Proceedings of the Eighth International Colloquium on Differential Equations, Plovdiv, Bulgaria, 18–23 August, 1997 [Reprint 2020 ed.] 9783112313923, 9783112302651

Table of contents :
CONTENTS
Preface
Professor Antonino Maugeri awarded with the prize of the International Colloquia on Differential Equations
50th Anniversary of Prof. P. Popivanov
Study of the Ince Equation
Trace Formula for the Sturm-Liouville Operator with Singularity
Inverse Spectral Problem for the Differential Equation of the Second Order with Singularity
Application of the Laplace, Mellin and Stieltjes Transformations in the Evaluation of Integrals
Asynchronous Multisplitting Algorithms for Differential-Algebraic Systems Discretized by Runge-Kutta Methods
Forced Oscillations of Solutions of Parabolic Equations of Neutral Type with Maxima
On the Stability of Solutions of Impulsive Parabolic Differential-Functional Equations
Approximate Analysis of Non-Cyclic Multi-Queue Systems with Batch Markovian Arrival Streams
Nonlocal Problems Modelling the Formation of Shear Bands
Solution of the Dirichlet Problem for Laplace Equation on Domains with Analytic Boundaries
Ill-Conditioning in Optimal Control Computation
Differentiability of Solutions of an Electro-Elasticity Model w.r.t. Parameters
Efficient and Accurate Local Time Stepping Algorithm for Multi-Rate Problems
Improving Laguerre's Method to Cope with Symmetry Pitfalls in Polynomial Root-Finding
A Nonlinear Wave Equation with Jumping Nonlinearity
Regular Solutions and Stability for the Nonlinear Thermoviscoelastic System with Nonlinear Damping on the Boundary
Analytical Solutions of the Navier-Stokes Equation for the Laminar and Turbulent Boundary Layer, Over a Flat Plate
Solving Singular Integral Equations
Asynchronous Multisplitting Methods with Flexible Communication for Pseudolinear P.D.E.
On the Approximation of an Infinite-Dimensional Linear Programming Problem
Partial HÖlder Continuity Results for Solutions of non Linear non Variational Elliptic Systems with Strictly Controlled Growth
A Structured Approach to Toeplitz and Toeplitz-like Linear Systems with Applications to (semi) Elliptic Problems
Finding Voxels Shared by a Ray and a Linear Octree
Existence and Uniqueness Theorem for an Oblique Derivative Problem for Nonlinear Elliptic Discontinuous Operators in the Plane
Special Functions in Fractional Relaxation-Oscillation and Fractional Diffusion-Wave Phenomena
Mathematical Modelling of Immunization Programs for Hepatitis A in Bulgaria
Complete-Integrable Calogero Dynamics for Random Matrix Theories
On Optimal Order of Approximation from Bivariate Spline Spaces
Solitary-wave Solution of Turbulence with Application to B ēnard Convection
Hermite Rational Matrix Approximation of Riccati Differential Equations
A Construction of Jordan Superalgebras from Triple Systems
Small Amplitude Fundamental Periodic Solutions on a Piecewise Linear Differential Equation
Monotonicity Methods for Stability Analysis of the Characteristic Polynomials Whose Coefficients are Polynomials
Employing Transform Orthogonality in the Determination of Integral Identities Involving Arbitrary Functions
Parallel Algorithm for Construction of Singular Surfaces in Linear Differential Games
Algorithms and Software for Selfadjoint ODE Eigenproblems
ʆ2λ-Theory for Nonvariational Basic Parabolic Systems
A New Approach to the Description of Deep Water Waves Using Fractional Derivatives
Application of a Spectral Finite Difference Scheme to a Complex Configuration
A New Minimum Curvature Multi-Step Method for Unconstrained Optimization
Towards a Real RealRAM: a Prototype Using C++
Asymptotic Behaviour of Solutions of an Impulsive Semilinear Parabolic Cauchy Problem
Error Estimates of Optimal Orders for One Dimensional Stefan Problems
Optimal Checking in Reliability Theory
Integration of Some Constitutive Relations of Plain Strain Elastoplasticity Using Block-Embedded Modified Runge-Kutta Methods
Numerical Study of the Homicidal Chauffeur Game
Nonlocal Regularization of Protter Problem for the 3-D Tricomi Equation
Blow Up and Singularities of the Solutions of Homogeneous Quasilinear Hyperbolic Systems in the Plane
Dirichlet Problem in Morrey Spaces for Elliptic Equations in Non Divergence Form with VMO Coefficients
On the Calculation of Lyapunov and Moment Lyapunov Exponents for Noise Driven Coupled Oscillators
A Unified Theory of Some Families of Elliptic Integrals Arising in Radiation Field and Other Problems
Computing Derivatives of Eigenvectors Corresponding to Multiple Eigenvalues of Nonsymmetric Matrix Functions
Error Estimate of Elliptic Integral by Chebyshev Collocation Method
Construction of Singularities for Nonlinear Wave Equations
Iterative Methods for Total Least Squares Problems
Theoretical Performance Analysis of the IQMR Method on Distributed Memory Computers with Different Network Topologies
Optimal Control of Certain M-D Systems
Optimal Control of Discrete-Time Approximations to Systems with Hysteresis

Citation preview

Proceedings of the Eighth International Colloquium on Differential Equations

A l s o available f r o m V S P Proceedings of the Seventh International Colloquium on Differential Equations Plovdiv, Bulgaria, 1 8 - 2 3 August 1996 Proceedings of the Sixth International Colloquium on Differential Equations Plovdiv, Bulgaria, 1 8 - 2 3 August 1995 Proceedings of the Fifth International Colloquium on Differential Equations Plovdiv, Bulgaria, 1 8 - 2 3 August 1994 Proceedings of the Fourth International Colloquium on Differential Equations Plovdiv, Bulgaria, 1 8 - 2 2 August 1993 Proceedings of the Third International Colloquium on Differential Equations Plovdiv, Bulgaria, 18 - 23 August 1992 Proceedings of the Third International Colloquium on Numerical Analysis Plovdiv, Bulgaria, 1 3 - 1 7 August 1994 Proceedings of the Second International Colloquium on Numerical Analysis Plovdiv, Bulgaria, 1 3 - 1 7 August 1993 Proceedings of the First International Colloquium on Numerical Analysis Plovdiv, Bulgaria, 1 3 - 1 7 August 1992

Proceedings of the Eighth International Colloquium on Differential Equations Plovdiv, Bulgaria, 18-23 August, 1997

Editor: D. Bainov

///VSP/// Utrecht, The Netherlands, 1998

VSP BV P.O. Box 346 3700 AH Zeist The Netherlands

© V S P BV 1998 First published in 1998 ISBN 90-6764-279-7

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

Printed in The Netherlands

by Ridderprint

bv,

Ridderkerk.

CONTENTS Preface

xi

Professor Antonino Maugeri awarded with the prize of t h e International Colloquia on Differential Equations 50th Anniversary of Prof. P. Popivanov D. Bainov

xiii xv

Study of the Ince Equation M. Amar and B. Tounsi

1

Trace Formula for the Sturm-Liouville Operator with Singularity R. Kh. Amirov and Y. Qakmak

9

Inverse Spectral Problem for the Differential Equation of the Second Order with Singularity R. Kh. Amirov and S. Giilyaz

17

Application of t h e Laplace, Mellin and Stieltjes Transformations in the Evaluation of Integrals A. Apelblat

25

Asynchronous Multisplitting Algorithms for Differential-Algebraic Systems Discretized by Runge-Kutta Methods J. Bahi and J.-C. Miellou

35

Forced Oscillations of Solutions of Parabolic Equations of Neutral Type with Maxima D. Bainov and E. Minchev

43

On the Stability of Solutions of Impulsive Parabolic Differential-Functional Equations D. Bainov and E. Minchev

51

Approximate Analysis of Non-Cyclic Multi-Queue Systems with Batch Markovian Arrival Streams D. Baum

57

Nonlocal Problems Modelling the Formation of Shear Bands J. Bebernes

67

Solution of the Dirichlet Problem for Laplace Equation on Domains with Analytic Boundaries S. Belov and N. Fujii

73

vi

Contents

Ill-Conditioning in O p t i m a l Control C o m p u t a t i o n F. Benyah and L. S. Jennings

81

Differentiability of Solutions of an Electro-Elasticity Model w.r.t. Parameters N.D. Botkin

89

Efficient and A c c u r a t e Local T i m e Stepping Algorithm for Multi-Rate Problems L. Cao and J. Zhu

97

Improving Laguerre's Method to Cope with S y m m e t r y Pitfalls in Polynomial Root-Finding Tien Chi Chen

105

A Nonlinear Wave Equation with J u m p i n g Nonlinearity Q.H. Choi and T. Jung

111

Regular Solutions and Stability for the Nonlinear Thermoviscoelastic System with Nonlinear D a m p i n g on the Boundary A.T. Cousin, C.L. Frota and N.A. Lar'kin

119

Analytical Solutions of t h e Navier-Stokes Equation for t h e Laminar and Turbulent B o u n d a r y Layer, Over a Flat P l a t e L. D'Attorre

127

Solving Integral Equations J.H. de Singular Klerk

137

Asynchronous Multisplitting Methods with Flexible Communication for Pseudolinear P.D.E. D. El Baz, P. Spiteri and J.-C. Miellou

145

On the Approximation of an Infinite-Dimensional Linear Programming Problem M. H. Farahi

153

Partial Holder Continuity Results for Solutions of non Linear non Variational Elliptic Systems with Strictly Controlled Growth L. Fattorusso and G. ¡done

161

A Structured Approach to Toeplitz and Toeplitz-like Linear Systems with Applications to (semi) Elliptic Problems G. Fiorentino

169

Finding Voxelsand Shared by a Ray and a Linear Octree I. Gargantini J-Y Liu

179

Existence and Uniqueness T h e o r e m for an Oblique Derivative Problem for Nonlinear Elliptic Discontinuous Operators in the Plane S. Giuffre

187

Contents

vii

Special Functions in Fractional Relaxation-Oscillation and Fractional Diffusion-Wave Phenomena R. Gorenflo, F. Mainardi and H.M. Srivastava

195

Mathematical Modelling of Immunization Programs for Hepatitis A in Bulgaria D. Greenhalgh

203

Complete-Integrable Calogero Dynamics for Random Matrix Theories H. Hasegawa

211

On Optimal Order of Approximation from Bivariate Spline Spaces D. Hong

219

Solitary-wave Solution of Turbulence with Application to Bënard Convection K. Ishibashi, S. Tsugë and T.M.S. Nakagawa

227

Hermite Rational Matrix Approximation of Riccati Differential Equations L. Jodar and E. Defez

237

A Construction of Jordan Superalgebras from Triple Systems N. Kamiya and S. Okubo

243

Small Amplitude Fundamental Periodic Solutions on a Piecewise Linear Differential Equation K. Katori, M. Otake, Y. Manome and T. Mishima

247

Monotonicity Methods for Stability Analysis of the Characteristic Polynomials Whose Coefficients are Polynomials T. Kawamura and M. Shima

253

Employing Transform Orthogonality in the Determination of Integral Identities Involving Arbitrary Functions V. Kowalenko

261

Parallel Algorithm for Construction of Singular Surfaces in Linear Differential Games S.S. Kumkov

and V.5. Patsko

275

Algorithms and Software for Selfadjoint ODE Eigenproblems C.R. Maple and M. Marietta

285

2,A £M. -Theory and for A. Nonvariational Basic Parabolic Systems Marino Maugeri A New Approach to the Description of Deep Water Waves Using Fractional Derivatives K. Matsuuchi

301

Application of a Spectral Finite Difference Scheme to a Complex Configuration Y. Mochimaru

311

293

Contents

A New Minimum Curvature Multi-Step Method for Unconstrained Optimization I.A.R. Moghrabi and S.E. Obeid

319

Towards a Real RealRAM: a Prototype Using C + + N. Th. Müller

327

Asymptotic Behaviour of Solutions of an Impulsive Semilinear Parabolic Cauchy Problem K. Nakagawa

335

Error Estimates of Optimal Orders for One Dimensional Stefan Problems M.-R. Ohm

341

Optimal Checking in Reliability Theory E. Omey

347

Integration of Some Constitutive Relations of Plain Strain Elastoplasticity Using Block-Embedded Modified Runge-Kutta Methods G. Papakaliatakis

and T.E. Simos

355

Numerical Study of the Homicidal Chauffeur Game V.S. Patsko and V.L. Turova

363

Nonlocal Regularization of Protter Problem for the 3-D Tricomi Equation N.I. Popivanov and M. Schneider

373

Blow Up and Singularities of the Solutions of Homogeneous Quasilinear Hyperbolic Systems in the Plane P. Popivanov and A. Slavova

379

Dirichlet Problem in Morrey Spaces for Elliptic Equations in Non Divergence Form with V MO Coefficients M.A. Ragusa

385

On the Calculation of Lyapunov and Moment Lyapunov Exponents for Noise Driven Coupled Oscillators N. Sri Namachchivaya and H.J. Van Roessel

391

A Unified Theory of Some Families of Elliptic Integrals Arising in Radiation Field and Other Problems H.M. Srivastava

399

Computing Derivatives of Eigenvectors Corresponding to Multiple Eigenvalues of Nonsymmetric Matrix Functions R. C. E. Tan and A. L. Andrew

409

Error Estimate of Elliptic Integral by Chebyshev Collocation Method K. Tsuji

415

Construction of Singularities for Nonlinear Wave Equations M. Tsuji

423

Contents

Iterative Methods for Total Least Squares Problems T. Yang Theoretical Performance Analysis of the I Q M R M e t h o d on Distributed Memory C o m p u t e r s with Different Network Topologies T. Yang and H.-X. Lin O p t i m a l Control of Certain M-D Systems S.A. Belbas O p t i m a l Control of Discrete-Time Approximations t o Systems with Hysteresis S.A. Belbas and I.D. Mayergoyz

Preface The Eighth International Colloquium on Differential Equations was organized by the Institute for Basic Science of Inha University, the International Federation of Nonlinear Analysts, the Mathematical Society of Japan, Pharmaceutical Faculty of the Medical University of Sofia, University of Catania and UNESCO, with the cooperation of the Association Suisse d'Informatique, the Canadian Mathematical Society, Kyushu University, the London Mathematical Society, Technical University of Plovdiv, and it was partially sponsored by the Bulgarian Ministry of Education, Science and Technologies under Grant MM-702. The Colloquim was held on August 18-23, 1997, in Plovdiv, Bulgaria. The subsequent colloquia will take place each year from 18 to 23 of August in Plovdiv, Bulgaria. The publishing of this volume is fully supported by the Bulgarian Ministry of Education, Science and Technologies under Grant MM-702. Organizing

Committee:

D. Bainov (Chairman), Q. H. Choi, J.Diblik, S. Hristova, J. Kajiwara (Vice Chairman), N. Kitanov (Secretary), D. Kolev (Secretary), V. Lakshmikantham, M. Marino, A. Maugeri (Vice Chairman), E. Minchev (Secretary), N. Popivanov, P. Popivanov, M. Robnik, M. Sambandham, H.M. Srivastava, R.U. Verma. Scientific

Committee:

D. Bainov (Bulgaria), C. Dafermos (USA), G. Da Prato (Italy), H. Fujita (Japan), R. Ivanov (Bulgaria), J. Kajiwara (Japan), A. Kaneko (Japan), N. Kenmochi (Japan), V. Lakshmikantham (USA), A. Maugeri (Italy), E. Minchev (Bulgaria), N. Pavel (USA), N. Popivanov (Bulgaria), P. Popivanov (Bulgaria), M. Robnik (Slovenia), M. Sambandham (USA), H.M. Srivastava (Canada), Fr. Stenger (USA), E. Titi (USA), M. Tsutsumi (Japan), R.U. Verma (USA). The address of the Organizing Committee is: Drumi Bainov, P.O.Box 45, 1504 Sofia, Bulgaria. The Editor

The Scientific and Organizing Committees of the E I G H T H I N T E R N A T I O N A L C O L L O Q U I U M ON D I F F E R E N T I A L EQUATIONS, PLOVDIV, BULGARIA, AUGUST 1 8 - 2 3 ,

1997

adjudged the prize of the International Colloquia on Differential Equations

to Professor Antonino Maugeri University

of Catania,

Italy.

The prize is awarded for the whole scientific activity and research of the recipient, and for his special merits in the organization and yearly carrying out of the Colloquia on Differential Equations.

50th Anniversary of Prof. P. Popivanov DRUMI

BAINOV

Higher Medical Institute, P.O. Box 45.

Sofia

1504

Bulgaria

The m e m b e r of the Organizing C o m m i t t e e of the International Colloquia on Differential Equations in Plovdiv — Bulgaria, Prof. Petar Popivanov, D.Sci., corr. member of the Bulgarian Academy of Sciences completed 50 years.

About 25 years of his life

he devoted to the Partial Differential Equations ( P D E ) , working in the domains of general theory of linear P D E , linear and nonlinear microlocal analysis, degenerate boundary value problems for semilinear elliptic equations, Monge-Ampere type equations, Cauchy problem for nonlinear hyperbolic equations and systems, P D E on the torus. Since 1990 he is regularly and actively participating in the activities of our International Colloquium on Differential Equations. Many of his new results were published for the first time in the Proceedings of the Colloquium - see [1,2,3,4] and more t h a n 12 scientific communications were given by him. Prof. P. Popivanov was a chairman of the Opening Ceremonies of several conferences, chairman of many round tables on Impulsive Differential Equations, on different aspects of the P D E and on new trends in the theory of Differential Equations. W i t h his talks, tact and authority he contributed too much in organizing of 8 conferences on Differential Equations on a high professional level. Despite the heavy crisis in our country each year more than 200 participants are proposing their new achievements in t h e domain of Differential Equations. As the biggest part of t h e m are coming f r o m abroad t h e Colloquia in Plovdiv give best possibilities for international scientific contacts and scientific cooperation.

D. Bainov

xvi

It is a pleasure for me as a chairman of the Organizing Committee and on behalf of it to congratulate Prof. P. Popivanov with his anniversary and to give him my best wishes for new successes in the domain of PDE.

References [1] P. R. Popivanov, Removable singularities of the solutions of fully nonlinear systems of partial differential equations, Proc. of the 2 Coll. on Diff. Eq., World Scientific, (1992), pp. 185-190. [2] P. R. Popivanov, Nonexistence of isolated singularities for semilinear systems of partial differential equations, Proc. of the 5 Coll. on D i f f . Eq., YSP, (1995), pp. 283286.

[3] P. R. Popivanov, Microlocal hypoellipticity of some classes of overdetermined systems of pseudodifferential operators, Proc. of the 4 Coll. on Diff. Eq., VSP, (1994), pp. 229-233. [4] P. R. Popivanov, D. K. Palagachev, On a degenerate boundary value problem for second order quasilinear elliptic operators, Proc. of the 6 Coll. on Diff. Eq., VSP, (1996), pp. 197-208. [5] P. R. Popivanov, D. K. Palagachev, The Degenerate Oblique Derivative Elliptic and Parabolic Equations,

Akademie-Verlag, Berlin, (1997).

Problem for

8th Int. Coll. on Differential Equations, pp. 1-8 D. Bainov (Ed) © VSP 1998

Study of the Ince Equation Makhlouf A M A R and B a h l o u l

Institute

of Mathematics,

University

TOUNSI

of Annaba,

B.P.

12, El-Hadjar,

Annaba,

Algeria

Abstract. In this work, we study the Ince equation: (*)

cos(v)y"(u) + psin(u)y'{u) + qcos(u)y(v)

= 0.

First, we give the fundamental solutions of (*) in the intervals

— + kir,— +

Gauss-Equation.

using the

We study the conditions for which (*) has the solutions: (CQ(V), 5 0 (f)), n n ( C > ) , 5 Î M ) , (Co(yj, Si(yj), where C^y) = £ A2p cos(2pu), S&j = £ B2p sin(2pu),Cjy) = p= 1 p=0 n n Y] A2p+i cos((2p+l)u), 5 i ( f ) = B2p+1 sin((2p+l)f). We give the necessary and sufficient conp=° p=i ditions for the equation (*) to have solutions ^Si(f), t/Si{v) + , (Co(V), uC0(u) + So(u)J and the closed, truncated solutions. Keywords: Ince equation, Gauss hypergeometric equation.

Introduction We consider the Kepler problem: r = — r'

r

= r(i),

r(0) = 1,

r(0) = 0.

The normal variational equation of this problem is the singular Hill equation:

A t ) § + m = o. By putting: r(t) = cos2(u),

£(i) = Z(v), this equation becomes:

cos2{v)Z"{v)

+ 2 sin(f) cos(u)Z'(u)

+ 2\Z{v)

= 0,

this equation is a particular case of a general equation which, we resolve: + [i? + gcos 2 (i/)] Z(v) = 0. We consider the following singular Ince Equation: (1)

[l + cos(2f)j Z"{u) + B sin(2u)Z'(u) + [C + D cos(2i/)] Z(u) = 0,

2

M. Amar and B.

Tounsi

where B, C, D are reals. It can be written in the form cos2(i/)Z"(i>) + P s i n ( i / ) c o s ( i / ) Z V ) + [R + Q cos2(i>)] Z{v) = 0,

(2)

where P = B, Q = D, R =

C-D

P r o p o s i t i o n 1. If Z(v) is a solution [cos(f)]**j/(i') verifies the equation:

of (2) then the function

y(v)

defined by Z{v)

=

c o s 2 ( i / ) < / » + Pi sinfi/) c o s [ v ) y ' ( v ) + [Ri + Qi cos 2 (i/)] y{v) = 0,

(3) where

Px = —2/i + P, Qi = -n2

(4)

+ Pn + Q,

Rx = fi2 - {P + 1)/^ + R.

•

Proof. By substitution. From equations (4), we deduce the differential system: dP1 dfj,

=

- 2

dQ

(5)

d/j.

A(0)

= P,

Qi(o)

= Q, = R.

This system has the next two first integrals: Xl

(6)

= (Pi + l)2 - ±Ri = (P + l)2 -

4R,

X2 = Pi + 4 Qx = P2 + 4 Q.

P r o p o s i t i o n 2 . If Xi > 0, then the function Z(v) = (cos(i/)) w j/(i/) where p. is a of the equation R\ — 0, transforms (2) to the equation:

s(f)i/"(f) + psin(f)y'(i/) + qcos(i/)y(v)

(7)

where p = P — 2/x, q = Q + P/i — ¡j,2. The equation equation.

solution

= 0,

(7) is called the reduced singular

Ince

Proof. = 0 implies /i2 - (P + l)fi + R = 0. We have A = (P + l) 2 - 4R = Xi- If Xi > 0 then it exist ^ such that Ri = 0, so the equation (2) becomes (7). • We consider a solution y(v) of (7). It is easy to verify that the function y(v) defined by: y(u) = [cos(t/)] P+1 y(i/),

v €

0

,

verifies the equation: (8)

cos(i/)j/"(i/) + Pi s i n ( f ) y ' ( i / ) + qi cos(i/)y(i/)

where pi = -p - 2, qi = q - [p + I).

= 0,

Study

/

of

7T

the

Ince

7T

S o l u t i o n of ( 7 ) in Ik = \

3

Equation

\

+ kit, — +

kirJ.

[c -

1 )z]y'{z)

We consider the Gauss equation: (9)

z( 1 -

where c = a + 6 +

+

z)y"

(a

+

b +

-

=

aby(z)

0,

which possesses the solution:

Fj(z)

=

F ( a , b,c,z)

ab a ( a + 1 ) 6 ( 6 + 1) , 1 + —z + I c ( c ^ i } V + ••• ,

=

F 0 2 (z) = z ^ ' F i a + l . c . b + l . z ) , F } { z )

=

F [ a , b , a +

F

=

(1

2

{ z )

-

b + l - c , l - z ) ,

z)c~a~bF(c

If we put 2 = cos J (i/), v 6

-

, c - b , c + l - a , l - z ) .

a

—, —) and Y(z)

= y{v), then the equation (9) becomes

[1]: (10)

cos(v)yH +

psm(i/)y' +

— 0

q cos(i/)y

with p = — 2(a + 6), q = - 4 a 6 . S o l u t i o n in t h e interval Iq. T h e equation (7) has two fundamental solutions f(i/)

and g(v) where v £

—, — ) such

that /(0) = l , / ' ( 0 ) = 0, (-2)

= 0,

B2( 2) + B44>(-4)

= 0,

- 2) = 0,

B2n(2n) + B2n+2(-2n

where if>(x) = x2 — px — q. We suppose that CQ(V) is closed with A2n+2 = 0. Then 4>(2n) = 0 and B2n+2 = 0 with 4>{-2n') = 0 for n' < n. Calculation of Xi and X2 gives 4>(2n) = (—2n') = 0 and thus p = Xi + X2 = 2 ( n — n') > 0 and q = —X\X2 = Ann'. Therefore, X, = (p + l ) 2 = [2(n - n') + l ] 2 = N'\

= [2(n + n ' ) ] ' =

N2.

Reciprocally, we consider N' = 2I 0 .

P r o p o s i t i o n 8 . If Xi = (2k' + l)2 and X2 = (2&)2 such that k < k' and k and k' have different parity, then the equation (7) has the two linearly independent solutions: yi(u) = 5 i ( v ) , y2(u) = uSi(v) + Ci(v).

M. Amar and B. Tounsi Proof.

W e consider a solution y(v)

of ( 7 ) and put y(v)

verify that y(i/) is a solution of ( 7 ) if Z(v) (11)

= vy(v)

+ Z(v).

It is easy to

verifies:

c o s ( i ' ) Z " + p s i n ( i / ) Z ' + qcos(i/)Z

= —2cos (v)y'

— psin(f)j/.

If {2n — 1) = 0, then (7) admits the solution: n

y(v)

= £ 5 2 p _ a s i n [ ( 2 p - 1)./]. P= I

If we put cos(i/)Z" + psin(i/)Z'

= M0 + M2 cos(2i/) + . . . + M2ncos(2nv)

+ qcos(v)Z

+ ... ,

by identification we have:

^

»

(

f

-

^

x

-

d

+

a ) ^

(12) (2m - 1 ) ) fl2ra_, -

M2m

M2n

If B2n+i B2n4.i

= ( J - (2n - l ) j B2n-i

= 0 then M 2 n =

-

( | + (2m + 1 ) )

( J + (2n + 1)J

fl2m+1,

B2n+l.

— (2n — l ) j 5 2 „ _ i and for m > n we have M2m

= B2n4.2 = • • • = 0. W e look for Z in the form: Z = C i ( f ) = J2 R2p+1 cos((2p + 1)1/). p=0

W e substitute Z in (11) and we identify - ^ ( - 1 ) ,

Mo = M2

= -1-(R^(l)

+

R3^2 — 0 because of (2ri — 2) ^ 0. T h e equation A2n^4{2ri - 4) + A 2n '_ 2 (2n' — 4) ^ 0 then A2n*^4 = 0. Proceeding in the same fashion, we find that: A2n'-6 = ^2n'-s = • • • = A4 = A2 = A0 = 0. To prove that A2n+2 = A2n+4 = ... = 0 we have A2n{2n) + A2n+2(—2n — 2) = 0, or we have 4>(2n) = 0. Then A2n+2 = 0 because A2n>-2

8

M. Amar and B. Tounsi

of (-2n - 2) yi 0. From the equation A2n+2(2n + 2) + A2n+4(-2n - 4) = 0 we have A2T1+4 = 0. In the same fashion we find A2n+2 = ->4.271+4 = . • • = 0. Calculation of the coefficient A2ni+2k '• From the equation A2n'^(2n') +A 2n i +2 (—2n'— 2) = 0 we have A2n,+2 p = In — 2n', q = inn'.

= -

— W e have (2n') = (2n') 2 - 2n'p - q, where . n + n' + 1

Generally, -4>(2k) =

,

{—2k — 2)

(n-A;)(n'+fc) =

,

(r» + Jfc + l)(Jfc + l - n ' )

'

We find that _ 2n'(2n' + l ) ( 2 n ' + 2 ) . . . ( 2 n ' + f c - l ) ( n - w , ) ( n - n ' - l ) . . . 2n

'+2t_ _ _

(n-n'-k+1)

(n+n'+l)(n+n'+2)...(n+n'+Jfc).1.2.3...Jfe

2n'

(2n' + k — l ) ! ( n — n')!(n + n')! (2n' - 1 )\k\(n - n> - k)\{n + n'+

k)}

2n''

where 2n' + 2k < 2n, k € N". By using an analogous proof, we obtain the proposition.

•

P r o p o s i t i o n 1 1 . Suppose that {2n + 1) = +1 cos((2n + l)i/) + . . . + A2n+1 cos((2n + l)t/), Si{i>) = B2n,+!

sin((2n' + l)i/) + . . . + B2n+l

sin((2n + 1)„).

References 1. M. Amar, Equation de Hill Dans le Domaine Complexe et Application à L'étude de L'équation aux Variations Normale au Voisinage des Solutions Périodiques Dans un Potentiel Homogène, Thèse d'état soutenue à l'université de Annaba en Septembre (1991). 2. M. Amar, C . R . Acad. Sci., Paris, série 2, 313, 1 8 3 - 1 8 6 (1991). 3. H. Yoshida, Celestial Mechanics, 40, 5 5 - 6 1 (1987).

8thlnt. Coll. on Differential D. Bainov (Ed) © VSP 1998

Equations,

pp. 9-16

Trace Formula for the Sturm-Liouville Operator with Singularity R . K H . A M I R O V AND Y . Q A K M A K Cumhuriyet University, Dept. of Mathematics,

Abstract.

58140,

SIVAS-TURKEY

Let ¡ii, fj.2, • • • > ^n, • • • be the Dirichlet spectrum of the operator —d2/dx2

acting on £ 2 ( 0 , n).

In the special case where q(x) = 0 , / i n = n 2 .

+

In the [ 1 ] and others

discovered the asymptotic formula

1 f Hn = n2 + - / q(x)dx + * Jo

0(n"2)

and the t r a c e formula

Er[tin-n

21 g(0) + j = -

,

n

provided that

/

= 0, where 9 ( 1 ) £ C 2 [ 0 , tt). These are beautiful formulas with many

q(x)dx

Jo

applications for example in solving inverse problems. In this work, the above mentioned problem has been studied for a Sturm-Liouville operator with A/x

( A is real ) singularity at 1 = 0.

K e y w o r d s : Trace formula, spectrum. A M S s u b j e c t classifications. 35J10, 35J40, 35K35

I n t r o d u c t i o n : Learning about the mation of electrons moving under Coulomb potential is of significance in quantum theory. Solving these type of problems provides us to find energy levels of not only Hidrogen atom but also single valance electron atoms such as Sodium. e2 For Hidrogen atom, Coulomb potential is given by U = , where r is the radius r of the nucleus, e is electronic charge. Accordingly, we use time dependent Schrodinger equation h—

%

dt

-

~

tC~ 9 1 ^ —

2m

dx C-2

+ U(x,y,z,t)*,

i

/

JR 3

n

|

|- dxdydz

= 1.

Here,*!' is the wave function, h is Planck's constant and m is the mass of electron. Here, of we make the necessary transformations we can get a Sturm-Liouville equation with Coulomb potential, —y" + ^

1- q(x)j

y = \y where A is a parameter which corresponds

to the energy. As for the class of function on which this type of singular expressions are defined y (0) is infinite, the class satisfying discrete boundary conditions y (0) = 0, y (k) - Hy{v) = 0

q(x)

10

R. Kh. Amirov and Y.

Qakmak

will be taken as the domain for the differential expressions under study. In consequence,

+

-y L =

+

= Ay 1/(0)

y »

= 0

- Hy(ir)

=0.

T h e domain of operator L is taken as = {y\Ly e L2[0,tt],

D(L)

q(x)

e L2{0,it},

y ( 0 ) = 0, y'(n)-Hy{%)

A, H real const.,

= 0}.

Eigenfunctions and eigenvalues of the operator L are given in the [1], Now,let A : H —• H be an operator. H is a Hilbert space and as it is separable we can use orthonormal systems. { e n } 6 His an orthonormal system and ||en||tf = 1. If we take { e n } as the eigenfunctions of A , (Aen = Xnen), we find oo

oo < Aen, en>

oo

= J 2
= ^

en > =

If ^ A n < +oo then it is the trace of A, shortly tr(A) n=1 As L : ¿2[0,7r] — • ¿2[0,7t] and is a Hilbert space, orthonormal system. Taking Lyn = A n y n , one obtains oo

oo

n=l

= ^ An. n=l can be thought of an

oo < ^nVn, Vn > =

< Lyn, yn > =

n=l

An.

oo K

< Vn,yn > =

n=l

n=l

However,this series is not convergent. Thus, we search for the regularized trace.To this end, we take out some terms from A„ . These traces are important in solving inverse problems with two spectrums. C a l c u l a t i o n of the regularized t r a c e of S t u r m - L i o u v i l l e operator with C o u lomb singularities. Given the differential equation +

+

=

(1)

2/(0) = 0 *

(2)

y'(*)-Hiy(*)=0

(3)

(y'OO - Hoyjn) = 0) where q(x)

(3')

€ C"[0,7r]; A, Hi and Ho real numbers.

Let { A n } and

{nn}respectively

represent the spectrum of ( l ) - ( 2 ) - ( 3 ) and ( l ) - ( 2 ) - ( 3 ' ) . F o r the problem of (l)-(2)-(3), the following theorem has been proved. T h e o r e m : If 9 ( 1 ) € C 2 [0,7r] then ^

,,

^

AHlnn

AN

2

+

AN

?(t)-9(0)

1 ("l

A

f •Kq(t)dt - AM

2?r ~ 7T L2 Jo

+

// f

A t I t J

-

Alnn

£

id)dt -

A M

-

H

)-

H

M J

AHN

Alnir

„

11

Trace Formula.

Here M, N and S are known constants,// = Hi — Ho • P r o o f : To find them we notice that the above summation is convergent and tp(x, A', is an entire function of ^ order with respect to A parameter. Weierstrass theorem gives 0 is satisfied then we can use the equation tp'(ir, —y) — Hip(n, —fx) = A$(—fi) . First, we find the asymptotic of ${—fJ.) .

°° / a\ = n=0 n \ 1 - r J)

°° / a.\ = n 0 Vi 1 + f ) = n=

n i n=0 V

^

i

n ( i + t ) -cosh vn*

TT / }

An + fJ. cosh y/Jhr n H + (n + 1 / 2 ) 2 ) n= o

'(n + 1 / 2 ) 2 > $(—fj.) = Ctf(fi) cosh ^JJiir , where c = J J ( ). is given by n=0 (n + 1/2) - An An +M 1 and ^(/i) is convergent. There= n n 2 ji + ( n + l / 2 ) 2 n=0 ¿1 + (n -h 1/2) fore, (n + 1/2)2 - An In 9 M In 1 |i + (n + l/2) a n=0 i n

1 -

1/4-A0^ M + 1/4

1 -

(n + 1/2)2 - An /i + (n + 1/2)-

Combining these, we find • t/ \

„ ^Inn As > — i s n

A0 - 1/4

+

£ln

(n + 1/2)2 — A„ /I + (n + 1/2)2

convergent, above series is convergent , too. For n —> 00 , general sum

goes to zero, enabling us to perform the following operators. We obtain Ap - 1/4

^

Ao - 1/4 _ ^

1 ((n + l/2)*-\n\k

1^

/(n + l / 2 ) 2 - A n y

^ ( 1

( 1_

Let us take the expression I - — — — ^ | . Adding and substracting the term /t + (n + 1/2)-

12

R. Kh. Amirov

W e

(n +

1/2)

4- - l n ( n + 7[

2

( n 4- 1 / 2 )

2

+

-

H +

1/2)

2

H

i » * 0 0

=

x

1/4

g

E fc= 1 "

g

-

OO

/>,

E E ^ E n = l fc=l K > = 2

+

0 ( M "

2

1/2)

we

*

2

(n

l/2)

2

+

n

take

F r o m (a

A„

M

1/2)

V

2

/

4- ^ l n ( r a +

2

V

+

2

(n +

Aq —

l n ( n 4- 1 / 2 ) (n +

1/2)

w

2

(n +

l/2)

+

1/2)

1/2) -

A

1/2) —

k-j

O

h

A„

2

k—1 n

4- ^ l n ( n 4- 1 / 2 ) — 1/2)

n

2

A

l n ( n 4-

7T n 4 - ( n 4 -

(n +

A

1/4

1/2)2

fi +

\J

4-1/2)

4- ^ l n ( n 4- 1 / 2 ) —

/i +

à ln(

1 / 2 )

y^i +

(n +

( n 4- 1 / 2 )

4- 1 / 2 )

+

2

1/2) 1/2)

A„

2

l n ( n 4W

2

l u

4-(n

1/2

4-1/2

)

" ( n 4 - l / 2 ) N o w

l n ( n

M +

ln(n +

+

1/2)

have

fk

fi + (n +

*

A

l n ( n 4- 1 / 2 ) —

rt=l

(n 4-1/2)

1

A ,

n + (n + 1/2)

*

- l n ( n -I7T

r E E n=lj=0

/ l y ' /

*=1

OC

2

-

ir/j 4 - ( n

7T

1 ~

00

OO

1/2)

H + (n+

C

fc= 1 n = 1

n

2

I n s e r t i n g its p l a c e in l n ^ ( ^ i ) ,

Ap -

A

1/2) -

1/2)

( n +

-

(n +

ln(n +

( n 4- 1 / 2 )

s

Çakmak

obtain

/x +

•

and Y.

the expression

—

2

+ ^ l n ( n 4 - l / 2 ) - A

/i +

the eigenvalue expression

( n 4-

we know

that

n

"

l n ( n 4-

1/2)

2

| A

— ^ n 4- - ^

n

/i +

(n +

k-1 E 71=1

2

4 — ln(n 4-1/2) 7T

H 4- ( n

4-1/2)

2

—

A„

l n ( n 4/i +

( n +

1/2

i:'

I n ( n 4- £ ) •
( n + 1/2) H— 7T n=l

/j + (n

Jt= 1 i=2 V ln(n

^

/ n — An

+ 1/2)

+ 1/2)

A0

- 1/4

^TT uk k= l71 ^ n= 1

=

_ y , /(n

+ 1/2)2

2-1

oc , (n

,

- E r £ k=2 * n = l

+ ^ln(n ^ + (n

+ 1/2)

1

fi + (n

(n

H

1/2)2

7T

/

+ 1/2)2+

- E E - è f f c n = l k = l k j=1 V 0

+

+0

_ A 1 ^

J

+ l/2)2 + -ln(n + l/2)-A n N

1

+Ai-Li/4 +

- Xn\

+ 1/2)2

j4\jY ln(n -f 1/2) 7T ) U + (r.

+ 1/2)2

— n-.

ln(n

+ l/2;

! + (=±^S)2

^Aa*-'^

ln(n + 1/2)

èi ^

, /n + 1/2

A»* ¿1

%/f

+1/2)-An

ln(ra + 1/2)

M + (n + 1/2)2

W

\ p — (n + 1/2J-

/ n Vf/

Therefore,

n=l » &

1 ~ /(n+l/2)a + £ln(n + l/2)-A„y k

ntî V

M + (n + 1/2)2

J

"

V

-r 1/2)2

Aa'-1 "

A. v f

£

ln(n - 1/2)

1 +

vf

14

R. Kh.

-

00 00 1 A E E ^K E n = l k = l j=2

and

Y.

Qakrnak

(n + l / 2 ) 2 + ^ l n ( n + l / 2 ) -

(k\

M

V

m

Amirov

V

+(n+1/2)

An1

2

l n f n + 1/2)

A\>

(

TtJ

\ n + (n + 1/2)2

)

M a k i n g suitable addings a n d substractions to this expression

ln*(M) = £

L

-

(n + 1/2)2 _ £

l n 2 ( n + 1/2) ~

71

(n + 1/2)2

, n

y °

, ,

n

_ 2 ^ " + / f (n + 1 2

-

7~TT (n + 1

l n ( n + 1/2)

2 +

'¿l(n+l/2)(/x +

1

(n+l/2)») l n 2 ( n + 1/2)

,

( n + 1/2)(/x + ( n + 1 / 2 ) 2 )

^

l n ( n + 1/2)

\ t l

2cq

1

, o,

y*

_

1 / 2 )

( n + 1 / 2 ) 2 J /i + ( n + 1 / 2 ) 2

1

^ + 7

+

l n ( n + 1/2) \ 72

"

t l M + (n + l / 2 )

^

l n ( n

( „ + 1/2)2(M + (n + 1/2:-)

Ap -

2

(n + l / 2 ) ( / x + (n + 1/2)2) +

1/4

Aln^JI +

^

2

^

If n e c e s s a r y processes a r e done,

1 00 In ¥ ( / . ) = _ £

(An -

(n + 1/2)2 -

ln2(N + 1/2)

2C2

1

fj,

(Ao -

71

1

- H 4

A

2i

(n + 1/2)

In2(n + 1/2) _

. o

2c0

- 2

(n + 1/2)2

2

- iln("+1/21 -

H + (n + 1/2)

, 2

9

„ v^

+

~(n + 1/2)

271 I n 2 2 + 471 In 2 *

2c

°-

(n+1/2)2

( n + 1 / 2 ) 2 J ¿t + ( n + 1 / 2 ) 2

1

4- 1 /91)

2"

Akk=0An'

These formula and (1.15) imply that

_^ . w « « (

\ ^

k=o\

_ oc

\

j - J ^n k=0

K

-Q

. 0 00

\

^kk=0

A" ~ ^fc

_ ClC2 p ~

R

0102

Q

nl

^r

K

-

n

^

Hk

^kk= 0

A_

A° fc=0 \

Hi

Mfc j - j An _

-

^k / fc=0 n

B1B2C1C21 H o -

- - s j ^ ^ n (i - 1 ) n ^

^

n fi——ì n XkJk=o\

Vk ) k=0

~ ^ fi Xn

-

Xk

~ ^k

k=0

-

^k

rk

~ fa T7 ^n ~ ^fc TT ^n ~ ^fc -pr An ~ Pk TT ^n ~ Hk \0 _ „0 I I \0 _ \0 1 1 \ _ \0 I I ;,0 _ „0 I I _„0'

An

Vnk=Q

An

Akk=0A"

Akk=0An

^kk=0A"

^k

We now show that To do this we observe that

c

lim \ = 1. Therefore A^-00 $ 0 i ( A )

^ is (• •- i) (' •- *r - M ^

We now show that

It follows from the asymptotic formulae that A^ = (A: + a o ( ( ) ) 2 + — In ( k + 4- 0 ( 1 ) t \ 2/ and

Therefore A^ - Ajj! = 0 ( 1 )

and

00 ^ _ the series ^ | —g k=0

Ak~

| converges uniformly in A A

for —oc < A < — 1. Therefore the infinite product (1.21) converges uniformly in A for —00 < A < —1, and so we can pass to the limit A —> —00 in this product. If we do this. 00 I A — C °° A" C we find that lim [ J 1 + ^ = 1. Therefore - ^ f ] = = 1. This means X~"xk=0

\

Ak~

/

A

k=0

Ak

Cl

C1C2 that . 0 B i B 2 — 1. This formula and the formula (1.20) imply (1.19), which proves CLC 2

a"

=

\ 11 °° \ °° A\ —A °° \n „0 OC j Ak TT O^n-^nn n k TT ^k TT ~ ^k n\ o _ uo 11 \o _ \0 11 < _ Ao 11 A0 _ »0 11 An

r'n k=0

n

Akk=0

A"

Akk=0An

LLkk=0A~

^k

T

10-

(122)

Inverse

Spectral

21

Problem.

2. Asymptotic formulae for the numbers a n . Suppose that we are given two sequences of numbers Ao, A j , . . . , ¡IQ, . . . , and that we know them to be the eigenvalues of the boundary value problems (1.1)-(1.2) and (1.1)-(1.3) with an unknown function q(x) and with the real numbers Hi, Ho. Suppose that the following asymptotic formulae hold: = (n + a0(())2

+

ln(n + 1/2) + — ln(n + 1/2) + 2 aQ + 2a 1 (n + 1/2) 2tt

+

2a 2 (n + 1/2)

.in , Aap\ ln(n + 1/2) ag + 2a 5 |2a 3 + ^ l n2 2 ( " + 1 / 22) + ^2a 4 + _ ) + 47T ; (n + 1/2)

+Q I

(ln2n — (2.1)

, „, A , . = (n + a,(i))' + ^ ln(n

+ 2a, +

4tt-

, , „ , „ , In(n + 1/2) 1/2) + 2a 0 + 2 +

+

In 2 (n + 1/2) + (n + 1/2)2

2a 4 +

Aa'0\ ln(n + 1/2) TT ) ( n + 1 / 2 ) 2

2a, J ^ aQ2 + 2al 'In n ' + 0 3 (n + 1/2) (2.2)

Then, by (2.1) and (2.2) a 0 = -

• »o = ~

7T

fore

77

Hi - H 2 = 7r(ao - a 0 )

+

q{t)dt

There(2.3)

By hypothesis the numbers {A„} and {fi n } are distinct spectra of one and the same equation. Hence it follows from the results of the previous section that the normalizing consants of the boundary value problem (2.1)-(2.2) are given by the formula (1.22) . In the rest of this section we shall find the asymptotic formula for the numbers an in terms of the known asymptotic formulae for the numbers An and /xn. We shall carry out this derivation in several stages. We first consider the infinite product ^(A,,) = Afc - A?\ ^ Clearly In $(A„) = j T In 1 + . For sufficiently large n and A1 — An Ä k=0 \ k - An c ^k - Afc (in what follows C will denote a constant, not necessarily A- - A* 1 ~ n + 1/2 the same one). Therefore 0

(-1Y ( Afc-A0^' Ajjl — XT

ln«I

(2.4)

We have L E M M A : Let | Xk - Ag |< a (k = 1, 2 , . . . ). Then

^

Jfe=l

Xk 1 Ak

-

_

A1

\

Än

IP < —1

k

1

I

Proof of Lemma is same as the proof of [2].

C

ln(n + 1/2) rrz—I p = 1 n +p 1/2 a , P > 2. (n + 1/2)P

(2.5)

22

R. Kh. Amirov and S. Giilyaz Let us study l n i ' ( A 7 l ) a little further. It follows from the estimate (2.5) that

I fc=l E p=3 E h Vr

^ r Ar V

I p=3 ^ Pjfc=l X ' a o -A nt

\ A fc

(2.6)

-

p ^ ( n + l/2)p

(n + l / 2 ) 3 £ 5 ( n + l/2)P

U v

U

This estimate and the formula (2.4) show that

W e now consider the behaviour of the sums appearing in this formula. Using the asymptotic formulae (2.1) and (2.2) we see that the first sum satisfies. If we use the expression A^ and Ag asymptotic, we get the expression

g

'

^

f

( i - i )

• < * - < * > (2.8)

3

+

TT 2 (2CO -

A ln(3/2)

2

3tt

C 0 °)

1

_ 2 + 0 (n +• 1/2)2

3TT

/Inn -5"

o\2

Therefore, the formulae (2.7), (2.8) and (2.9) imply that

In ^

= A(C0 - C0°)(J - ¿ )

2AIn(3/2)l 3tt

+

(c0-c0)Q + ^ ( C o - C g )

{

2

M x

)! }

+

1

_

[3 +

1 ( n

+

1 / 2 )

C

c}

, _ /ln2n\ , + 0 ^ — J -

Hence *(A„)

= 1 + A(Co -

+

- ¿ )

2A ln(3/2) 1 .

0

m + (n + 1/2 ) 2

+

I [3

+

.»(Co -

C°)

^

3tt (2.10)

Therefore

fi

^

T

1

2A ln(3/2) 3tt

+

-

cS)

(i - ¿ ) I ^ W

+

{m»

+

5[3

+

"' 0.

The

first condition of the theorem shows that An and ¡in are interlaced. Therefore, for n = 0, 1 , . . . ¡j.n > An or ¡j.n < A n . a „ ' s are given by formula (1.22) or

Qn

_ ~

0 a n

— A4" ^n ~ An — fio t t An — Afc iS. An — fj.fr \0 — „0 \0 _ A\0A \0 _ „0 1 1 \o _ \0 1 1 To A ToAkk= n i*n\ k n A'O k=l An 1 n~Llk

Here, we can write J } ^ ^ > 0, J | ^ — ^ > 0. k= 1 Xn ~ Ak k=l An ~ Pk Since > 0 and A n , fi n and A°, are interlaced. It is obvious from a n formula that n = 0, 1 , . . . , Q n > 0. Since Sturm-Liouville operators can be built according to { A n } and { n n } sequences. As a result, we have proven that for given { A n } and

sequences

Sturm-Liouville operators can be built [4j.

References 1. V. A. Marchenko,

Tr. Mosk. Mat. Obsh., 1, 327 (1952).

2. M. G. Gasimov, B . M. Levitan, 3. M. G. Gasimov,

Matem. sborn., 63 (105), no. 3, 445 (1964).

Dokl. Akad., Nauk S S S R , vol. 161. no. 2 (1965).

4. M. G. Gasimov, R. Kh. Amirov, Dokl. Akad., Nauk A z . S S R , vol. 41., no.8 (1985).

8th Int. Coll. on Differential D. Bainov (Ed) © VSP 1998

Equations,

pp. 25-34

A p p l i c a t i o n of t h e Laplace, Mellin and Stieltjes Transformations in t h e E v a l u a t i o n of Integrals ALEXANDER Department

APELBLAT

of Chemical

Engineering,

Ben Gurion

University

of the Negev, Beer Sheva,

Israel

A b s t r a c t . The evaluation of definite and improper integrals using the Laplace, Mellin and Stieltjes transformations is discussed. The proposed procedures are illustrated by considering a large number of logarithmic, trigonometric, Bessel functions and other integrals. K e y w o r d s : Integral

1.

transformations,

integrals

of elementary

and special

functions

INTRODUCTION

Integral transforms serve as a starting point in the evaluation of definite and improper integrals of elementary and special functions by applying various mathematical operations under the integral sign, frequently in conjunction with rules and theorems of operational calculus (see for example [1]). For some classes of integrals [2-4], special techniques were developed, mainly applying the Laplace transformation while the Mellin and Stieltjes transformations are only scarcely considered in the literature. In this work these integral transforms are used to obtain integrals which are not tabulated in standard compilations of integrals and integral transforms [5-11],

2.

EVALUATION OF I N T E G R A L S USING THE LAPLACE TRANSFORMATION

If the function f ( t ) possesses the Laplace transform F(s) oo (1)

£{/(«)}

= / e-stf(t)dt

o

=

F(s)

and is of the form / ( f ) = u(t — a)h(t), where u(t — a) is the shifted step unit function, then by changing the integration variable in (1) Of a > 0,

(2)

a =

e'sa.

0 The case for which a = 0 and a = 1 was already considered in [4j. Similarly, if f ( t ) = [u(t) — u(t — a)]/i(t), equation (1) gives l

(3)

a > 0,

a

a =

e~sa.

A. Apelblat

26

These relations are illustrated by a number of examples. If f(t) = u(t — a)t" and f(t) [u(t) - u(t - a)]t", then from [9, p. 137, (4.3.2) and (4.3.3)] it follows that

(4a)

/[In ( i ) P *

(4b)

/Kä

= I > + 1, A),

=

A = ln(i)

dx = 7(1/ + 1, A),

0 < a < 1,

Re

> —1

where T(a, z) and 7(a, 2) are the incomplete gamma functions [12]. If the same functions are shifted, f(t) = u(t - a)(t — a ) " and }(t) o)](t - a)", then from [9, p. 137, (4.3.5) and (4.3.6)] one has

(5a)

dx = a r ( f + 1),

(5b)

dx = aj(i/

0 < a < 1,

= [u(i) — u(t

Rei/>-l,

+ 1, — A).

Similarly derived, without going into details about functions f(t) and F(s), the following group of logarithmic integrals is expressed in terms of the incomplete gamma function T(a, z) and the exponential integral Ei(z) [12].

(6a)

?K?)r

/ I

{

y / J dx = I > + l ) A T ( - y , A), (x)

ln

a /

(6b)

dx

—ßEi I n i ?

=

,

0 0,

W )

(6c)

[ ^ l ln(/J*)ln(§)

2 In/?

- -Ei[ln(a/?)] L

ßEi

a/3 ^ 1

a /

(6d)

In x

'X0

X

0-dx=

dx = £¿[(1 ± ß ) l n a ]

/ i n f i l l + Ei[(l

0 < a < 1,

- ß)lna]

0
/Ä)l

V

i

ln

(x)

where A is defined in (4a). The last group of logarithmic integrals, related to the Laplace transform, is associated with the modified Bessel functions Iv(z) and Ku(z) and the Struve function L„{z) [12] a f

(8a)

n

(8c)

(8d)

. 0 < q < 1,

dx = —XKi(\),

RX

*

J

j k i ( a x ) In

j

[in Q

-1/2

In ( - L ) ] "

dx =

( 2 A ) T (v + i )

" dx = 2 " " 1 V / F r ( „ + ^

A),

Rei/ > —1/2,

A"[/„(A) - L„(A)].

a

3.

EVALUATION OF INTEGRALS USING THE MELLIN TRANSFORMATION

If in the Mellin transform of the function /(f) oo

(9)

M{f(t)}

=

J ts~1f(t)dt = A/(s), o

the integration variable is changed to x = e - t , then equation (9) becomes

0 ,

h(— l n x ) d i = M (s, A).

These relations are illustrated by considering the function-Mellin transform pairs, e.g., if f(t) = l n ( l + e~at) [11, p. 38, (4.33) and (4.34)] then (12a)

l l [

l n ( l + xa)dx

ln(i)

IIHÏ)}

(12b)

In the second example, f(t) (13)

yV"

1

InQ)

= - ^ ( 1 - 2-K(ß

l n ( l - xa)dx

=

bt

[11, p. 25, (3.2)] and using ( l i b )

A= l n ( £ ) ,

dx =

a, 0 > 0,

+ 1).

= [u(t) — u(t — a)]e

a

+ 1),

Re/x, u > 0.

^

If v = 1/2, the integrals (13) can be expressed in terms of the error functions rM-l

(14a)

l /

(14b)

I

In the case of f(t)

J

r

0-1 ^ - u i l + xa .

=

J-eT(c(J\fi).

= 1/(1 + e Q t ) [11, p. 28, (3.26)], using (10) leads to

l (15)

rf-l I——dx

)

r(ß) dx =

} \xj.

-

a > 0,

ß > 1.

However, applying the operational rule of the Mellin transformation (16)

M{f^(t)}

= (-1)"

, r ( 5 ) ,M(s T(s-n)

— n),

to the first and second derivatives of the function (17)

fit) =

-

(1 + eat)2

'

f"(t)

»» = 0 , 1 , 2 , 3 , . . .

f(t) = af'(t)

+

2a2e

(!

+eai)3

it follows from (16) that (18)

M

l -

(1 + e«*)2

-(s -

l)M(s-

1)

Application

of the Laplace, Meilin and. Stieltjes

(19)

(s -

2)M(s

where M(s)

= M(ß)

29

Transformations...

and l ) ( s -

- 2) =

0-i

/öf^hG)

dx =

(20b) / T

I T ^ [ln © P

=

riß) Or

)M(s -

2 q 2„2at e

)+M

1

(H-e0')^

- 2 2 -")C(/? - 1),

a > 0, ß > 2,

+ (1

-

- 4

'

Similarly, introducing f(t)

(21b)

1

- 22 ' 0)c{ß - 1 } a > 0,

(21a)

-

is given in (15). Thus, from (10), (15), (18) and (19) one has

(20a)

0 ^

- q ( s

l / 1 - xa

ra-l

7 ( 1 -

Xc)2

,ln

=

l/(eat

ß-i

ß-i

— 1) [11, p. 28, (3.19)] it is possible to obtain

dx =

(z).

In

ß > 3.

dx =

a >

0,

ß > 1,

rtß) 1),

a

>

Y(ß) dx = - ^ [ ? ( / 3 - l ) + C ( 0 - 2 ) ] ,

ß >

0,

2,

a>0,

ß > 3.

The results in (20) and (21) can be generalized

°> e-d > 1

and (23a)

/(¡T^hO)

c— 1 dx ~

b + a-

,

a -)- b\

(

2a+ b

30

A.

(23b)

x^"-1

f

i

J

(23c)

Apelblat

_a&—1 (d -

ln

xby-

dx =

£(c)

b'd2

a, b > 0,

iCb

C_1,

* G ' )"*G' °)

c > 2,

d > 1

a

where C(z, ) a n d (z, a , /?) are t h e generalized zeta and the Lerch zeta functions [12], In t h e next example, the operational rule of the Mellin transformation (24)

M {/(t)

+ tf'(t)}

= —(5 —

l)M(s)

is applied to t h e function f ( t ) = e ~ ' l n t [9, p. 315, (6.4.13)]. After a number of steps,

/['«G)P»K;)]

(25)

dx = r ( a ) [ l +

aip(a)},

a > 0

where tp(z) is t h e psi (digamma) function [12]. If the Mellin transforms can b e separated into the real and imaginary parts, M (a + iw) = A(a, oj) +iB(a, ui), then it follows from (9) t h a t OC (26a)

J t"~l c o s ( u l n t ) / ( £ ) d t = A(a, 0 00

(26b)

J i"'1 sin(u\nt)f(t)dt

By introducing t = e

1

u),

=

B(a,u).

=

A(a,uj),

into (26) one has 00 J e~ax cos(ux)f(e~x)dx

(27a)

— OC OO

J e~°x s'm(ujx)f (e~z)dx

(27b)

= -

B(a,u)

-00

which has t h e form of a two-sided Laplace transformation [13] (in the case of the finite Mellin transforms, equations (27) become the ordinary Laplace transforms). These sets of equations will be illustrated by three examples. For f ( t ) = 1 /(a + t) [11, p. 13, (2.4)] after long b u t r a t h e r elementary operations it is possible to obtain (28a)

r ee~axalcos(wi) / J a

(28b)

/ J

r

e™ singer + u In a) 4- e ™ sin(7rij — u) In a) dx = 7T — a 1_iT [cosh(27ra;) — cos(27ra)]

e-> sin(a;i) — dl = 7Ta + e-

' cos(na + u;lna)

' cos(7r 0,

For f(t)

r(l - v ) 3

¡3 > 1,

a n

( cc V „

/a,

\

Re i/ < 1.

= l/(a + t) [9, p. 216, (14.2.2)] and using repeatedly, the operational rule,

(35)

5{/'(t)} =

- ^ " / ( 0 ) as

s

it is possible to obtain a general result dx

/

o —

x(l

(36b) ' 1

J

/

1

— - = -,

— lni)n+1

n=

n

dx

x(l - lnx)(a - lnx)n+1

1,2,3,...,

= L D l ^ ln ( l n a n! da \a - 1

Similarly, applying /(t) = e~at [9, p. 217, (14.2.11)] and 00

(37a)

S{tf(t)}

=

j

f(t)dt

-

sG{s),

o (37b)

Sp{tf(t)}

= G{s\p

— \) — sG(s]

p)

a. > 0.

Application

of the Laplace, Mellin

and Stieltjes Transformations.

33

..

one has /•x Q_1 lna; / dx = -eaEi(-a) J 1 — In x o

(38a)

1

, a

a > 0,

l (38b)

= a ' - 2 e a [ a r ( l -p,a)~

J o

T(2 - p, a

'

If both Stieltjes transforms of f(t) and t / ( f ) are known and the integral in (37a) can be independently determined, then new functional special functions can be derived. In the case of the Bessel function Jv(at) (39a)

[9, p. 224, (14.3.2)], it follows that

SiJJai)} = — ^ ^ - \ j j a s ) - j j a s ) ] , sin(7ri/) L i jrc

(39b)

S{tJu(at)}

=

,

1

/

+ -tF2

J„(as)

sin(7ri/) "

a

a > 0,

Rei/>-l,

1 — it 1 4- u

1;

V '

s I 1 — v 2 + 1/ ~lF2 1 ; — , — ; -

2

'

2

n-o^ 4

a2s2N —

oo (39c)

[ Jv{at)dt J

=

-, a

where Jn{z) is the Anger function [12]. From (37a) and (39) the Anger function can be expressed in terms of the generalized hypergeometric series J„(z) ^

=

sin(7ri/) i\z

F(v,z),

V * „ ( 2 - 1 / 2 + 1/ z 2 \ / 1 - 1 / 1 + 1/ F ( „ , z ) = 1 + 7 1 F 2 1; — , — ; - - !F 2 1; 2 2 v \ 1 1 4/ V

2 4

Comparison of the integral representations of the Anger function [12] with (40) yields *• y cos(i/0 - zsin9)d9

(41a)

o oo (41b)

• i

\

= iH^ZJ'^(i/, z),

1

Je~Z3'mh$'ued9

= -F(v,z)

- 7rcosec(7TU)J V (Z).

0 For u = 0, the integral (42b) is reduced to the well-known result [11, p. 17, (2.38)] °°

(42)

°°

-zy

J e — ^ ' d O = J - j L = f d y = I [ f f 0 ( z ) - Vo(z)], 0

o

R e z > 0,

y

where HQ(Z) and YQ(Z) are the Struve and Bessel functions of the second kind and of the zero order [12]. For v = 1, a new Laplace transform is derived OO (43)

/ I v

/

J'** -==fdy = — - ^ [ t f i ( O - n ^ ) ] . I T F [y + \ / T T p J 2 2l J

R e s > 0.

34

A.

Apelblat

References 1. M. Parodi, Introduction a l'Etude de l'Analyse Symbolique, Gauthier-Villars, Paris (1957). 2. A. Apelblat, IMA J. Appl. Math., 27, 481-496 (1981). 3. A. Apelblat, IMA J. Appl. Math., 34, 173-186 (1985). 4. A. Apelblat, J. Math. Anal. Appl., 186, 237-286 (1994). 5. I. S. Gradshtein and I. Ryzhik, Tables of Series, Products and Integrals, Verlag Harri Deutsch, Frankfurt (1964). 6. A. Apelblat, Table of Definite and Infinite Integrals, Elsevier Sei. Publ. Co., Amsterdam (1983). 7. A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and Series, Gordon and Breach Sei. Publ., New York (1988). 8. A. Apelblat, Tables of Integrals and Series, Verlag Harri Deutsch, Frankfurt (1996). 9. A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Tables of Integral Transforms, McGraw-Hill Book Co., Inc., New York (1954). 10. F. Oberhettinger and L. Badii, Table of Laplace Transforms, Springer-Verlag, Berlin (1973). 11. F. Oberhettinger, Table of Mellin Transforms, Springer-Verlag, Berlin (1974). 12. A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendential Functions, McGraw-Hill Book Co., Inc., New York (1955). 13. B. van der Pol and H. Bremmer, Operational Calculus Based on the Two-Sided Laplace Integral, Cambridge University Press, Cambridge (1964).

8th Int. Coll. on Differential Equations, pp. 35-41 D. Bainov (Ed) © VSP 1998

A s y n c h r o n o u s M u l t i s p l i t t i n g A l g o r i t h m s for Differential-Algebraic S y s t e m s D i s c r e t i z e d by Runge-Kutta Methods J. B A H I AND J.-C. M I E L L O U Scientific Calculus Laboratory. U.M.R. BP 527, 90016 Belfort cedex, France [email protected],

- C.N.R.S.

6623, I.U.T

B elfort-Month

¿Hard,

[email protected]

Abstract. We present iterative parallel asynchronous multisplitting algorithms in order to treat index one differential-algebraic systems discretized by suitable Runge-Kutta methods. The theoretical framework is based on El tarazi's theorem which concerns general fixed point iterative methods in the case of a product space and here include nonlinear waveform relaxation methods (Jacobi, Gauss-Seidel), and also various parallel extensions mixing point and block relaxation implemented in synchronous or asynchronous mode. Keywords: Asynchronous algorithms, multisplitting methods, differential algebraic systems, Runge-Kutta methods.

1

Introduction

We present iterative parallel asynchronous multisplitting algorithms in order to treat index one differential-algebraic systems discretized by suitable R u n g e - K u t t a methods. Differential-algebraic systems arise in many scientific problems, particularly from electrical network modeling, where the computer simulation of the time behavior requires the numerical integration of very large differential-algebraic equations, see [1], [2]. In [3) we defined a continuous fixed point m a p p i n g which the associated asynchronous algorithms converge to the solution of the considered differential-algebraic problem. In this paper we are interested with the discretization by Runge K u t t a m e t h o d s of the above mentioned continuous fixed point mapping. We prove, not only the convergence of the associated asynchronous algorithms b u t also the convergence of associated asynchronous multisplitting algorithms. This aim is achieved by using a general formulation of asynchronous multisplitting methods which we introduced in [4], we o b t a i n in this way t h e convergence of the particular cases corresponding to O'Leary and W h i t e kind multisplittings and Schwarz alternating methods. T h e theoretical framework is based on El tarazi's t h e o r e m which concerns general fixed point iterative m e t h o d s in the case of a p r o d u c t space a n d here include nonlinear waveform relaxation methods (Jacobi, Gauss-Seidel), a n d also various parallel extensions mixing point and block relaxation implemented in synchronous or asynchronous mode.

36

2

J. Bahi and J.-C.

Miellou

T h e problems and their discretizations

We consider differential-algebraic systems in t h e form: ( f(x'(t), (1)

I

x(t), y(t), t) = 0

g(x(t),y(t),t) = 0

[ x(t0) te[io,io + T],

= XQ

X = (ll,-.,I«),

/ = (/l, •••,/«),

9 =

V = (i/li •••! J/m)

(,91,-,9m)

Consider the following continuous fixed point m a p p i n g the fixed point of which coincides with t h e solution of the continuous problem (1), see [3] T(w,u,v)

= (z,x,y)

such

A M O . . . . , Wj_i(t), Zi{t), wi+1(t),..., (2)

that

w n ( t ) , u(t), v(t), t) = 0

f

" l ( ) . •••> V i - l ( t ) . y«. v t + l ( i ) > •••. v m ( i ) .

P i ( « l ( t ) > •••,

0 = 0

* î ( 0 = «i(0

Xi(to) = Xoi [to, to + T) is discretized as follows: to < ii < ••• < i/v-i = to + T, with the constant time window h = ti — ti_i for i € {1,..., N — 1} . Denote:

(3)

^ = (ûy)i •••> 7m(t)i |Si(t)-»i(0| .

^

V, t)| then an 0,

< 0

for

(x,t) 6 G, tj < 0, f > 0.

c(x,t,r},Z)

,m.

H4. /(x,t) 6 G(G,R). H 5 . A/(f) is a closed and bounded set for each t € [0, oo) and lim min s = oo. H 6 . ¥>(x,i) 6 C(dQ, x R + , R ) . H 7 . g(x,t) e C(dQ x R + . R ) . H8. 7(x,t) e x R+,R+). Definition 1 The solution u € C 2 ( G ) n G ^ G ) of problem (1), (2) ((1), (3)) is said to oscillate in the domain G = fl x (0, oo) if for any positive number ¡j, there exists a point (xo, to) € fi x [i-i, oo) such that the equality u(xo, fo) = 0 holds. It is known that the first eigenvalue Qo of the problem AU 4- aU = 0 i7=0

in ft,

on Sfi

is positive and the corresponding eigenfunction $ is positive in fi. Without loss of generality we may assume that $ is normalized, i.e., J $>(x)dx = 1.

3 3.1

Main results Oscillation of solutions of problem ( 1 ) , ( 2 )

T h e o r e m 1 Let the following conditions hold: 1. Assumptions H1-H6 are fulfilled. 2. For every sufficiently large number to > 0 we have liminf t—»oo

(4)

J f(x, to

-Ian

I

s)$(x)di

D

a(s)y?(x,s) +

>=i

Yjaj(s) t

n From (2) and Green's formula it follows that 'du d S + J Au$dx = J ( g-* Au

If u(x, t) < 0 for (x, t) 6 fi x [to, oo), then by H3 the function — u(x, t) satisfies the relations 9 dt

+

Y^\i{t)u(x,t-Ti)


0 we have

large number

liminf J

I J

J ds =

s)$(x)dx

f(x,

to V n

-c

/

and limsup J to

Then

every solution d

+

dt

I J f ( x , s ) $ ( x ) d sx I ds = oo. \

u g C2{G)

i(t)u(x,t-

n fl C1(G)

Ti)

of the

= a(t)Au

problem

+

Pj(t)) j= 1

c ( x , t, u ( x , t ) , m a x |u(i, s)|) + f ( x , t ) , sÇM(t) u = 0 oscillates

3.2

in

(0, oo),

on 3 f t x

G.

O s c i l l a t i o n o f s o l u t i o n s of p r o b l e m

T h e o r e m 2 Let the following

conditions

H 1 - H 5 , H7, H8 are

1. Assumptions 2. For

(i. ( j g i l x

every sufficiently

w

/

+J

(14)

(3)

hold: fulfilled.

large number

/ to k an

(1),

a(s)g{x,s)

to > 0 we have

+

^2aj(s)g(x,pj(s))

dS

+

3=1

f(x,

s)dax >ds = — oc

and

a(s)g[x,s)

+J

(15)

Then

every solution

f ( x,

+J2aj(s)9{X:Pj{s))

dS-

j=i

to k an

s)dix yds = oc.

u € C 2 ( G ) fl C1(G)

of problem

(1), (3) oscillates

P r o o f . S u p p o s e t h a t the p r o b l e m ( 1 ) , ( 3 ) has a solution u 6 C2(G) has no zero in ft x [i 0 , o o ) f o r some io > 0.

in G.

fl C1(G)

which

48

D. Bainov

and E.

Minchev

Assume that u(x, t) > 0 in fl x [io, oo). Then there exists Ti) >

0, u(x,pj(t))

>

0 for ( x , t)

> t 0 such that u(x, t —

€ fi x [ti, oo), w h e r e i =

1 , . .. , m ; j

=

1 , . . . , k.

Therefore, from (1) and (3) we obtain

J

udx

4- ^

Ln

Ai(t)

+ ]Ta (t) J i

}=l

a(t)

J

u(x,

t -

Au(x,

4-

pj(t))dx

n

J{--yu

Ti)dx

n

i=1

< a(t)

J

Audx

J

n

J f{x,t)dx

=

+

n

+ g)dS

+ JJaj(t)

an

J

( - i ( x , Pj(t))u(x,

Pj(t))

+ g(x,

dS +•

Pj

an

(16) + J

f ( x , t)dx


to, J

satisfies

L dt

¿=l
K+ is the maximal solution of the problem \u(x,y)\ < w(x\T])

Then

\u(x,y)\ < U>(X;T])

for for

(x,y)£d0r. (x,y)&TDQ.

(7)-(9) and

On the Stability of

55

Solutions.

P r o o f . Define 7 ( x ) = m a x ||u(x, ¡/)| : T h e n y € Cimp[Io

3/S [ ~ M ] } ,

xe/oU/.

U / , K + ] . W e will prove that 7 satisfies all conditions of L e m m a 1.

It follows from (13) and from condition 2 of the theorem t h a t the e s t i m a t e for the impulses (11) a n d t h e initial inequality (12) are fulfilled. Suppose t h a t x e E. T h e n there exists y e [-&,&] such t h a t 7 ( 1 ) = |u(i,j7)|. Assume t h a t 7(2?) = u(x, y). It follows from (14) t h a t

y

6

(—b,b)

and consequently = 0,

UIi(X,Y)

UYY(X,Y]