n + 1)(s + 2n + 1) (α − n)  α + 1 2 n=0

⎤ 1−γ 2

−n

 x 2n ⎦ . (11.21)

We obtain p = 3, q = 4, 1−γ , α1 = 2, α2 = 2, α3 = −1, 2 1−γ , b1 = s1 + 1, b2 = s2 + 1, b3 = α, b4 = α + 2 β1 = 2, β2 = 2, β3 = −1, β4 = −1,

a1 = s1 , a2 = s2 , a3 =

4 

βj −

j =1

3 

αl = 2 + 2 − 1 − 1 − 2 − 2 + 1 = −1,

l=1

δ = 1. So the series (11.21) is absolutely convergent for |x| < 1 and   ⎡ 1−γ ∞ n (s + 2n)(s + 2n) − n  1 2 2 ⎣ (−1)  n! (s + 2n + 1)(s + 2n + 1) (α − n)  α + 1 2 n=0    (s1 , s2 , 1−γ  2 2 ; 2, 2, −1) −x . 3 4  (s1 + 1, s2 + 1, α, α + 1−γ 2 ; 2, 2, −1, −1)

⎤ 1−γ 2

−n

Therefore,  s  s1 −γ +1 1 1 1 −s1 2, 2  s + x Gα,γ ,1 (x) = 2α s1 −γ +1 1 (s1 − s2 ) 2 +α 2 + α, 2  s  s2 −γ +1 2 1 −s2 2, 2 x  s + s2 −γ +1 2 (s2 − s1 ) +α 2 + α, 2     (s1 , s2 , 1−γ  2 2 ; 2, 2, −1) , −x 3 4  (s1 + 1, s2 + 1, α, α + 1−γ 2 ; 2, 2, −1, −1)

 x 2n ⎦ =

498

where

Transmutations, Singular and Fractional Differential Equations

 1 s1 = (γ − 1 − 4α − (γ − 1)2 − 4A0 ), 2  1 s2 = (γ − 1 − 4α + (γ − 1)2 − 4A0 ). 2

So for x < t,

 s  s1 −γ +1  x −s1 1 , 1 1 2 Gα,γ ,1 = 2α  s 2 + s1 −γ +1 1 t (s1 − s2 ) t 2 + α, +α 2 2   s s2 −γ +1  x −s2 2 1 2, 2  s + s2 −γ +1 2 (s2 − s1 ) t +α 2 + α, 2     x2 ; 2, 2, −1) (s1 , s2 , 1−γ  2 , − 3 4 (s1 + 1, s2 + 1, α, α + 1−γ ; 2, 2, −1, −1)  t 2 x 

2

and ∞ f (x) =

Gα,γ ,1

x  t

h(t)

dt . t

x

11.3 Hyperbolic Riesz B-potential and its connection with the solution of an iterated B-hyperbolic equation M. Riesz [472,475] has created a new method for solution to nonhomogeneous linear equations by generalization of the fractional Riemann–Liouville integral. We generalize and apply this method to solution to some linear equations with Bessel operators acting by all variables. This method overcomes difficulties within the theory of differential equations which are due to the occurrence of divergent integrals. Namely, in some cases (for example, for hyperbolic equations) it is necessary to use the analytical continuation of a potential which depends analytically on a parameter.

11.3.1 General algorithm Let us start from the presentation of the general algorithm of construction of the solution to the nonhomogeneous equation Lu = f with some linear operator L. An algorithm for constructing the Riesz potential generalized by the operator L and application to the solution of differential equations with this operator L follow next. 1. An integral transform FL convenient for working with operator L is chosen (for example, FL is the Fourier transform when L = , FL is the Hankel transform when L = γ ). For suitable functions f we have FL Lf = P FL f , where P is a symbol of operator L.

Fractional differential equations with singular coefficients

499

2. The fractional negative power of L or the Riesz potential is constructed by the α α α formula L− 2 f = FL−1 P − 2 FL f . Here P − 2 can be a generalized function, for example when P is an indefinite quadratic form. 3. An integral representation of the Riesz potential for operator L is realized in the α form of the convolution I α f = (FL P − 2 ∗ f )L . The convolution (· ∗ ·)L must correspond to the chosen integral transform FL . 4. The obtained integral I α f is studied for absolute convergence for some class of functions f . It is examined at what values of α this integral converges absolutely. Other properties, such as boundedness, semigroup property, etc., can also be studied. 5. Additional conditions on the function f for which the equality I α+k Lf = I α f for some natural k (for example, k = 2 when P is a quadratic form) is true are clarified. 6. By constructing an analytic continuation (or without it if possible) it should be shown that for α = 0 the potential I α f is the identity operator I 0 f = f for some class of functions. 7. Using obtained results one can easily write a solution to the equation Lu=f for some class of functions f . It is just necessary to apply I α+k to both sides of Lu = f : I α+k Lu = I α u = I α+k f . Then putting α = 0, we get u = I k f . Here an analytic continuation I α f is used if needed. It is easy to see that using this scheme, we can also obtain a solution to the equation Lm u = f with the iterated operator L. The algorithm for constructing the Riesz potential is close to the composition method (see Chapter 6) developed by S. M. Sitnik (see [146,230,231,535]). Remark 24. It we would like to construct a solution to Lu = f with initial conditions (for example, in parabolic or hyperbolic cases), it is better to start from the case when the time variable acts by the first or the second derivative. Then, we apply a transmutation operator preserving initial conditions and obtain a solution to the problem under consideration.

11.3.2 Definition Consider the potential generalizing the Riesz potential of the form  α−n−|γ | 1 α (I f )(x) = (y12 −y22 −...−yn2 ) 2 (γ Ty f )(x)y γ dy, γ Hn,γ (α)

(11.22)

K+

where y γ =

n $ i=1

γ

yi i ,

  %      n 2α−n γi + 1 γ1 + 1 α α − n − |γ | Hn,γ (α) =  sin π   +1 , π 2 2 2 2 i=1

500

Transmutations, Singular and Fractional Differential Equations

K + ={y∈Rn+ :y12 ≥y22 +...+yn2 }, γ1 =2k −1, k∈N, and (γ Ty f )(x)=(γ1 Tx11 ...γn Txnn f )(x) is the multi-dimensional generalized translation (3.169). Operator (11.22) will be called hyperbolic B-Riesz potential. Up to a constant, this operator coincides with the first term in formula (10.3); therefore, for (11.22) the same statements about absolute convergence (Theorem 127) and about boundedness (Theorem 129) are true. The reason for considering this operator is its convenience in finding a solution to the iterated nonhomogeneous general Euler–Poisson–Darboux equation of the form m γ u = f , m ∈ N. By constructing an analytic continuation we show that for α = 0 α is the identity operator I 0 f = f for some class of functions. the potential I γ γ Note that (11.22) can be written as  | 1 α 2 2 2 α−n−|γ 2 f )(x) = (y −y −...−y ) (γ Ty f )(x)y γ dy = (I n 1 2 γ Hn,γ (α) y

y

K+

1 Hn,γ (α) 1 Hn,γ (α)

∞



γ

y1 1 dy1 0

(y12 −|y  |2 )

α−n−|γ | 2

(γ Ty f )(x)(y  )γ dy  = {y  = y1 z } =

(1−|z |2 )

α−n−|γ | 2

(γ1 ,γ Ty1 ,y1 z f )(x)(z )γ dz .

|y  |