Theory and numerical approximations of fractional integrals and derivatives
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Table of contents :
Contents......Page 5
List of Figures......Page 7
List of Tables......Page 9
Preface......Page 11
1.1 Riemann-Liouville integral......Page 14
1.2 Fractional integrals of other types......Page 25
2.1 Riemann-Liouville derivative......Page 29
2.2 Some remarks on the Riemann-Liouville derivative......Page 43
2.3 Caputo derivative......Page 46
2.4 Some remarks on the Caputo derivative......Page 55
2.5 Riesz derivative......Page 56
2.6 The fractional Laplacian......Page 64
2.7 Fractional derivatives of other types......Page 74
2.8 Definite conditions for fractional differential equations......Page 81
3.1 Numerical methods based on polynomial interpolation......Page 94
3.2 Fractional linear multistep method......Page 99
3.3 Spectral approximations......Page 102
3.4 Diffusive approximation......Page 108
4.1 L1, L2, and L2C methods......Page 111
4.2 High-order methods based on polynomial interpolation......Page 125
4.3 Fractional linear multistep method......Page 176
4.4 Spectral approximations......Page 184
4.5 Diffusive approximation......Page 189
5.1 L1, L2, and L2C methods......Page 191
5.2 Approximation based on spline interpolation......Page 193
5.3 Grünwald-Letnikov type approximations......Page 200
5.4 Fractional backward difference formulae with modifications......Page 209
5.5 Fractional average central difference method......Page 237
5.7 Numerical method based on finite-part integrals......Page 242
6 Numerical Riesz differentiation......Page 245
6.1 Indirect approximations to the fractional diffusion operator......Page 247
6.2 Direct approximations to the fractional diffusion operator......Page 259
6.3 Indirect approximations to the fractional convection operator......Page 274
6.4 Direct approximations to the fractional convection operator......Page 284
7 Numerical fractional Laplacian......Page 288
7.1 Approximations based on regularization and interpolation......Page 292
7.2 Approximation based on the weighted trapezoidal rule......Page 296
7.3 Some remarks on the numerical fractional Laplacian......Page 299
Bibliography......Page 301
Index......Page 313
Citation preview
Theory and Numerical Approximations of Fractional Integrals and Derivatives
OT163_LI_FM_V4.indd 1
9/26/2019 3:49:45 PM
OT163_LI_FM_V4.indd 2
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Theory and Numerical Approximations of Fractional Integrals and Derivatives Changpin Li Shanghai University Shanghai, China
Min Cai
Shanghai University Shanghai, China
Society for Industrial and Applied Mathematics Philadelphia
OT163_LI_FM_V4.indd 3
9/26/2019 3:49:45 PM
Copyright © 2020 by the Society for Industrial and Applied Mathematics 10 9 8 7 6 5 4 3 2 1 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 Market Street, 6th Floor, Philadelphia, PA 19104-2688 USA. Trademarked names may be used in this book without the inclusion of a trademark symbol. These names are used in an editorial context only; no infringement of trademark is intended. MATLAB is a registered trademark of The MathWorks, Inc. For MATLAB product information, please contact The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098 USA, 508-647-7000, Fax: 508-647-7001, [email protected], www.mathworks.com. Publications Director Executive Editor Acquisitions Editor Developmental Editor Managing Editor Production Editor Copy Editor Production Manager Production Coordinator Compositor Graphic Designer
Kivmars H. Bowling Elizabeth Greenspan Paula M. Callaghan Mellisa Pascale Kelly Thomas Ann Manning Allen Matthew Bernard Donna Witzleben Cally A. Shrader Cheryl Hufnagle Doug Smock
Library of Congress Cataloging-in-Publication Data Please visit www.siam.org/books/ot163 to view the CIP data.
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Contents List of Figures
vii
List of Tables
ix
Preface
xi
I
Background and Theory
1
1
Fractional integrals 3 1.1 Riemann-Liouville integral . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Fractional integrals of other types . . . . . . . . . . . . . . . . . . . . . . . 14
2
Fractional derivatives 2.1 Riemann-Liouville derivative . . . . . . . . . . . . . 2.2 Some remarks on the Riemann-Liouville derivative . 2.3 Caputo derivative . . . . . . . . . . . . . . . . . . . 2.4 Some remarks on the Caputo derivative . . . . . . . . 2.5 Riesz derivative . . . . . . . . . . . . . . . . . . . . 2.6 The fractional Laplacian . . . . . . . . . . . . . . . . 2.7 Fractional derivatives of other types . . . . . . . . . . 2.8 Definite conditions for fractional differential equations
II 3
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Numerical Approximations
19 19 33 36 45 46 54 64 71
85
Numerical fractional integration 3.1 Numerical methods based on polynomial interpolation 3.2 Fractional linear multistep method . . . . . . . . . . 3.3 Spectral approximations . . . . . . . . . . . . . . . . 3.4 Diffusive approximation . . . . . . . . . . . . . . . .
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87 87 92 95 101
Numerical Caputo differentiation 4.1 L1, L2, and L2C methods . . . . . . . . . . . . . . . . 4.2 High-order methods based on polynomial interpolation 4.3 Fractional linear multistep method . . . . . . . . . . . 4.4 Spectral approximations . . . . . . . . . . . . . . . . . 4.5 Diffusive approximation . . . . . . . . . . . . . . . . .
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105 105 119 170 178 183
v
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6
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Contents
Numerical Riemann-Liouville differentiation 5.1 L1, L2, and L2C methods . . . . . . . . . . . . . . . . . . 5.2 Approximation based on spline interpolation . . . . . . . . 5.3 Grünwald-Letnikov type approximations . . . . . . . . . . 5.4 Fractional backward difference formulae with modifications 5.5 Fractional average central difference method . . . . . . . . 5.6 Spectral approximations . . . . . . . . . . . . . . . . . . . 5.7 Numerical method based on finite-part integrals . . . . . .
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185 185 187 194 203 231 236 236
Numerical Riesz differentiation 6.1 Indirect approximations to the fractional diffusion operator . 6.2 Direct approximations to the fractional diffusion operator . . 6.3 Indirect approximations to the fractional convection operator 6.4 Direct approximations to the fractional convection operator .
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239 241 253 268 278
Numerical fractional Laplacian 283 7.1 Approximations based on regularization and interpolation . . . . . . . . . . 287 7.2 Approximation based on the weighted trapezoidal rule . . . . . . . . . . . . 291 7.3 Some remarks on the numerical fractional Laplacian . . . . . . . . . . . . . 294
Bibliography
297
Index
309
List of Figures 2.1
A plot illustrating the consistency of fractional derivatives [90]. . . . . . . . . 39
4.1
The spectral radius α ≤ 1 [100]. . . . . The spectral radius α ≤ 2 [100]. . . . .
4.2
5.1 5.2 5.3 5.4
Plots of Bl Plots of Bl Plots of Bl Plots of Bl
associated . . . . . . associated . . . . . .
with . . . with . . .
the . . the . .
Caputo fractional . . . . . . . . . . Caputo fractional . . . . . . . . . .
and m(l) with l = 5, 6, 7, 8. . . . and m(l) with l = 8, 9, 10, 11. . . and m(l) with l = 11, 12, 13, 14. and m(l) with l = 14, 15, 16, 17.
vii
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operator . . . . . operator . . . . .
for . . for . .
0 < . . . . 183 1 < . . . . 184
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208 208 209 209
List of Tables 2.1
The one-half order derivatives of the most used functions. . . . . . . . . . . . 35
3.1 3.2
The absolute errors for Example 3.2 with u = v = 0 and µ = 3.5. . . . . . . . 101 The absolute errors for Example 3.2 with u = v = − 21 and µ = 3.5. . . . . . . 101
4.1 4.2 4.3 4.4
Numerical results for Example 4.6. . . . . . . . . . . . . . . . . . . . . . . Numerical results for Example 4.14. . . . . . . . . . . . . . . . . . . . . . . Numerical results for Example 4.15. . . . . . . . . . . . . . . . . . . . . . . The truncated errors and convergent orders for Example 4.18 by scheme (4.229). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The truncated errors and convergence orders for Example 4.19 by scheme (4.229). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The absolute errors and convergence orders of Example 4.25 with p = 0. . . The absolute errors and convergence orders of Example 4.25 with p = 1. . . The absolute errors and convergence orders of Example 4.26 with p = 1. . . The absolute errors for Example 4.27 with u = v = 0 and µ = 3.5. . . . . . The absolute errors for Example 4.27 with u = v = − 12 and µ = 3.5. . . . . The absolute errors for Example 4.28 with u = v = 0. . . . . . . . . . . . . The absolute errors for Example 4.28 with u = v = − 21 . . . . . . . . . . . .
4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 5.1 5.2
5.3 6.1 6.2 6.3 6.4 6.5 6.6
. 127 . 153 . 154 . 165 . . . . . . . .
166 177 178 179 181 181 182 182
The maximum errors, L2 errors, and their convergence rates for Example 5.7 with α = 1.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 The maximum errors and convergence orders for schemes (5.231) and (5.243), respectively. The parameters α = 1.2 and (p, q, r, s, p¯, q¯, r¯, s¯) = (1, 2, 1, 0, 1, 2, 1, −2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 Fractional derivatives associated with finite-part integrals. . . . . . . . . . . . 238 The absolute errors and convergence orders of Example 6.4 by numerical scheme (6.46). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The absolute errors and convergence orders of Example 6.8 by numerical scheme (6.48). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The absolute errors and convergence orders of Example 6.8 by numerical scheme (6.49). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The absolute errors and convergence orders of Example 6.8 by numerical scheme (6.50). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The absolute errors and convergence orders of Example 6.11 by numerical scheme (6.52). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The absolute errors and convergence orders of Example 6.12 by numerical scheme (6.54). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
. 248 . 250 . 250 . 251 . 252 . 253
x
List of Tables
6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 6.16 6.17 6.18
The absolute error E(h) and the experimental convergence order (ECO) of function f2 (x) by numerical scheme (6.111). . . . . . . . . . . . . . . . . . The absolute error E(h) and the experimental convergence order (ECO) of function f3 (x) by numerical scheme (6.112). . . . . . . . . . . . . . . . . . The absolute error E(h) and the experimental convergence order (ECO) of function f4 (x) by numerical scheme (6.113). . . . . . . . . . . . . . . . . . The absolute error E(h) and the experimental convergence order (ECO) of function f5 (x) by numerical scheme (6.114). . . . . . . . . . . . . . . . . . The numerical results of Example 6.21 by using the 4th order fractionalcompact formula (6.111). . . . . . . . . . . . . . . . . . . . . . . . . . . . The absolute errors and convergence orders of Example 6.23 by numerical scheme (6.130) with p = n = 2. . . . . . . . . . . . . . . . . . . . . . . . . The absolute errors and convergence orders of Example 6.23 by numerical scheme (6.130) with p = n = 3. . . . . . . . . . . . . . . . . . . . . . . . . The absolute errors and convergence orders of Example 6.23 by numerical scheme (6.130) with p = n = 4. . . . . . . . . . . . . . . . . . . . . . . . . The absolute errors and convergence orders of Example 6.23 by numerical scheme (6.130) with p = n = 5. . . . . . . . . . . . . . . . . . . . . . . . . The absolute errors and convergence orders of Example 6.23 by numerical scheme (6.130) with p = n = 6. . . . . . . . . . . . . . . . . . . . . . . . . The absolute errors and convergence orders of Example 6.26 by numerical scheme (6.147). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The absolute errors and convergence orders of Example 6.27 by numerical scheme (6.149). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 265 . 266 . 266 . 267 . 267 . 271 . 272 . 273 . 274 . 275 . 277 . 278
Preface Fractional calculus, which has two main features—singularity and nonlocality from its origin—means integration and differentiation of any positive real order or even complex order. It has a history of at least three hundred years, since it can be traced back to a letter from Gottfried Wilhelm Leibniz to Guillaume de l’Hôpital, dated 30 September 1695, in which the meaning of the one-half order derivative was first discussed and some remarks about its possibility were made. It is generally accepted that fractional calculus underwent two stages: from its beginning to the 1970s, and after the 1970s. At the first stage, fractional calculus was studied mainly by mathematicians as an abstract field containing only pure mathematical manipulations of little or no use. At the second stage, the paradigm began to shift from pure mathematical research to applications in various realms, such as anomalous diffusion, anomalous convection, power laws, allometric scaling laws, history dependence, long-range interactions, and so on. Although numerical methods for fractional integrals and fractional derivatives have been collected and remarked in two review articles (Int. J. Bifurcation Chaos, 22 (4), 1230014, 2012; Int. J. Comput. Math., 95 (6-7), 1048–1099, 2018), and Chapter 2 of the book Numerical Methods for Fractional Calculus (CRC Press, Boca Raton, 2015), novel algorithms keep emerging and are widely scattered through many technical and scientific journals. A comprehensive book is required to collect and summarize the recent advances in numerical fractional calculus as well as the traditional and also most used algorithms. This book aims at collecting and sorting out these studies, and includes two parts. One is about the background and theory of fractional calculus, which are presented in Chapters 1 and 2. The other is the major element of this book focusing on numerical approximations to fractional integrals and fractional derivatives, from Chapter 3 to Chapter 7. In the first chapter, background and theory of fractional integrals are covered. Starting with introducing the Riemann-Liouville integral out of the description of the fractional diffusion equation, Chapter 1 conveys to the reader comprehensive knowledge on fractional integrals, by virtue of asymptotical derivation of anomalous diffusion and nonexponential relaxation patterns from basic random walk models and a generalized master equation. As a kind of frequently utilized fractional integral, the Riemann-Liouville integral is the protagonist of this chapter. The definition, existence conditions, and main properties, especially its relationship with the integer-order integral, are introduced. They are complementary instruments for theoretical analysis of fractional differential systems. Fractional integrals of some other types are also presented, along with the corresponding basic knowledge. What follows is fundamental knowledge on fractional derivatives presented in the coming chapter. In Chapter 2, heavily utilized fractional derivatives (Riemann-Liouville derivative, Caputo derivative, Riesz derivative, and fractional Laplacian) are introduced. Some other well-known fractional derivatives are also mentioned. Definitions and properties of these fractional derivatives are routinely presented, as well as their correlations. These aspects are considered and the results help the reader in understanding fractional derivative operators as pseudo-differential operators, together with their tremendous application potential in applied science and engineering, xi
xii
Preface
not merely mathematical generalizations of the classical derivative operators. As an adequate tool describing unusual diffusion processes due to random displacements and Lévy flights, the fractional Laplacian is especially mentioned. The relationship between the fractional Laplacian and Riesz derivative is clarified in detail, elucidating that the fractional Laplacian seems to possess a larger range of applications in characterizing anomalous diffusion and anomalous convection, which can be seen from the fact that the fractional Laplacian has attracted increasing interest. The Riesz derivative seems to be neglected. If one carefully and meticulously reads the encyclopedic book Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach Science Publishers, Amsterdam, 1993), he/she may find its usefulness, profoundness, and beauty. In effect, the Riesz derivative defined on R is essentially the fractional Laplacian defined on R. And the fractional Laplace operator is also known as the Riesz fractional derivative operator (Fract. Calc. Appl. Anal., 20 (1), 7–51, 2017). The partial Riesz derivative in multiple dimensions seems inconvenient for applications. The fractional Laplacian in the multidimensional case is therefore adopted. An alternative definition of the fractional Laplacian, i.e., the so-called spectral definition, is a simple generalization of a positive definite operator in finite dimensions (i.e., the symmetric and positive definite matrix in finite dimensional linear space). This spectral definition is different from the aforementioned fractional Laplacian in the sense of Riesz. On the other hand, different from integer order derivative, semigroup properties for fractional derivatives generally do not hold. The equalities πα (1) sin(α) (x) = sin x + 2 and πα cos(α) (x) = cos x + (2) 2 generally do not hold either when the derivative order α is not a positive integer. Definite conditions for fractional differential equations are also carefully described. These three aspects have often been misused and so are highlighted in Chapter 2. Once acquainted with fractional calculus, it remains to study fractional differential equations. The reality is that most fractional differential equations are difficult or even impossible to analytically solve. Consequently, numerical solving fractional differential equations becomes a preferred alternative. And there is no doubt that techniques evaluating fractional integrals and fractional derivatives are fundamental in this regard. Part II collects and presents almost all the existing numerical approximations to fractional integrals and fractional derivatives. Numerical approximations to fractional integrals, the Caputo derivative, Riemann-Liouville derivative, Riesz derivative, and fractional Laplacian are included in Chapters 3–7, respectively. Riemann-Liouville integrals are evaluated in Chapter 3. Numerical approximations based on polynomial interpolation, spectral methods, the fractional multistep method, and diffusive approximation are derived in detail. Some methods have convergence orders depending on the integral order α, while others have convergence orders for which α is irrelevant. Numerical examples are presented to directly display the effect of these numerical methods. Viewing the Caputo derivative as a Riemann-Liouville integral of the integer-order derivative, we introduce a series of numerical approximations to the Caputo derivative in Chapter 4, based on the ideas introduced in Chapter 3. Replacing the given function by its polynomial interpolation, the L1, L2, and L2C methods, high-order methods based on polynomial interpolation, and spectral approximations can be readily obtained after direct calculating the Riemann-Liouville integral of integer-order derivatives of these interpolation functions. Diffusive approximation to the Caputo derivative can be derived analogously to the Riemann-Liouville integral. In view of the relationship between Caputo and Riemann-Liouville derivatives, fractional backward difference formulae for Caputo derivatives are also derived in Chapter 4. Apart from fractional
Preface
xiii
backward difference formulae, which are of integer-order accuracy, numerical approximations introduced in Chapter 4 have error estimates depending on the derivative order α. This can be verified by numerical examples displayed in that chapter. Numerical evaluations of Riemann-Liouville derivatives are introduced in Chapter 5. In view of numerical approximations to the Caputo derivative, the link between Caputo and RiemannLiouville derivatives yields L1, L2, and L2C methods and spectral approximations to RiemannLiouville derivatives. Other approaches approximating the Riemann-Liouville derivative in direct ways, including Grünwald-Letnikov type approximations, fractional backward difference formulae and their modifications, the fractional average central difference method, and the numerical method based on finite-part integrals, are also introduced, along with numerical examples. Chapter 6 introduces numerical approximations to Riesz derivatives. According to the range of α, say 0 < α < 1 and 1 < α < 2, numerical methods are divided into two cases: one for the fractional convection operator and another for the fractional diffusion operator. Indirect methods follow from the observation that the Riesz derivative is a linear combination of left- and right-sided Riemann-Liouville derivatives. Direct methods are mainly derived from the asymmetric centered difference operator and its variants. The corresponding numerical examples are presented as well. To emphasize the ubiquity of the integral definition of fractional Laplacian, the continuous time random walk process is considered in Chapter 7 to show its physical interpretations. Then the numerical methods for the fractional Laplacian in one space dimension are presented. Some relevant remarks are also included. Overall, fundamental knowledge on fractional calculus along with comprehensive ideas introduced in this book provide the reader with better understanding of this subject, in terms of both pure theories and applications. The audience may benefit from the detailed derivation processes of the fruitful numerical approximations, since these derivation processes can give some hints on establishing some other novel numerical schemes. Accordingly, this book may indeed be a genuine guide to the central ideas of fractional calculus. In view of the fact that almost all the existing results on numerical approximations to fractional integrals and fractional derivatives are included in this book, it is appropriate to use it in educational classes as a detailed introduction to fractional calculus, as we move forward into a new era of the fractional world. Last but not least, we thank Professors Qinsheng Bi, Guanrong Chen, YangQuan Chen, Jinqiao Duan, Boling Guo, Haiyan Hu, George Em Karniadakis, Abdul Q. M. Khaliq, Jürgen Kurths, Jie Shen, Zhizhong Sun, Yifa Tang, Hong Wang, Yubin Yan, and Weiqiu Zhu for their support and encouragement, and/or for providing suggestions for updating. We greatly appreciate Paula M. Callaghan for her support, especially for her quick response and humorous replies when we communicate with each other via email. We wish to also thank Ning Cai for his careful editing and input. The first author particularly thanks his Ph. D. student Min Cai (who is the second author of this book) for her active cooperation. He also thanks his former Ph. D. students Hengfei Ding and Fanhai Zeng, and his present postgraduate student Caiyu Jiao for their careful proofreading. The first author acknowledges the financial support from National Natural Science Foundation of China (grant nos. 11671251 and 11632008). Changpin Li and Min Cai February 2019
Chapter 1
Fractional integrals
Our goal is to show how fractional integrals emerged in real-world applications, which of them are most used nowadays, and which properties are easily confused. First we focus on introducing anomalous diffusion that is characterized by the Riemann-Liouville fractional integral (called the Riemann-Liouville integral for brevity). Some of the assertions concentrating on the right-sided fractional integral naturally resemble those of the left-sided one, and so are omitted here.
1.1 Riemann-Liouville integral It is recorded that the proper history of fractional calculus in applications began with the papers by Abel [1, 2], where the integral equation Z
x
a
ϕ(t) dt = f (x), x > a, 0 < µ < 1, (x − t)µ
(1.1)
was solved for given f (x) in connection with a tautochrone problem [122]. Although Abel was not intending to generalize differentiation to a noninteger version, the investigation played an enormous part in the development of this idea, because the formulation of this integral equation led to the fractional integral operator of order 1 − µ [148]. As time passed, definitions of fractional integrals were modified to meet practical requirements. When studying anomalous diffusion, Metzler and Klafter introduced a generalized diffusion equation of fractional order on the basis of the continuous time random walk (CTRW) scheme [127]. CTRW processes can be classified according to the characteristic waiting time ∞
Z T =
tw(t)dt
(1.2)
x2 λ(x)dx
(1.3)
0
and the jump length variance 2
Z
∞
Σ = −∞
being finite or divergent, respectively. Here λ(x) and w(t) are the jump length probability density function (PDF) and the waiting time PDF, determined by a jump (joint) PDF ψ(x, t), in the forms Z
∞
λ(x) =
ψ(x, t)dt 0
3
(1.4)
4
Chapter 1. Fractional integrals
and
Z
∞
ψ(x, t)dx.
w(t) =
(1.5)
−∞
A particular case is the decoupled CTRW process in which the jump length and waiting time are independent random variables, i.e., ψ(x, t) = w(t)λ(x). In general, a CTRW process can be characterized by the equation [25, 61] Z ∞Z ∞ η(x0 , t0 )ψ(x − x0 , t − t0 )dt0 dx0 + δ(x)δ(t), (1.6) η(x, t) = −∞
0
which relates the PDF η(x, t) of having just arrived at position x at time t to the event of having just arrived at x0 and at instant t0 . The second term δ(x)δ(t) in the above equation represents the initial condition of the random walk. As a result, the PDF W (x, t) of being at position x at time t can be expressed as Z t
η(x, t0 )Ψ(t − t0 )dt0 ,
W (x, t) =
(1.7)
0
with Z Ψ(t) = 1 −
t
ω(t0 )dt0 .
(1.8)
0
Here η(x, t0 ) denotes the PDF of having already arrived at x and at instant t0 , while Ψ(t − t0 ) is the PDF of not having left since arrival. The Fourier-Laplace transform of the PDF W (x, t) satisfies the relation [76] c f (k, u) = 1 − ω(u) W0 (k) , W (1.9) e u) u 1 − ψ(k, c0 (k) being Fourier transform of the initial condition W0 (x). with W At this stage, we consider a decoupled CTRW process with the characteristic waiting time T being divergent while the jump length variance Σ2 remains finite. We can choose a Gaussian jump length PDF and a long-tailed waiting time PDF with the asymptotic behavior [129, 130, 131] w(t) ∼ Aα (τ /t)1+α , 0 < α < 1. (1.10) Then Fourier-Laplace transform for the PDF W (x, t) takes the form f (k, u) = W
c0 (k)/u] [W 1 + Kα u−α k 2
(1.11)
2
in the (k, u) → (0, 0) diffusion limit. Here Kα = τσα is the generalized diffusion constant. Consequently, an inverse Fourier-Laplace transform yields the fractional integral equation Z t 1 ∂2 W (x, t) − W0 (x) = (t − s)α−1 Kα 2 W (x, s) ds. (1.12) Γ(α) 0 ∂x Here Γ(·) denotes the Gamma function. During systematic generalization of the above fractional integral equation, the Riemann-Liouville integral is introduced. Definition 1.1. The left- and right-sided Riemann-Liouville integrals of a suitably smooth function f (x) on (a, b) are defined by Z x 1 f (t) −α dt, a < x < b, α > 0, (1.13) RL Da,x f (x) = Γ(α) a (x − t)1−α
1.1. Riemann-Liouville integral
5
and −α RL Dx,b f (x) =
1 Γ(α)
Z
b
x
f (t) dt, a < x < b, α > 0, (t − x)1−α
(1.14)
respectively. Normally, the fractional integral is a proxy of the left-sided Riemann-Liouville integral if no confusion arises. Using the notation defined by (1.13), the fractional integral equation (1.12) reads as ∂2 −α (1.15) W (x, t) − W0 (x) = RL D0,t Kα 2 W (x, t) . ∂x Our next step is to consider under what circumstances the integrals in (1.13) and (1.14) make sense. In effect, it is crucial to study the existence of fractional integrals for given functions. The existence of fractional integrals was investigated in [148], which revealed that the fractional integrability is related to the absolute continuity. Definition 1.2. A function f (x) is absolutely continuous on an interval Ω ⊂ R if for any > 0 there exists a δ > 0 such that P for any finite set of pairwise nonintersecting intervals [ak , bk ] ⊂ Pn n Ω, k = 1, 2, . . . , n, such that k=1 (bk − ak ) < δ, the inequality k=1 |f (bk ) − f (ak )| < holds. The space of these functions is denoted by AC(Ω). Remark 1.1. The space AC(Ω) coincides with the space of primitives of Lebesgue summable functions [77], i.e., Z x f (x) ∈ AC(Ω) ⇔ f (x) = c + ϕ(t)dt, (1.16) a
Rb
where c is a constant and a |ϕ(t)|dt < ∞. Corresponding to AC(Ω), the notation AC m (Ω) denotes the space of functions f (x) which have continuous derivatives up to order m − 1 on Ω with f (m−1) (x) ∈ AC(Ω), where m = 1, 2, . . ., and Ω ⊂ R. Theorem 1.3. Let f (x) ∈ AC([a, b]) and 0 < α < 1. Then the fractional integral RL D−α a,x f (x) exists and is absolutely continuous on [a, b]. Furthermore, it holds that −α RL Da,x f (x) =
Z x 1 (x − a)α f (a) + (x − t)α f 0 (t)dt . Γ(1 + α) a
Proof. Since f ∈ AC([a, b]), we may substitute the function f (t) = f (a) + (1.13) so that −α RL Da,x f (x)
f (a) 1 = (x − a)α + Γ(1 + α) Γ(α)
x
Z a
dt (x − t)1−α
Z
(1.17)
Rt a
f 0 (s)ds into
t
f 0 (s)ds.
(1.18)
a
The on the right-hand side of (1.18) is an absolutely continuous function as (x − a)α = R xfirst term α−1 α a (t − a) dt. Since Z a
x
dt (x − t)1−α
Z
t 0
Z
x
Z
f (s)ds = a
a
a
t
f 0 (s)ds (t − s)1−α
dt,
(1.19)
6
Chapter 1. Fractional integrals
which may be verified through direct interchanging the order of integration on both sides of the equation, the second term on the right-hand side of equation (1.18) is also a primitive of a summable function, and hence it is absolutely continuous. The representation in (1.17) follows from (1.18) after the interchange of the order of integration. This completes the proof. Remark 1.2. Similarly to the above proof, we can readily show that the fractional integral −α m RL Da,x f (x) (α > 0) exits almost everywhere on [a, b] whenever f (x) ∈ AC ([a, b]), m = [α] + 1. Here [α] denotes the largest integer no bigger than α. As the regularity of solutions to fractional differential equations usually relies on mapping properties of the governing fractional operators, now we present mapping properties of the Riemann-Liouville fractional integral, which were considered in [148]. First, we introduce the following Hölder spaces as preliminaries. Definition 1.4. Let Ω be a finite interval. The function f (x) given on Ω is said to satisfy the Hölder condition of order λ on Ω if |f (x1 ) − f (x2 )| ≤ A|x1 − x2 |λ
(1.20)
for any x1 , x2 ∈ Ω, where A is a constant and λ is the Hölder exponent. In the cases of the whole line or a half-line, a function f (x) is said to satisfy the Hölder condition in the neighborhood of infinity if λ 1 1 (1.21) |f (x1 ) − f (x2 )| ≤ A − x1 x2 for all x1 and x2 with sufficiently large absolute values. Definition 1.5. For the finite interval Ω, we denote by H λ = H λ (Ω) the space of all functions which in general are complex valued and satisfy the Hölder condition (1.20) of a fixed order λ on Ω. If Ω is a line or a half-line, we denote by H λ = H λ (Ω) the space of functions satisfying the Hölder condition (1.20) for any finite interval of Ω and the condition (1.21) in the neighborhood of infinity. Remark 1.3. Interestingly, conditions (1.20) and (1.21) are equivalent to a single condition or a global Hölder condition |f (x1 ) − f (x2 )| ≤ A
|x1 − x2 |λ , (1 + |x1 |λ ) (1 + |x2 |λ )
(1.22)
which can be verified directly. Let C m = C m (Ω) denote a space of functions which are m times continuously differentiable on Ω with the norm kf kC m =
m X k=0
kf (k) kC =
m X k=0
sup f (k) (x) , m = 0, 1, 2, . . . .
x∈Ω
The following definition characterizes the Hölder space of another type.
(1.23)
1.1. Riemann-Liouville integral
7
Definition 1.6. Let λ = m + σ, where m = 0, 1, 2, . . . , and 0 < σ ≤ 1. We say that f (x) ∈ H λ,k = H λ,k (Ω), k > 0, if f (x) ∈ C m (Ω) and k 1 1 (m) (m) σ ≤ A|h| , |h| < . (x + h) − f (x) ln (1.24) f |h| 2 And the corresponding norm is defined as kf kH λ,k = kf kC m
(m) f (x + h) − f (m) (x) . + sup k x,x+h∈Ω, σ ln 1 |h| |h|≤1/2 |h|
(1.25)
Now we present a series of mapping properties of the fractional integration operator on Hölder spaces and the Lp space. Theorem 1.7. Let f (x) ∈ H λ ([a, b]), λ ≥ 0, and α > 0. Then the fractional integral −α RL Da,x f (x) has the form −α RL Da,x f (x)
=
m X k=0
f (k) (a) (x − a)α+k + ψ(x), Γ(α + k + 1)
where m is the maximal integer such that m < λ, and λ+α , if λ + α is not an integer or if λ H and α are integers, ψ(x) ∈ λ+α,1 H , if λ + α is an integer but λ and α are not integers.
(1.26)
(1.27)
Proof. We first prove the case of 0 ≤ λ ≤ 1 and 0 < α < 1. Representing RL D−α a,x f (x) as Z x Z x f (a) 1 f (t) − f (a) −α (x − t)α−1 dt + dt, (1.28) RL Da,x f (x) = Γ(α) a Γ(α) a (x − t)1−α we obtain (1.26) with ψ(x) given by 1 ψ(x) = Γ(α)
Z a
x
f (t) − f (a) dt. (x − t)1−α
(1.29)
It remains to show that ψ(x) satisfies (1.27). If λ + α ≤ 1, we set g(x) = f (x) − f (a), so that |g(x)| ≤ A(x − a)λ . Let h > 0, x, x + h ∈ [a, b]. We have ψ(x + h) − ψ(x) Z x−a Z x−a g(x − t) g(x − t) 1 dt − dt = Γ(α) (t + h)1−α t1−α −h 0 Z 0 1 g(x − t) − g(x) g(x) = [(x − a + h)α − (x − a)α ] + dt Γ(1 + α) Γ(α) −h (t + h)1−α Z x−a 1 + (t + h)α−1 − tα−1 [g(x − t) − g(x)] dt Γ(α) 0 =J1 + J2 + J3 .
(1.30)
(1.31)
8
Chapter 1. Fractional integrals
If h ≥ x − a, then (1.30) yields |J1 | ≤
A (x − a)λ |(x − a + h)α − (x − a)α | ≤ chλ+α . Γ(1 + α)
(1.32)
If 0 < h < x − a, combining (1.30) with the inequality (1 + t)α − 1 ≤ αt which holds for arbitrary t > 0, we have α h A λ+α |J1 | ≤ (x − a) − 1 1+ x−a Γ(1 + α) (1.33) λ+α−1 λ+α ≤ ch(x − a) ≤ ch . We can also readily observe that Z
A |J2 | ≤ Γ(α)
0
|t|λ dt ≤ chλ+α . (t + h)1−α
−h
(1.34)
For J3 , it holds that |J3 | ≤
A Γ(α)
x−a
Z
tλ tα−1 − (t + h)α−1 dt
0
Ahλ+α = Γ(α)
Z
x−a h
(1.35) λ
t tα−1 − (t + 1)α−1 dt.
0
As a result, the estimate |J3 | ≤ chλ+α , λ + α ≤ 1, holds if x − a ≤ h. If x − a > h and λ + α < 1, we have |J3 | ≤ chλ+α also, in view of the convergence of the second integral on the right-hand side of equation (1.35) at infinity, since " α−1 # α−1 1 α−1 α−1 t − (t + 1) =t 1− 1+ ≤ ctα−2 , t > 1. (1.36) t If λ + α = 1, then (1.35) yields the estimate λ+α |J3 | ≤ Ah
Z
x−a h
! t
c+
λ+α−2
dt
1
(1.37)
x−a 1 ≤ c1 h + c2 h ln ≤ ch ln h h
provided that 0 < h < 21 . Collecting the estimates for J1 , J2 , and J3 yields the conclusion that ψ ∈ H λ+α if λ + α 6= 1, and ψ ∈ H λ+α,1 when λ + α = 1. In a similar manner, the result ψ ∈ H λ+α still holds when λ + α > 1. Now we consider the case with arbitrary λ ≥ 0 and α > 0. Take into account that the Beta function given by Z 1 B(α, β) = (1 − z)α−1 z β−1 dz (1.38) 0
can be expressed through the Gamma function in the form B(α, β) =
Γ(α)Γ(β) . Γ(α + β)
(1.39)
1.1. Riemann-Liouville integral
An affine transformation z =
t−a x−a
9
with respect to t yields that
m X f (k) (a) −α (x − a)α+k ψ(x) = D f (x) − RL a,x Γ(α + k + 1) k=0 Z m x X f (t) 1 B(α, k + 1) dt − = f (k) (a)(x − a)α+k 1−α Γ(α) (x − t) Γ(α)Γ(k + 1) a k=0 Z x 1 f (t) = dt Γ(α) a (x − t)1−α Z m X f (k) (a)(x − a)α+k 1 k − z (1 − z)α−1 dz Γ(α)Γ(k + 1) 0 k=0 Z Z x m x X f (k) (a) 1 f (t) (t − a)k dt − dt = 1−α Γ(α) a (x − t) Γ(α)Γ(k + 1) a (x − t)1−α k=0 # Z x" m (k) k X 1 f (a)(t − a) = f (t) − (x − t)α−1 dt. Γ(α) a k!
(1.40)
k=0
By virtue of the Taylor expansion f (t) =
m X f (k) (a)(t − a)k k=0
k!
+
(−1)m m!
Z
t
(s − t)m f (m+1) (s)ds,
(1.41)
a
we have ψ(x) given by Z xZ t (−1)m ψ(x) = (s − t)m f (m+1) (s)(x − t)α−1 dsdt Γ(α)Γ(m + 1) a a Z x Z x 1 (m+1) f (s) (t − s)m (x − t)α−1 dtds = Γ(α)Γ(m + 1) a s Z x B(m + 1, α) = f (m+1) (s)(x − s)m+α ds Γ(α)Γ(m + 1) a Z x 1 = f (m+1) (s)(x − s)m+α ds. Γ(m + 1 + α) a
(1.42)
Let {α} = α − [α]. Then the function ψ(x) has a derivative of order m + [α] in the form ψ (m+[α]) (x) =
1 Γ({α})
Z
x
h
i f (m) (t) − f (m) (a) (x − t){α}−1 dt.
(1.43)
a
In view of the proof for the case with 0 ≤ λ ≤ 1 and 0 < α < 1, the equality (1.26) remains valid in this case. The proof is thus completed. Corollary 1.8. Let 0 < α < 1 and 0 ≤ λ ≤ 1. It follows from Theorem 1.7 that the operator R x f (t)−f (a) 1 λ λ+α if λ + α 6= 1 and into H λ+α,1 if λ + α = 1. Γ(α) 0 (x−t)1−α dt is bounded from H into H
10
Chapter 1. Fractional integrals
Theorem 1.9. [75] The fractional integration operator Lp (a, b) (1 ≤ p ≤ ∞):
−α RL Da,x
(b − a)α , Γ(1 + α)
(1.44)
, 1 ≤ p < ∞,
(1.45)
kRL D−α a,x f kLp ≤ Kkf kLp , K = where Z
! p1
b
kf kLp =
with α > 0 is bounded in
p
|f (x)| dx a
and kf kLp = esssupa≤x≤b |f (x)|, p = ∞
(1.46)
are the Lp -norms. Proof. The proof of this theorem can be verified by simple operations using the generalized Minkowski inequality Z Ω1
Z dx
Ω2
p p1 Z f (x, y)dy ≤
Z
|f (x, y)| dx
dy
Ω2
p1
p
;
(1.47)
Ω1
the details are omitted. Theorem 1.10 (Hardy-Littlewood Theorem). [148] If 0 < α < 1 and 1 < p < p p q fractional integration operator RL D−α a,x is bounded from L into L with q = 1−αp . Theorem 1.11. [148] If α > 0 and p >
1 α,
bounded from Lp (a, b) into H (a, b) if α − p1 6= 1, 2, . . . and into H 1, 2, . . ., and 1 α− p −α as x → a. RL Da,x f (x) = o (x − a) 1 p
+
1 p0
then the
then the fractional integration operator RL D−α a,x is
1 α− p
Here p0 satisfies
1 α,
1 1 α− p , p0
(a, b) if α −
1 p
=
(1.48)
= 1.
Proof. We first consider the case with α −
1 p
≤ 1. For x, x + h ∈ [a, b], we have
−α −α RL Da,x+h f (x + h) − RL Da,x f (x) Z x+h 1 (x + h − t)α−1 f (t)dt = Γ(α) x Z x 1 + (x + h − t)α−1 − (x − t)α−1 f (t)dt Γ(α) a =I1 + I2 . Utilizing the Hölder inequality, we have ! p1 Z x+h 1 |I1 | ≤ |f (t)|p dt Γ(α) x 1 ≤ chα− p kf kLp
Z
x+h
x
(1.49)
! 10 p
0
(x + h − t)(α−1)p dt (1.50)
1.1. Riemann-Liouville integral
11
and 10 Z x p 0 kf kLp α−1 α−1 p dt (x + h − t) − (x − t) |I2 | ≤ Γ(α) a
≤ |Γ(α)|
1 −1 α− p
h
Z kf kLp
x−a h
p0 α−1 t − (t + 1)α−1 dt
(1.51)
! 10 p
.
0
If x − a ≤ h, then an estimate for I2 is clear. If x − a > h, the inequalities |xµ − y µ | ≤ |µ|(x − y)y µ−1 , x ≥ y > 0, µ ≤ 1,
(1.52)
and 1
1
1
(A + B) p0 ≤ A p0 + B p0
(1.53)
" # 10 Z (x−a)/h p 1 α− p (α−2)p0 p |I | ≤ h kf k c + c t dt L 1 2 2 1 " 1−α+ p1 # 1 h α− ≤ h p kf kLp c3 + c4 x−a
(1.54)
yield that
10 p with c4 being additionally multiplied by ln x−a in the case with α − h derive the estimate chα− p1 kf kLp , α − p1 < 1, |I2 | ≤ 1 ch ln 1 p0 kf kLp , α − 1 = 1. h p
1 p
= 1. Hence we
(1.55)
Collecting the estimates for I1 and I2 , we complete the proof of the theorem when α − p1 ≤ 1. Now let α − p1 > 1. Then k < α − p1 ≤ k + 1, k = 1, 2, . . ., and this case is reduced to the previous one by direct differentiation, dk −α −(α−k) f (x), 0 < α − k ≤ 1, RL Da,x f (x) = RL Da,x dxk
(1.56)
using the definition of the spaces H λ and H λ,k in the case with λ > 1. This ends the proof.
∞ Remark 1.4. (I) The boundedness of the operator RL D−α into H α stated in Corollary a,x from L 1.8 corresponds to the case with p = ∞ in Theorem 1.11. (II) Mapping properties of the fractional integration operator RL D−α x,b can be similarly summarized as Theorems 1.7–1.11.
We continue presenting other fundamental properties of fractional integrals. One of the reasons for proposing fractional integrals is to consider them as a generalization of the classical integer-order integration. Fractional integrals (1.13) and (1.14) appear as a unification of the integer-order one, which is elucidated by the following assertion.
12
Chapter 1. Fractional integrals
Property 1.1. [139] Definitions (1.13) and (1.14) coincide with the n-th fold integrals of the forms −n RL Da,x f (x) Z tn−1 Z t1 Z x f (tn )dtn dt1 dt2 · · · = (1.57) a a a Z x 1 (x − t)n−1 f (t)dt = (n − 1)! a and −n RL Dx,b f (x) Z b Z b Z b = dt1 dt2 · · · f (tn )dtn x t1 tn−1 Z b 1 = (x − t)n−1 f (t)dt (n − 1)! x
(1.58)
whenever α = n ∈ Z+ in (1.13) and (1.14). Apart from coinciding with the integer-order integration, fractional integrals also inherit the following semigroup property, which implies that the commutative law of the Riemann-Liouville integration operators is also valid. Property 1.2. [146] The left- and right-sided Riemann-Liouville fractional integral operators satisfy the semigroup properties (
−β −α RL Da,x RL Da,x f (x)
= RL D−α−β f (x), a,x
−β −α RL Dx,b RL Dx,b f (x)
−α−β = RL Dx,b f (x),
(1.59)
where α, β > 0. If f (x) is continuous on [a, b], then lim
x→a
−α RL Da,x f (x)
= lim RL D−α x,b f (x) = 0, α > 0. x→b
(1.60)
Proof. For a suitably smooth function f (x), it holds that −α −β RL Da,x RL Da,x f (x) Z x 1 = (x − t)α−1 RL D−β f (t) dt a,t Γ(α) a Z x Z t 1 α−1 = (x − t) dt (t − τ )β−1 f (τ )dτ Γ(α)Γ(β) a a Z x Z x 1 f (τ )dτ (x − t)α−1 (t − τ )β−1 dt = Γ(α)Γ(β) a τ Z x 1 = (x − τ )α+β−1 f (τ )dτ Γ(α + β) a =RL D−(α+β) f (x). a,x
(1.61)
1.1. Riemann-Liouville integral
13
Here the equality Z
x
(x − t)α−1 (t − τ )β−1 dt =
τ
Γ(α)Γ(β) (x − τ )α+β−1 Γ(α + β)
(1.62)
is utilized. The second equality in equation (1.59) can be similarly derived. Equation (1.60) can be proved directly from the definition of fractional integrals. Similar to the integer-order integration, Riemann-Liouville integrals (1.13) and (1.14) of the power functions (x − a)β−1 and (b − x)β−1 yield power functions of the same form. The following assertion states this in detail. Property 1.3. [75] If α > 0 and β > 0, then −α RL Da,x (x
− a)β−1 =
Γ(β) (x − a)α+β−1 Γ(β + α)
(1.63)
−α RL Dx,b (b
− x)β−1 =
Γ(β) (b − x)α+β−1 . Γ(β + α)
(1.64)
and
Proof. According to the definition of the left-sided Riemann-Liouville integral, we have Z x 1 −α β−1 = (x − t)α−1 (t − a)β−1 dt, (1.65) RL Da,x (x − a) Γ(α) a and the integral converges for β − 1 > −1 and α − 1 > −1. Performing the substitution t = a + ξ(x − a) and using the definition of Beta function yield that −α β−1 RL Da,x (x − a) Z 1 1 α+β−1 = (x − a) ξ β−1 (1 − ξ)α−1 dξ Γ(α) 0 (1.66) = 1 B(α, β)(x − a)α+β−1 Γ(α) Γ(β) = (x − a)α+β−1 . Γ(β + α) A similar result for the right-sided fractional integral can be derived in the same manner.
When considering linear integral/differential equations, we often use the Fourier transform and the Laplace transform. Now we present the Fourier transform and the Laplace transform formulae for Riemann-Liouville integral, which will be useful in the coming chapters. Theorem 1.12. [139] The Fourier transform and the Laplace transform of the Riemann-Liouville integral are given by −α F{RL D−∞,x f (x); ω} Z ∞ (1.67) −α b eiωx RL D−α f (ω), α > 0, = −∞,x f (x)dx = (−iω) −∞
14
Chapter 1. Fractional integrals
and L{RL D−α 0,x f (x); p}
∞
Z
−α e−px RL D−α F (p), α > 0. 0,x f (x)dx = p
=
(1.68)
0
Here fb(ω) and F (p) are the Fourier transform and Laplace transform of f (x), respectively. Proof. Define the auxiliary function ( h+ (x) =
xα−1 Γ(α) ,
x > 0,
0,
x ≤ 0.
(1.69)
Then we have −α RL D−∞,x f (x) =
Z
1 Γ(α)
(x − t)α−1 f (t)dt = h+ (x) ∗ f (x).
(1.70)
−∞
Since the identity 1 Γ(α)
x
∞
Z
xα−1 e−px dx = p−α
(1.71)
0
gives the Fourier transform formula for h+ (x) in the form Z ∞ Z ∞ 1 F{h+ (x); ω} = eiωx h+ (x)dx = eiωx xα−1 dx = (−iω)−α , Γ(α) 0 −∞
(1.72)
the relation F{h+ (x) ∗ f (x); ω} = F{h+ (x); ω}F{f (x); ω} yields −α b f (ω). F{RL D−α −∞,x f (x); ω} = (−iω)
(1.73)
Note that −α RL D0,x f (x) =
1 Γ(α)
α−1
Z
x
tα−1 f (x − t)dt =
0
1 α−1 x ∗ f (x). Γ(α)
(1.74)
α−1
Combining the relation L{ xΓ(α) ∗ f (x); p} = L{ xΓ(α) ; p}L{f (x); p} and identity (1.71), we can readily obtain −α L{RL D−α F (p). (1.75) 0,x f (x); p} = p Hence the proof is completed.
1.2 Fractional integrals of other types Apart from the Riemann-Liouville fractional integral, there are fractional integrals of other types worth mentioning, especially the Riesz fractional integral. We introduce these fractional integrals in the present section. Definition 1.13. [148] The Riesz fractional integration Iα is realized via the Riesz potential, which is defined as the Fourier convolution of the form Z α (I f ) (x) = kα (x − y)f (y)dy, x ∈ Rd , α > 0. (1.76) Rd
Here the function kα (x), called the Riesz kernel, is given by ( |x|α−d , α − d 6= 0, 2, 4, . . . , 1 kα (x) = 1 γd (α) |x|α−d log |x| , α − d = 0, 2, 4, . . . ,
(1.77)
1.2. Fractional integrals of other types
and the constant γd (α) has the form α d Γ( α 2) 2 π 2d−α , Γ( 2 ) γd (α) = d−α d (−1) 2 2α−1 π 2 Γ 1 +
15
α − d 6= 0, 2, 4, . . . , (1.78) α−d 2
Γ
α 2
,
α − d = 0, 2, 4, . . . .
Normally, the Riesz fractional integral is considered in the setting of the Lizorkin space Φ(Rd ) , Φ = ϕ ∈ S(Rd ) Dk ϕ(0) = 0 , |k| ∈ {0} ∪ Z+ , (1.79) which is a subspace of the Schwartz space S(Rd ), X |N| k ∞ d d D f (x) < ∞ S(R ) = f ∈ C (R ) sup (1 + |x|) x∈Rd |k|≤|N|, + |N|∈{0}∪Z
(1.80)
and is invariant with respect to the Riesz potential Iα . Here N = (N1 , N2 , . . . , Nd ) and k = Pd Pd (k1 , k2 , . . . , kd ) are the multi-indexes with |N| = i=1 Ni and |k| = i=1 ki . Constrained on the Lizorkin space, the Riesz fractional integral has the following properties. Property 1.4. [75] If α > 0, then Fourier transform of the Riesz potential Iα in (1.76) reads F {Iα f (x); ω} = |ω|−α fb(ω), ω ∈ Rd , f ∈ Φ,
(1.81)
with fb(ω) being the Fourier transform of f (x). The property above instantly gives the following assertion. Corollary 1.14. [75] If α > 2, then it holds that ∆Iα f = −Iα−2 f
(1.82)
whenever f ∈ Φ. Here ∆ denotes the classical Laplace operator. Property 1.5. [75] The Lizorkin space Φ is invariant with respect to the Riesz potential Iα . Moreover, Iα (Φ) = Φ and Iα Iβ f = Iα+β f, α > 0, β > 0, f ∈ Φ.
(1.83)
For the existence condition and mapping properties of Riesz fractional integral, we present the following results. Property 1.6. [148] If 0 < α < d ∈ Z+ and 1 < p < defined for f ∈ Lp (Rd ).
d α,
then the Riesz potential (Iα f )(x) is
Theorem 1.15 (Sobolev Theorem). [148] Let α > 0, 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞. The operator Iα is bounded from Lp (Rd ) into Lq (Rd ) if and only if 0 < α < d ∈ Z+ , 1 < p
0, (1.85) D f (x) = −∞ x Γ(α) −∞ (x − t)1−α 1 Γ(α)
−α x D∞ f (x) =
∞
Z
x
f (t)dt , x ∈ R, α > 0. (t − x)1−α
(II) Liouville fractional integral on the half axis: Z x f (t)dt 1 −α , x > 0, α > 0, D f (x) = 0 x Γ(α) 0 (x − t)1−α 1 Γ(α)
−α x D∞ f (x) =
Z
∞
x
f (t)dt , x > 0, α > 0. (t − x)1−α
(III) Limit expression of the Grünwald-Letnikov fractional integral: [ x−a h ] X −α α j −α D f (x) = lim h (−1) f (x − jh) GL a,x h→+0 j j=0 x−a
[ h ] X Γ(j + α) 1 = lim hα f (x − jh), a < x < b, Γ(α) h→+0 Γ(j + 1) j=0
[ b−x h ] X −α α j −α lim h (−1) f (x + jh) GL Dx,b f (x) = h→+0 j j=0
[ b−x h ]
(1.86)
(1.87)
(1.88)
(1.89)
(1.90)
X Γ(j + α) 1 = lim hα f (x + jh), a < x < b. Γ(α) h→+0 Γ(j + 1) j=0
(IV) Hadamard fractional integral: Z x 1 x α−1 dt −α log f (t) , 0 < a < x < b, α > 0, H Da,x f (x) = Γ(α) a t t −α H Dx,b f (x) =
1 Γ(α)
Z
b
x
α−1 t dt log f (t) , 0 < a < x < b, α > 0. x t
(1.91)
(1.92)
(V) Erdélyi-Kober type fractional integral: Iα a,x; σ,η f (x)
σx−σ(α+η) = Γ(α)
Iα x,b; σ,η f (x) =
σxση Γ(α)
Z
x ση+σ−1
Z a
t f (t)dt , α > 0, σ σ (x − t )1−α
(1.93)
b σ(1−α−η)−1
x
t
f (t)dt , α > 0, (tσ − xσ )1−α
where 0 ≤ a < x < b ≤ ∞ for any real σ or −∞ ≤ a < x < b ≤ ∞ for any integer σ.
(1.94)
1.2. Fractional integrals of other types
(VI) Fractional integrals of a function f (x) by a function g(x) of order α: Z x g 0 (t)f (t)dt 1 α Ia,x; g f (x) = , α > 0, −∞ ≤ a < b ≤ ∞, Γ(α) a [g(x) − g(t)]1−α α Ix,b; g f (x) =
1 Γ(α)
Z
b
x
g 0 (t)f (t)dt , α > 0, −∞ ≤ a < b ≤ ∞, [g(t) − g(x)]1−α
17
(1.95)
(1.96)
where f (x) ∈ L1 (a, b) and g(x) is a monotone increasing function with a continuous derivative. Remark 1.6. It should be noted that Grünwald-Letnikov fractional integrals coincide with Riemann-Liouville integrals (1.13) and (1.14) when certain smooth conditions are satisfied. Therefore, Grünwald-Letnikov fractional integrals can be used for evaluating Riemann-Liouville integrals numerically, which will be discussed in Chapter 3. Readers can also refer to [75] and [148] for more details of various fractional integrals.
Chapter 2
Fractional derivatives
In this chapter, we mainly concentrate on introducing typical fractional derivatives, such as Riemann-Liouville, Caputo, and Riesz derivatives, and the fractional Laplacian. Their differences and relationships are described in detail. In addition, fractional derivatives of other types, such as the Hadamard derivative and Grünvald-Letnikov derivative, are also introduced.
2.1 Riemann-Liouville derivative Consider again the decoupled CTRW process introduced in the previous chapter. In order to ∂ spot features of subdiffusion, we apply the differential operator ∂t on both sides of the fractional integral equation (1.12) and obtain the fractional diffusion equation (FDE) t
∂2 Kα 2 W (x, s) ds, 0 < α < 1. ∂x
(2.1)
Z t ∂2 1 ∂ ∂2 α−1 Kα 2 W (x, t) = (t − s) Kα 2 W (x, t) ds. ∂x Γ(α) ∂t 0 ∂x
(2.2)
∂ 1 ∂ W (x, t) = ∂t Γ(α) ∂t
Z
α−1
(t − s) 0
Denote 1−α RL D0,t
Then it holds that ∂ ∂2 W (x, t) = RL D1−α K W (x, t) . α 0,t ∂t ∂x2
(2.3)
The integro-differential nature of RL D1−α 0,t according to equation (2.2), with the integral kernel α−1 M (t) ∝ t , ensures the non-Markovian nature of this diffusive process characterized by the fractional differential equation (2.1) [127]. In this case, the fractional differential equation (2.1) can be rewritten in the equivalent form α RL D0,t W (x, t)
−
∂2 t−α W (x, 0) = Kα 2 W (x, t) Γ(1 − α) ∂x
(2.4)
due to RL D−α 0,t W (x, t) t=0 = 0 if W (x, t) is assumed to be continuous with respect to t. It is evident that the second term of the left-hand side of equation (2.4) decays with inverse power-law form (i.e., with negative power) with respect to t, not exponentially fast as in the case of standard 19
20
Chapter 2. Fractional derivatives
diffusion [10, 127, 191]. Furthermore, in the limit α → 1, the fractional differential equation (2.1) reduces to the classical Fick’s second law, as expected. The notation RL Dα 0,t , which is called the Riemann-Liouville differential operator, turns out to be essential in fractional differential equations characterizing anomalous phenomena. It is therefore necessary to study fractional derivatives of this type. Definition 2.1. The left- and right-sided Riemann-Liouville derivatives of order α are defined by dm −(m−α) α f (x) RL Da,x f (x) = m RL Da,x dx Z x 1 dm f (t) = dt, x > a, m α−m+1 Γ(m − α) dx a (x − t)
(2.5)
m −(m−α) α m d f (x) RL Dx,b f (x) =(−1) dxm RL Dx,b Z b (−1)m dm f (t) = dt, x < b, m α−m+1 Γ(m − α) dx x (t − x)
(2.6)
and
respectively, where m is a positive integer satisfying m − 1 ≤ α < m. Remark 2.1. (I) The Riemann-Liouville derivative usually refers to the left-sided RiemannLiouville derivative if no confusion arises. (II) In the special cases with a = −∞ and b = ∞, the Riemann-Liouville derivatives (2.5) and α (2.6) reduce to Liouville fractional derivatives −∞ Dα x f (x) and x D∞ f (x), respectively. Similar to the case of classical differentiation, there exist functions nondifferentiable in the Riemann-Liouville sense. For fractional differentiability in the Riemann-Liouville sense, we present the following result, which is also related to the absolute continuity. Theorem 2.2. [102] If f (x) ∈ AC m ([a, b]), then the Riemann-Liouville derivatives RL Dα a,x f , α + D f (m − 1 ≤ α < m ∈ Z ) exist almost everywhere on [a, b] and may be represented in RL x,b the forms α RL Da,x f (x)
= =
m−1 X k=0
f (k) (a)(x − a)k−α 1 + Γ(k − α + 1) Γ(m − α)
Z
x
f (m) (t) dt, (x − t)α+1−m
a
(2.7)
α RL Dx,b f (x) m−1 X k=0
(−1)m (−1)k f (k) (b)(b − x)k−α + Γ(k − α + 1) Γ(m − α)
Z
b
x
f (m) (t) dt. (t − x)α+1−m
(2.8)
Proof. Since f (x) ∈ AC m ([a, b]), we have the representation 1 f (x) = (m − 1)!
Z a
x
(x − t)m−1 ϕ(t)dt +
m−1 X k=0
ck (x − a)k
(2.9)
2.1. Riemann-Liouville derivative
21
with ϕ(x) = f (m) (x) ∈ L1 ([a, b]) and ck = formula for the m-fold integral, Z x Z xn Z x2 dxn dxn−1 · · · ϕ(x1 )dx1 = a
a
a
f (k) (a) k! ,
which follows from the well-known
1 (m − 1)!
Z
x
(x − t)m−1 ϕ(t)dt.
(2.10)
a
Substituting the representation (2.9) into equation (2.5) gives α RL Da,x f (x) "Z Z t x 1 1 dm m−α−1 = (x − t) (t − s)m−1 ϕ(s)dsdt Γ(m − α) dxm a Γ(m) a # Z x m−1 X m−α−1 k + (x − t) ck (t − a) dt a k=0 " Z x Z x m 1 d 1 ϕ(s)ds (x − t)m−α−1 (t − s)m−1 dt = Γ(m − α) dxm Γ(m) a s # m−1 X Z x m−α−1 k + ck (x − t) (t − a) dt a k=0 " Z m d B(m − α, m) x 1 (x − s)2m−α−1 ϕ(s)ds = m Γ(m − α) dx Γ(m) a # m−1 X + ck B(m − α, k + 1)(x − a)m−α+k k=0 " Z x 1 dm Γ(m − α) = ϕ(s)(x − s)2m−α−1 ds Γ(m − α) dxm Γ(2m − α) a # m−1 X Γ(m − α)f (k) (a)(x − a)m+k−α + Γ(m − α + k + 1) k=0 Z x m−1 X f (k) (a)(x − a)k−α 1 f (m) (t) = + dt Γ(k − α + 1) Γ(m − α) (x − t)α+1−m
(2.11)
a
k=0
after simple transformations. We can also verify (2.8) in a similar manner. The proof is thus completed. As mentioned before, the fractional differential equation (2.1) reduces to the classical equation in the limit α → 1. This is a result of the following consistency for the Riemann-Liouville derivative, which asserts that the Riemann-Liouville derivative is reduced to the integer-order derivative when the derivative order α approaches an integer for any fixed x. Property 2.1. [90] Let f (x) be suitably smooth. For a positive integer m satisfying m − 1 < α < m, the following relations hold: lim
α→(m−1)+
lim
α→m−
α RL Da,x f (x)
α RL Da,x f (x)
= f (m−1) (x),
(2.12)
= f (m) (x).
(2.13)
22
Chapter 2. Fractional derivatives
Proof. For the suitably smooth function f (x), integration by parts gives α lim RL Da,x f (x) α→(m−1)+ Z x f (t) dm 1 dt = lim α→(m−1)+ Γ(m − α) dxm a (x − t)α+1−m m−1 X f (k) (a)(x − a)k−α = lim Γ(k − α + 1) α→(m−1)+ k=0 ! Z x 1 f (m) (t) + dt Γ(m − α) a (x − t)α+1−m Z x =f (m−1) (a) + f (m) (t)dt = f (m−1) (x).
(2.14)
a
In a similar manner, we also have lim RL Dα a,x f (x) α→m− ! Z x m−1 X f (k) (a)(x − a)k−α 1 f (m) (t) = lim + dt α→m− Γ(k − α + 1) Γ(m − α) a (x − t)α+1−m k=0 m−1 X f (k) (a)(x − a)k−α f (m) (a)(x − a)m−α = lim− + Γ(k − α + 1) Γ(m − α + 1) α→m k=0 ! Z x (m+1) 1 f (t) + dt α−m Γ(m − α + 1) a (x − t) Z x =f (m) (a) + f (m+1) (t)dt = f (m) (x).
(2.15)
a
Here the relation
1 Γ(−z)
= 0, z = 0, 1, 2, 3, . . . is utilized. This completes the proof.
Resembling the case of the Riemann-Liouville integral, Riemann-Liouville derivatives of the power functions (x − a)β−1 and (b − x)β−1 yield power functions of the same form. Property 2.2. [75] If m − 1 ≤ α < m ∈ Z+ and β > m, then α RL Da,x (x
− a)β−1 =
Γ(β) (x − a)β−α−1 Γ(β − α)
(2.16)
and
Γ(β) (b − x)β−α−1 . (2.17) Γ(β − α) In particular, if β = 1, then the Riemann-Liouville derivative of a constant is, in general, not equal to zero. Instead, it holds that α RL Dx,b (b
α RL Da,x 1
(x) =
− x)β−1 =
(x − a)−α , Γ(1 − α)
α RL Dx,b 1
(x) =
(b − x)−α , Γ(1 − α)
0 < α < 1.
(2.18)
On the other hand, for j = 1, 2, . . . , [α] + 1, α RL Da,x (x
− a)α−j = 0,
α RL Dx,b (b
− x)α−j = 0.
(2.19)
2.1. Riemann-Liouville derivative
23
Proof. Here we give the proof for (2.16). For m − 1 ≤ α < m ∈ Z+ , we have Z x 1 dm (t − a)β−1 α β−1 D (x − a) = (x − t)m−α−1 dt, RL a,x Γ(m − α) a dtm
(2.20)
since all the nonintegral terms on the right-hand side of (2.7) are equal to 0. Taking into account m d (t − a)β−1 dtm =(β − 1)(β − 2) · · · (β − m)(t − a)β−m−1 (2.21) Γ(β) (t − a)β−m−1 = Γ(β − m) and substituting t = a + ξ(x − a) yield α β−1 RL Da,x (x − a) Z x Γ(β) (x − t)m−α−1 (t − a)β−m−1 dt = Γ(β − m)Γ(m − α) a Γ(β)B(m − α, β − m) (x − a)β−α−1 = Γ(β − m)Γ(m − α) Γ(β) = (x − a)β−α−1 . Γ(β − α)
(2.22)
Equality (2.17) can be similarly derived. The other assertions are direct results of (2.16) and (2.17), so the remaining proofs are omitted. Existence and uniqueness of solutions to fractional differential equations are important in theory and are also necessary for numerically analyzing fractional dynamics, which have been studied in [50] and [121]. As a direct result of Property 2.2, the following corollary is frequently utilized in the study of existence and uniqueness of solutions to fractional differential equations in the Riemann-Liouville sense. Corollary 2.3. [75] Let α > 0 and n = [α] + 1. (I) The equality RL Dα a,x y(x) = 0 holds if and only if y(x) =
n X
cj (x − a)α−j ,
(2.23)
j=1
where cj ∈ R (j = 1, 2, . . . , n) are arbitrary constants. In particular, when 0 < α ≤ 1, α−1 the relation RL Dα with the constant a,x y(x) = 0 holds if and only if y(x) = c(x − a) c ∈ R. (II) The equality RL Dα x,b y(x) = 0 holds if and only if y(x) =
n X
dj (b − x)α−j ,
(2.24)
j=1
where dj ∈ R (j = 1, 2, . . . , n) are arbitrary constants. In particular, when 0 < α ≤ 1, α−1 the relation RL Dα with the constant x,b y(x) = 0 holds if and only if y(x) = d(b − x) d ∈ R.
24
Chapter 2. Fractional derivatives
It is known that ordinary differentiation operations if the lat and integration are reciprocal d n n n ter is applied first. In other words, dx Ia+ ϕ = ϕ(x) but Ia+ ϕ(n) differs from ϕ(x) by a n polynomial of the order n − 1. Here Ia+ denotes the nth-fold integration [148]. We want to know whether or not this relation keeps valid for fractional integral and fractional derivative in −α the Riemann-Liouville sense. In the following, we shall see that RL Dα a,x RL Da,x ϕ = ϕ while α −α RL Da,x RL Da,x ϕ does not necessarily coincide with ϕ(x). Property 2.3. [75] If α ≥ β > 0, then for f (x) ∈ Lp ([a, b]) (1 ≤ p ≤ ∞) the relations −α β RL Da,x RL Da,x f (x)
−(α−β) = RL Da,x f (x)
β −α RL Dx,b RL Dx,b f (x)
= RL Dx,b
and
−(α−β)
(2.25)
f (x)
(2.26)
hold almost everywhere on [a, b]. In particular, when β = k ∈ Z+ and α > k, k −α RL Da,x RL Da,x f (x)
= RL D−(α−k) f (x) a,x
−α k RL Dx,b RL Dx,b f (x)
= RL Dx,b
and
−(α−k)
(2.27)
f (x).
(2.28)
Proof. Suppose that n−1 ≤ β < n ∈ Z+ . Then β = n+(β −n) with β −n < 0. Consequently, the semigroup property of the Riemann-Liouville integral gives β −α RL Da,x RL Da,x f (x) dn β−n −α = n RL Da,x RL Da,x f (x) dx (2.29) n d β−n−α = n RL Da,x f (x) dx =RL D−(α−β) f (x). a,x Equation (2.26) can be verified in a similar manner. The remaining equalities are direct results of (2.25) and (2.26), where RL Dka,x = a) and
k RL Dx,b
=
dk (−1)k dx k
dk dxk
(x >
(x < b) are used. The proof is thus completed.
We can readily obtain the following result from Property 2.3. Corollary 2.4. Let α > 0 and f (x) ∈ Lp ([a, b]) with 1 ≤ p ≤ ∞. Then the relations α −α RL Da,x RL Da,x f (x)
= f (x)
and
−α α RL Dx,b RL Dx,b f (x)
= f (x)
(2.30)
hold almost everywhere on [a, b]. As a result, differentiation in the Riemann-Liouville sense is also a reciprocal operation of integration in the same sense. Property 2.4. [139, 146] Let m−1 ≤ α < m ∈ Z+ . The left- and right-sided Riemann-Liouville derivatives satisfy m (x − a)α−j D−α Dα f (x) = f (x) − X Dα−j f (x) , RL RL RL a,x a,x a,x x=a Γ(α − j + 1) j=1 (2.31) m h i α−j X (b − x) α−j −α α D D f (x) = f (x) − . RL Dx,b f (x) RL x,b RL x,b x=b Γ(α − j + 1) j=1
2.1. Riemann-Liouville derivative
25
For more general cases, let β > 0; then α −β RL Da,x RL Da,x f (x) m X (x − a)β−j α−β α−j , = D f (x) − RL a,x RL Da,x f (x) x=a Γ(1 + β − j) j=1 −β α RL Dx,b RL Dx,b f (x) m h i X (b − x)β−j α−β α−j . = D f (x) − D f (x) RL RL x,b x,b x=b Γ(1 + β − j) j=1
(2.32)
In particular, when α = m is a positive integer, the following equalities hold: m−1 X f (j) (a)(x − a)β+j−m −β m m−β D D f (x) = D f (x) − , RL RL RL a,x a,x a,x Γ(1 + β + j − m) j=0
m−1 X f (j) (b)(b − x)β+j−m m−β −β m . RL Dx,b RL Dx,b f (x) = RL Dx,b f (x) − Γ(1 + β + j − m)
(2.33)
j=0
Proof. We now prove the first composition formula in (2.31). On one hand, we have −α α RL Da,x RL Da,x f (x)
Z x d 1 α α = (x − t) RL Da,t f (t)dt . dx Γ(α + 1) a
(2.34)
On the other hand, repeatedly integrating by parts and recalling (1.59), we have Z x 1 (x − t)α RL Dα a,t f (t)dt Γ(α + 1) a Z x dm 1 −(m−α) (x − t)α m RL Da,t = f (t)dt Γ(α + 1) a dt Z x 1 −(m−α) = (x − t)α−m RL Da,t f (t)dt Γ(α − m + 1) a m m−j X d (x − a)α−j+1 −(m−α) − D f (x) RL a,x dxm−j x=a Γ(2 + α − j) j=1 −(m−α) =RL D−(α−m+1) f (x) RL Da,x a,x m X (x − a)α−j+1 α−j − D f (x) RL a,x x=a Γ(2 + α − j) j=1 m X (x − a)α−j+1 α−j =RL D−1 f (x) − . RL Da,x f (x) x=a a,x Γ(2 + α − j) j=1
(2.35)
26
Therefore,
Chapter 2. Fractional derivatives
−α α RL Da,x RL Da,x f (x) Z x 1 α α = d (x − t) RL Da,t f (t)dt dx Γ(α + 1) a m X (x − a)α−j α−j =f (x) − . RL Da,x f (x) x=a Γ(1 + α − j) j=1
(2.36)
We can similarly derive that the second equality in (2.31) is also valid. Now we give the proof of (2.32). In view of (1.59) and (2.25), the equality −β RL Da,x f (x)
α−β −α = RL Da,x RL Da,x f (x)
holds for arbitrary α, β > 0. Then the composition formula (2.31) gives −β α RL Da,x RL Da,x f (x) −α α =RL Dα−β RL Da,x RL Da,x f (x) a,x m α−j X (x − a) α−j =RL Dα−β f (x) − RL Da,x f (x) x=a a,x Γ(1 + α − j) j=1 m X (x − a)β−j α−β α−j = D f (x) − D f (x) . RL RL a,x a,x x=a Γ(1 + β − j) j=1
(2.37)
(2.38)
The other equality in (2.32) can be proved in a same manner. Relations in (2.33) can be viewed as direct results of (2.32). The proof is thus completed.
Remark 2.2. [99] In view of (2.31), the equalities −α α RL Da,x RL Da,x f (x)
= f (x)
(2.39)
−α α RL Dx,b RL Dx,b f (x)
= f (x)
(2.40)
and hold for m − 1 ≤ α < m provided that α−j α−j RL Da,x f (x) x=a = 0, RL Dx,b f (x) x=b = 0, j = 1, 2, . . . , m.
(2.41)
When f (x) has a sufficient number of continuous derivatives, conditions in (2.41) may be replaced by the following classical homogeneous conditions: f (j) (a) = 0, f (j) (b) = 0, j = 0, 1, . . . , m − 1.
(2.42)
Generally speaking, conditions (2.41) and (2.42) are not equivalent. The later are usually chosen to take the place of the former in studying differential equations of the Riemann-Liouville type, mainly for convenience. In view of Properties 2.3 and 2.4, the Riemann-Liouville integral and derivative are generally not commutative unless homogeneous conditions (2.41) are satisfied, since the functions (x − a)α−k , k = 1, 2, . . . , [α] + 1, may arise. The linear combinations of these functions play the same role of polynomials given by the difference Ian ϕ(n) − ϕ in the integer-order case.
2.1. Riemann-Liouville derivative
27
We shall also demonstrate that the semigroup property and commutativity are generally invalid for the Riemann-Liouville derivative, which is different from the case of the fractional integral. Property 2.5. [101, 102, 139] Let m − 1 < α < m, n − 1 < β < n with m, n being positive α+β β α β α integers. If RL Dα+β a,x f (x), RL Da,x RL Da,x f (x), RL Dx,b f (x), and RL Dx,b RL Dx,b f (x) exist, then β α RL Da,x RL Da,x f (x) n X (2.43) (x − a)−α−j α+β β−j , = D f (x) − D f (x) RL RL a,x a,x x=a Γ(1 − α − j) j=1
β α RL Dx,b RL Dx,b f (x)
α+β =RL Dx,b f (x) −
n X
β−j RL Dx,b f (x) x=b
j=1
(b − x)−α−j . Γ(1 − α − j)
Furthermore, the relations ( α+β α β β α RL Da,x RL Da,x f (x) = RL Da,x RL Da,x f (x) = RL Da,x f (x), β α RL Dx,b RL Dx,b f (x)
α+β = RL Dβx,b RL Dα x,b f (x) = RL Dx,b f (x)
hold if f (x) satisfies the homogeneous conditions β−j RL Dβ−j a,x f (x) x=a = RL Dx,b f (x) x=b α−k = Dα−k f (x) = RL Dx,b f (x) x=b = 0, RL a,x x=a
(2.44)
(2.45)
(2.46)
where j = 1, . . . , n and k = 1, . . . , m. If α = m is a positive integer, then the equations m β RL Da,x RL Da,x f (x)
= RL Dm+β a,x f (x)
(2.47)
β m RL Dx,b RL Dx,b f (x)
= RL Dm+β x,b f (x)
(2.48)
β m RL Da,x RL Da,x f (x)
6= RL Dm+β a,x f (x)
(2.49)
β m RL Dx,b RL Dx,b f (x)
6= RL Dm+β x,b f (x).
(2.50)
and hold while, in general, and In fact, β m RL Da,x RL Da,x f (x) m−1 X f (j) (a) m+β (x − a)j−β−m , = D f (x) − RL a,x Γ(1 + j − β − m) j=0 β m RL Dx,b RL Dx,b f (x) m−1 X f (j) (b) m+β = D f (x) − (b − x)j−β−m . RL x,b Γ(1 + j − β − m)
(2.51)
j=0
As a result, the commutativity of Riemann-Liouville differentiation operators generally does not hold.
28
Chapter 2. Fractional derivatives
Proof. Since m − 1 < α < m, the relation (2.32) gives β α RL Da,x RL Da,x f (x) i dm h β −(m−α) D f (x) D = RL a,x RL a,x dxm dm β−(m−α) f (x) = dxm RL Da,x n X (x − a)(m−α)−j β−j − D f (x) RL a,x x=a Γ(1 + m − α − j) j=1 n X (x − a)−α−j α+β β−j = D f (x) − D f (x) RL RL a,x a,x x=a Γ(1 − α − j)
(2.52)
β α RL Dx,b RL Dx,b f (x) m −(m−α) β m d =(−1) D RL Dx,b f (x) m RL x,b dx dm β−(m−α) m =(−1) f (x) RL Dx,b dxm n X (b − x)(m−α)−j β−j − D f (x) RL a,x x=b Γ(1 + m − α − j) j=1 n X (b − x)−α−j α+β β−j . = D f (x) − RL Dx,b f (x) x=b RL x,b Γ(1 − α − j)
(2.53)
j=1
and
j=1
Thus (2.43) and (2.44) are proved to be valid. The remaining results can be readily derived from these two equations. Note that equalities 1 (2.47) and (2.48) are valid since Γ(1−m−j) = 0 with m ≥ 1 and j ≥ 1. This ends the proof.
The Fourier transform and Laplace transform are commonly used when we deal with differential equations. Here we present the following assertions about those two integral transforms for the Riemann-Liouville derivative. Theorem 2.5. [139] Let m − 1 ≤ α < m ∈ Z+ . The Fourier transform of the Riemann-Liouville derivative with lower terminal a = −∞ is given by F{RL Dα −∞,x f (x); ω}
Z
∞
=
αb eiωx RL Dα −∞,x f (x)dx = (−iω) f (ω),
(2.54)
−∞
provided that f (x) along with its integer-order derivatives up to order (m − 1) vanish at x = −∞. Here fb(ω) denotes the Fourier transform of f (x). The Laplace transform of the Riemann-
2.1. Riemann-Liouville derivative
29
Liouville derivative with lower terminal a = 0 is L{RL Dα 0,x f (x); p} Z ∞ = e−px RL Dα 0,x f (x)dx
(2.55)
0
m−1 X α−k−1 α pk RL D0,x f (x) x=0 , =p F (p) − k=0
in which F (p) denotes the Laplace transform of f (x). Proof. When f (x) and its integer-order derivatives vanish at x = −∞, we can rewrite its Riemann-Liouville derivative RL Dα −∞,x f (x) as α RL D−∞,x f (x) =
1 Γ(m − α)
Z
x
−∞
f (m) (t) −(m−α) dt = RL D−∞,x f (m) (x) (x − t)α+1−m
(2.56)
through integrating by parts. The Fourier transform formulae for the Riemann-Liouville integral and integer-order derivative lead to F{RL Dα −∞,x f (x); ω} =(−iω)−(m−α) F{f (m) ; ω} (2.57) −(m−α) (−iω)m fb(ω) =(−iω) =(−iω)α fb(ω). When considering the Laplace transform formula for the Riemann-Liouville derivative, we can first express RL Dα 0,x f (x) in the form α RL D0,x f (x)
= g (m) (x)
(2.58)
with −(m−α)
g(x) = RL D0,x
f (x).
(2.59)
Then the Laplace transform formula for the integer-order derivative, L{f (n) (x); p} = pn F (p) −
n−1 X
pk f (n−k−1) (0),
(2.60)
k=0
yields m L{RL Dα 0,x f (x); p} = p L{g(x); p} −
m−1 X
pk g (m−k−1) (0).
(2.61)
k=0
Note that −(m−α)
L{g(x); p} = L{RL D0,x
f (x); p} = p−(m−α) F (p),
(2.62)
and, for k = 0, 1, . . . , m − 1, g (m−k−1) (x) =
dm−k−1 −(m−α) α−k−1 f (x) = RL D0,x f (x). RL D0,x dxm−k−1
(2.63)
30
Chapter 2. Fractional derivatives
We obtain α L{RL Dα 0,x f (x); p} = p F (p) −
m−1 X
pk
α−k−1 f (x) x=0 RL D0,x
.
(2.64)
k=0
This ends the proof. The above transform formulae may also help us to observe the formulation of initial value conditions, which need to be posed for differential equations with Riemann-Liouville derivative. For instance, the Laplace transform formula in (2.55) implies that the Riemann-Liouville approach results in initial conditions containing the limit values of the Riemann-Liouville derivative/integral at the lower terminal in the forms [139] lim RL Dα−1 a,x f (x) = b1 , x→a lim RL Dα−2 a,x f (x) = b2 , x→a (2.65) . .. lim α−m RL Da,x f (x) = bm , x→a
where bk , k = 1, 2, . . . , m, are given constants. As a result, special attention should be paid to behaviors of the Riemann-Liouville integral and derivative near and/or far from the lower terminal. Property 2.6. [139] If f (x) is analytic at least in the interval [a, a + ] for some small positive , and can be represented by the Taylor series f (x) =
∞ X f (k) (a) k=0
in this interval, then
k!
(x − a)k
0, α f (a), lim RL Da,x f (x) = x→a+ ∞,
(2.66)
α < 0, α = 0,
(2.67)
α > 0.
If we allow f (x) to have an integrable singularity at x = a, then it can be written in the form f (x) = (x − a)β f∗ (x) with f∗ (x) 6= 0 and β > −1. Suppose that f∗ (x) can be represented by its Taylor series; then α < β, 0, f∗ (a)Γ(β+1) lim+ RL Dα f (x) = (2.68) a,x Γ(β−α+1) , α = β, x→a ∞, α > β. Proof. Equation (2.67) can be derived directly from term-by-term differentiation of (2.66) using the formulae (1.63) and (2.16) for power functions. When f (x) has an integral singularity at x = a, we can write β f (x) = (x − a) f∗ (x) ∞ (k) X f∗ (a)(x − a)k β = (x − a) k! (2.69) k=0 ∞ (k) X f∗ (a)(x − a)β+k = . k! k=0
2.1. Riemann-Liouville derivative
31
Performing term-by-term Riemann-Liouville fractional differentiation gives α RL Da,x f (x) =
∞ (k) X f∗ (a) k=0
k!
Γ(β + k + 1) (x − a)β+k−α , Γ(β + k − α + 1)
(2.70)
from which it follows that (2.68) is valid. Here we have to pause to introduce the Leibniz rule for the Riemann-Liouville derivative, which is of help for considering behaviors of the Riemann-Liouville derivative far from the lower terminal. Lemma 2.6. [139] Let f (x) and ϕ(x) along with all their derivatives be continuous on [a, b]. Then the Leibniz rule for fractional derivatives takes the form α RL Da,x
∞ X α (k) (ϕ(x)f (x)) = ϕ (x)RL Dα−k a,x f (x), k
(2.71)
k=0
which is somewhat similar to that for the nth order derivative of the product ϕ(x)f (x) with n ∈ Z+ . Property 2.7. [139] Suppose that f (x) is analytic. If x is far from the lower terminal a with |x| |a|, then it holds that α RL Da,x f (x)
≈ RL Dα 0,x f (x).
(2.72)
If a → −∞, then, for a fixed value of x with |a| |x|, α RL Da,x f (x)
≈ RL Dα x+a,x f (x)
(2.73)
under certain conditions on f (x), for instance, f (x) is bounded. Proof. Let ϕ(x) = H(x − a), where H(x) is the Heaviside function with its fractional derivative given by (x − a)−α , x > a. Γ(1 − α)
(2.74)
α RL Da,x f (x) =RL Dα a,x (f (x)ϕ(x)) ∞ X α (k) = f (x)RL Dα−k a,x H(x − a) k k=0 ∞ X α (x − a)k−α (k) = f (x). k Γ(k − α + 1)
(2.75)
α RL Da,x H(x
− a) =
Then the Leibniz rule (2.71) yields that
k=0
32
Chapter 2. Fractional derivatives
Using the definition of binomial coefficients and properties of the Gamma function, we have α RL Da,x f (x) ∞ X (x − a)k−α (k) Γ(α + 1) f (x) = Γ(k + 1)Γ(α − k + 1) Γ(k − α + 1) k=0 ∞ X (x − a)k−α Γ(α + 1) = f (k) (x) k!(α − k)Γ(α − k) Γ(1 − (α − k)) k=0 ∞ X (2.76) Γ(α + 1) sin(π(α − k)) k−α (k) = (x − a) f (x) k!(α − k) π k=0 ∞ X sin(πα) Γ(α + 1) (−1)k (x − a)k−α f (k) (x) = k!(α − k) π k=0 ∞ Γ(α + 1) sin(πα) X (−1)k (x − a)k−α (k) = f (x). π (α − k)k! k=0
When |x| |a|, we can write k−α
(x − a)
=x
k−α
2 a k−α (k − α)a a k−α 1− =x 1− +O , x x x2
and therefore (x − a)k−α ≈ xk−α +
(α − k)axk , |x| |a|. xα+1
Substituting (2.78) into (2.76), we obtain α RL Da,x f (x) (∞ Γ(α + 1) sin(πα) X (−1)k xk−α (k) ≈ f (x) π (α − k)k! k=0 ) ∞ a X (−1)k xk f (k) (x) . + xα+1 k!
(2.77)
(2.78)
(2.79)
k=0
Recalling (2.76) gives α RL Da,x f (x)
≈ RL Dα 0,x f (x) +
aΓ(α + 1) sin(πα)f (0) , |x| |a|. πxα+1
Taking x → ∞, we obtain (2.72) for large x. When a → −∞, we have |a| |x| for fixed x. In this case, x k−α k−α =(−a)k−α 1 − (x − a) a 2 (k − α)x x =(−a)k−α 1 − +O , a a2
(2.80)
(2.81)
from which it follows that (x − a)k−α ≈ (−a)k−α −
(α − k)x(−a)k , |a| |x|. (−a)α+1
(2.82)
2.2. Some remarks on the Riemann-Liouville derivative
33
Substitution of (2.82) in (2.76) gives α RL Da,x f (x) (∞ X (−1)k (x − (x + a))k−α Γ(α + 1) sin(πα) ≈ f (k) (x) π (α − k)k! k=0 ) ∞ X (−1)k (x − (x + a))k f (k) (x) x . − (−a)α+1 k!
(2.83)
k=0
Using (2.76) yields α RL Da,x f (x)
≈ RL Dα x+a,x f (x) −
xΓ(α + 1) sin(πα)f (x + a) , |a| |x|. π(−a)α+1
(2.84)
Therefore, we may conclude that, under certain conditions on f (x) (e.g., f (x) is bounded), equality (2.73) is valid with a → −∞. Remark 2.3. Property 2.7 is of significance in applications. In fact, relation (2.72) shows that, for large x, the Riemann-Liouville derivative with lower terminal x = a can be replaced, for example, by the fractional derivative with the lower terminal x = 0. Consequently, the impact of the instant at which the dynamical process f (x) started (and therefore the impact of the transient effects) vanishes as x → ∞ [139]. Meanwhile, (2.73) reveals that, under certain conditions on f (x), the Riemann-Liouville derivative with a fixed large negative lower terminal can be replaced by the fractional derivative with a moving lower terminal. In view of the definition and properties of the Riemann-Liouville derivative, it is clear that the Riemann-Liouville derivative is well established in terms of pure mathematics. Nevertheless, it is in a weak position when faced with the demands of modern technology. Applications of fractional differential equations naturally lead to the necessity of the formulation of initial value conditions for such equations. It has been shown that the Riemann-Liouville approach results in initial value conditions in the form of (2.65). However, there is generally no known physical interpretation for such types of initial value conditions. Definitions of fractional derivatives allowing the utilization of physically interpretable initial value conditions, which contain integer-order derivatives like f (a), f 0 (a), etc., are more appealing. Besides, the Riemann-Liouville derivative of an arbitrary nonzero constant is not equal to zero in the case of a finite lower terminal, which contradicts the common integer-order sense. This also leads to the utilization of the RiemannLiouville definition with a = −∞ [135], whose physical meaning is that the starting time of the physical process is set to −∞. In such a case transient effects cannot be studied [139].
2.2 Some remarks on the Riemann-Liouville derivative In the above section, the definition, existence, and properties of the Riemann-Liouville derivative were introduced. In this section, we show the exact expressions of the Riemann-Liouville derivatives of some elementary functions, where we can see the great difference between the Riemann-Liouville derivative and the integer-order derivative. In the following, we only consider the case with 0 < α < 1.
34
Chapter 2. Fractional derivatives
It is known that ex =
∞ X xk k=0
cos x =
sin x =
,
∞ X (−1)k k=0
and
k!
(2k)!
(2.85)
x2k ,
∞ X (−1)k 2k+1 x . (2k + 1)!
(2.86)
(2.87)
k=0
Using the equalities α RL D0,x
and α RL D0,x
one has
xβ−1 =
1=
x−α , 0 < α < 1, Γ(1 − α)
Γ(β) xβ−α−1 , 0 < α < 1, β > 1, Γ(β − α)
∞ k+1−1 X α x α x RL D0,x e = RL D0,x k! k=0 ∞ X 1 Γ(k + 1) x−α + xk+1−α−1 = Γ(1 − α) k! Γ(k + 1 − α) k=1
∞
X x−α xk−α = + Γ(1 − α) Γ(k + 1 − α)
(2.88)
(2.89)
(2.90)
k=1
=
∞ X k=0
k−α
x , 0 < α < 1. Γ(k + 1 − α)
It is obvious that α x RL D0,x e
6= ex , 0 < α < 1.
(2.91)
x It can be easily verified that, for example, limx→0+ RL Dα 0,x e does not exist due to the term x−α Γ(1−α) ,
x x but limx→0+ ex = 1. In effect, RL Dα 0,x e = e does not hold for every x > 0 and λx 0 0 and 0 < α ∈ / Z+ . 0,x e By using (2.86), one has α RL D0,x cos x ∞ X (−1)k α 2k α = D 1 + RL D0,x x RL 0,x (2k)! k=1 ∞ −α X (2.92) x (−1)k Γ(2k + 1) = + x2k−α Γ(1 − α) (2k)! Γ(2k + 1 − α) k=1 ∞ X x−α (−1)k = + x2k−α , 0 < α < 1. Γ(1 − α) Γ(2k + 1 − α) k=1
It is evident that α RL D0,x
πα cos x 6= cos x + , 0 < α < 1. 2
(2.93)
2.2. Some remarks on the Riemann-Liouville derivative
35
It is also very easily verified that, for example, limx→0+ RL Dα 0,x cos x does not exist but limx→0+ cos x + πα makes sense. So, 2 α RL D0,x
πα cos x 6= cos x + 2
(2.94)
for every x > 0 and 0 < α ∈ / Z+ . One can also show that α RL D0,x
πα sin x 6= sin x + 2
(2.95)
for every x > 0 and 0 < α ∈ / Z+ . In addition, we list the one-half order derivatives of the mostly used functions in Table 2.1. Table 2.1. The one-half order derivatives of the most used functions. f (x)
√1 , x
√
1 2
RL D0,x f (x),
0
0
1
√1 πx
x>0
0 √ 1
x, x > 0
x>0
x2
π px 2 π 3√ 4 πx
x2
8x √2 3 π
xn , n = 0, 1, 2, . . .
(n!)2 (4x)n √ (2n)! πx √ (2n+1)! π x n 2(n!)2 4
2
x 3
1
xn+ 2 , n = 0, 1, 2, . . .
3
Γ(p+1) p− 1 x 2 Γ(p+ 12 )
xp , p > −1 ex e−x √ ex erf( x), x > 0 √ daw( x), x > 0 sin x cos x log x, x > 0 √ x log x, x > 0 log √ x, x
x>0
√ + ex erf( x) √ − √2π daw( x)
√1 πx √1 πx
ex 1√ −x 2 πe q √ 2x sin x + π4 − 2gres π q √ 2x √1 + cos x + π − 2fres 4 π πx log(4x) √ πx √
π 2
log √
x 4
π x
+2
36
Chapter 2. Fractional derivatives
Here erf(x) denotes the error function given by Z x 2 2 erf(x) = √ e−t dt, π 0 √ daw( x) is Dawson’s integral Z x 2 −x2 et dt, daw(x) = e
(2.96)
(2.97)
0
and gres(x) and fres(x) are Fresnel integrals defined by Z x gres(x) = sin(t2 )dt
(2.98)
0
and
Z fres(x) =
x
cos(t2 )dt.
(2.99)
0
Remark 2.4. (I) In [166], Trenˇcevski and Tomovski reconsidered the fractional derivatives of cos x and sin x. They gave an initial assumption (axiom) that cos x and sin x can be defined by cos x =
∞ X k=−∞
ak
xk , k!
sin x =
∞ X k=−∞
bk
xk , k!
(2.100)
with a2k+1 = 0, a2k = (−1)k , b2k+1 = (−1)k , b2k = 0, (k ∈ Z) and (−1)! = (−2)! = (−3)! = · · · = ±∞. They showed that πα α (2.101) RL D0,x cos x = cos x + 2 and
πα α D sin x = sin x + . (2.102) RL 0,x 2 This generalization may be useful in the “new” axiom system. However, all mathematical models are built up in the formal axiom system in use. If we use the conclusions in the “new” axiom system to deal with the mathematical models derived in the formal axiom system, the results are very likely fallacious or even fatally erroneous. (II) Definite conditions of Riemann-Liouville type differential equations are carefully introduced in the last section of this chapter.
2.3 Caputo derivative The Caputo derivative was believed to be proposed by Caputo [15, 16] to consider a relatively complete description of actual elastic fields. It was also considered by El-Sayed [47, 48] (in Banach space). Fractional derivatives of this type frequently appear in fractional differential equations characterizing anomalous phenomena. To see this, we apply the fractional integration −(1−α) on both sides of equation (2.1) and obtain RL D0,t Z t 1 ∂ ∂2 (t − s)−α W (x, s)ds = Kα 2 W (x, t) (2.103) Γ(1 − α) 0 ∂s ∂x due to RL D−α 0,t W (x, t) t=0 = 0 when W (x, t) is assumed to be continuous with respect to t. In this case, the temporal and spatial derivatives are separated on the left- and right-hand sides of the
2.3. Caputo derivative
37
equation. The left-hand side of equation (2.103) happens to be the αth order Caputo derivative of W (x, t) with respect to t when 0 < α < 1. Definition 2.7. The left- and right-sided Caputo derivatives of order α > 0 are defined by α C Da,x f (x) =
x
1 Γ(m − α)
Z
(−1)m Γ(m − α)
Z
a
f (m) (t)dt , x > a, (x − t)α−m+1
(2.104)
f (m) (t)dt , x < b, (t − x)α−m+1
(2.105)
and α C Dx,b f (x)
=
b
x
respectively, where m is a positive integer satisfying m − 1 < α ≤ m. It follows from equations (2.104) and (2.105) that the mth order differentiation is required for the Caputo derivative. Precise results on the existence of Caputo derivative are presented as follows. Theorem 2.8. [75] Suppose that α ≥ 0. Let m = [α] + 1 for α ∈ / Z+ , and m = α for α ∈ Z+ . f (x) and C Dα If f (x) ∈ AC m ([a, b]), then the Caputo derivatives C Dα a,x x,b f (x) exist almost everywhere on [a, b]. Theorem 2.9. [102] If f (m−1) (x) is of bounded variations and continuous from the right, then 1 α C Da,x f (x) is in L (a, b), and also has the following form: α C Da,x f (x)
=
1 Γ(m − α)
x
Z
(x − t)m−α−1 df (m−1) (t),
(2.106)
a
where m − 1 < α ≤ m ∈ Z+ . In addition, we can define the Caputo derivative via the RiemannLiouville derivative if f (x) ∈ AC m (a, b), in the following forms: α α C Da,x f (x) = RL Da,x f (x) −
m−1 X
f (k) (a)(x − a)k−α , Γ(1 + k − α)
(2.107)
(−1)k f (k) (b)(b − x)k−α . Γ(1 + k − α)
(2.108)
k=0
α C Dx,b f (x)
= RL Dα x,b f (x) −
m−1 X k=0
Remark 2.5. In view of equations (2.107) and (2.108), it is evident that the Caputo and RiemannLiouville derivatives are different in general. The equivalence between these two types of fractional derivatives holds if a = −∞ and b = +∞, or the homogeneous conditions f (k) (a) = 0 (k = 0, 1, . . . , m − 1) and f (k) (b) = 0 (k = 0, 1, . . . , m − 1) are satisfied. Differently from the Riemann-Liouville derivative, the fractional derivative in the Caputo sense does not coincide with the integer-order one in general. Instead, we have the following result. Property 2.8. [90] Let m − 1 < α ≤ m. If f (x) has an (m + 1)th order derivative, then lim
α→m−
α C Da,x f (x)
= f (m) (x),
(2.109)
38
Chapter 2. Fractional derivatives
while lim
α→(m−1)+
α C Da,x f (x)
= f (m−1) (x) − f (m−1) (a).
Proof. According to the definition of the Caputo derivative, it holds that α lim C Da,x f (x) α→(m−1)+ Z x 1 m−α−1 (m) (x − t) f (t)dt = lim Γ(m − α) a α→(m−1)+ Z x = f (m) (t)dt a (m−1) =f (x) − f (m−1) (a). In addition, if f (x) is suitably smooth, integration by parts gives lim C Dα a,x f (x) α→m− (m) f (a)(x − a)m−α = lim Γ(m − α + 1) α→m− Z x 1 m−α (m+1) + (x − t) f (t)dt Γ(m − α + 1) a Z x (m) =f (a) + f (m+1) (t)dt a =f (m) (x).
(2.110)
(2.111)
(2.112)
The proof is thus completed. Recalling that the Riemann-Liouville derivative can be reduced to the integer-order derivative when the fractional order α approaches an integer for any fixed x, we conclude that the Caputo derivative possesses a consistency different from that of the Riemann-Liouville derivative. Figure 2.1 visualizes the difference. In this case, we say that the Caputo derivative has fractional-order upper consistency while it does not have fractional-order lower consistency for m − 1 < α < m ∈ Z+ . Regarding compositions related to the Caputo derivative, we present the following assertions. Property 2.9. [99] Let α > 0, n − 1 < β ≤ n ∈ Z+ , and f ∈ C n ([a, b]). Then the following equalities hold: −α β β−α (2.113) RL Da,x C Da,x f (x) = C Da,x f (x), β −α RL Dx,b C Dx,b f (x)
where
β−α C Da,x
=
β−α RL Da,x
and
β−α C Dx,b
−β β RL Da,x C Da,x f (x)
=
= C Dβ−α x,b f (x),
β−α RL Dx,b
= f (x) −
if β < α. Especially,
n−1 X k=0
−β β RL Dx,b C Dx,b f (x)
= f (x) −
(2.114)
n−1 X k=0
f (k) (a) (x − a)k , Γ(k + 1)
(2.115)
f (k) (b) (b − x)k . Γ(k + 1)
(2.116)
2.3. Caputo derivative
39
Figure 2.1. A plot illustrating the consistency of fractional derivatives [90]. Reprinted from Appl. Math. Comput., 187 (2007), pp. 777–784, C. P. Li and W. H. Deng, Remarks on fractional derivatives, with permission from Elsevier. Proof. We first prove (2.113) and (2.115). For a suitably smooth function f (x), the Taylor series expansion gives (n) f (x) = φ(x) + RL D−n (x) (2.117) a,x f with φ(x) =
n−1 X
f (k) (a) (x − a)k . Γ(k + 1)
(2.118)
= RL Dβa,x [f (x) − φ(x)] .
(2.119)
k=0
It can be verified directly that β C Da,x f (x)
Consequently, the composition formula (2.32) yields −α β RL Da,x C Da,x f (x) β =RL D−α a,x RL Da,x [f (x) − φ(x)] =RL Dβ−α a,x [f (x) − φ(x)] n X (x − a)α−j β−j − RL Da,x (f (x) − φ(x)) x=a Γ(1 + α − j) j=1 =RL Dβ−α a,x [f (x) − φ(x)] n h i X (x − a)α−j β−j−n (n) − D f (x) RL a,x x=a Γ(1 + α − j) j=1 = Dβ−α [f (x) − φ(x)] , RL
(2.120)
a,x
where RL Dβ−j−n f (n) (x) x=a = 0 is utilized since β − j − n < 0 and f (n) (x) is bounded. If a,x β−α β = α, it is clear that RL Dβ−α a,x [f (x) − φ(x)] = f (x) − φ(x). If β 6= α, RL Da,x [f (x) − φ(x)] β−α = C Da,x f (x) due to (2.119).
40
Chapter 2. Fractional derivatives
Equations (2.114) and (2.116) can be shown in a similar manner. The proof is thus completed.
Compared to Property 2.4, the following assertion shows big differences between the Riemann-Liouville derivative and the Caputo derivative. Property 2.10. [99] Let α, β > 0 and f (x) be suitably smooth. Suppose that C Dβa,x RL D−α a,x f (x), β β−α −α β−α D f (x), D D f (x), and D f (x) exist. Then C a,x C x,b RL x,b C x,b −(α−β) f (x), β ≤ α, RL Da,x β−α + C Da,x f (x), α < β, α ∈ Z , n−m P β −α f (j) (a) β−α (2.121) C Da,x RL Da,x f (x) = (x − a)j+α−β , D f (x) + C a,x Γ(1+j+α−β) j=0 α < β, m − 1 < α < m, n − 1 < β < n, m, n ∈ Z+ , and
β −α C Dx,b RL Dx,b f (x)
=
−(α−β) f (x), β ≤ α, RL Dx,b β−α α < β, α ∈ Z+ , C Dx,b f (x), n−m P f (j) (b) β−α C Dx,b f (x) + Γ(1+j+α−β) (b j=0
− x)j+α−β ,
(2.122)
α < β, m − 1 < α < m, n − 1 < β < n, m, n ∈ Z+ .
Proof. We first give the proof of (2.121). For β ≤ α, we have n ≤ m. Therefore, it holds that β −α C Da,x RL Da,x f (x) n −α (2.123) =RL D−(n−β) RL Da,x RL Da,x f (x) a,x = D−(n−β) Dn−α f (x). RL a,x RL a,x When α > n, we have n − α < 0. It is evident that β −α C Da,x RL Da,x f (x)
n−α β−α = RL D−(n−β) RL Da,x f (x) = RL Da,x f (x). a,x
(2.124)
When α < n, we have 0 < n − α < 1 since n ≤ m and m − 1 < α < m. It follows from (2.32) that β −α C Da,x RL Da,x f (x) −(n−β) n−α RL Da,x f (x) =RL Da,x h i (2.125) (x − a)n−β−1 (n−α)−(n−β) (n−α)−1 = D f (x) − D f (x) RL RL a,x a,x Γ(n − β) x=a =RL Dβ−α a,x f (x). As a result, the relation β −α C Da,x RL Da,x f (x)
holds for any 0 < β ≤ α.
= RL Dβ−α a,x f (x)
(2.126)
2.3. Caputo derivative
41
For the positive integer α = m < β, β −m C Da,x RL Da,x f (x) n −m =RL D−(n−β) RL Da,x RL Da,x f (x) a,x = D−(n−β) Dn−m f (x). RL a,x RL a,x The relation n > m and the composition formula (2.33) yield −m β C Da,x RL Da,x f (x) n−m =RL D−(n−β) RL Da,x f (x) a,x n−m−1 X f (j) (a)(x − a)m−β+j β−m = D f (x) − RL a,x Γ(m − β + j + 1) j=0 β−m =RL Dβ−m a,x [f (x) − φ(x)] + RL Da,x φ(x) n−m−1 X f (j) (a)(x − a)m−β+j − Γ(m − β + j + 1) j=0 =RL Dβ−m a,x [f (x) − φ(x)] =C Dβ−m a,x f (x),
(2.127)
(2.128)
Pn−m−1 f (j) (a)(x−a)j . where φ(x) = j=0 Γ(j+1) For α < β, n − 1 < β < n, and m − 1 < α < m, we have n − m < n − α < n − m + 1. The Taylor series expansion gives f (x) = φ1 (x) + RL D−(n−m+1) f (n−m+1) (x) a,x with φ1 (x) =
n−m X k=0
f (k) (a)(x − a)k . Γ(k + 1)
−(n−m+1) (n−m+1)
Hence, the relation f (x) − φ1 (x) = RL Da,x
f
(2.129)
(2.130)
(x) yields
−α RL Da,x
[f (x) − φ1 (x)] = RL D−(n−m+1)−α f (n−m+1) (x), (2.131) a,x which implies that RL Dka,x RL D−α a,x (f (x) − φ1 (x)) x=a = 0 for k = 0, 1, . . . , n − 1. Consequently, β −α C Da,x RL Da,x f (x) = Dβ D−α [f (x) − φ1 (x)] + C Dβa,x RL D−α a,x φ1 (x) C a,x RL a,x =RL Dβ RL D−α [f (x) − φ1 (x)] + C Dβ RL D−α φ1 (x) a,x a,x a,x a,x (2.132) β−α β −α =RL Da,x [f (x) − φ1 (x)] + C Da,x RL Da,x φ1 (x) n−m X f (k) (a)(x − a)k+α−β =C Dβ−α f (x) + . a,x Γ(1 + k + α − β) k=0
Equality (2.121) is therefore proved. Relation (2.122) can be similarly verified. This ends the proof.
42
Chapter 2. Fractional derivatives
Remark 2.6. In view of Properties 2.9 and 2.10, the Caputo derivative is not commutative with the fractional integral unless necessary homogeneous initial conditions are met. Compared to Property 2.3, the following property implies great differences between the Riemann-Liouville derivative and the Caputo derivative. Property 2.11. [99] If m and n are positive integers, then, for n − 1 < β ≤ n, β m C Da,x C Da,x f (x)
= C Dm+β a,x f (x),
(2.133)
while m β m+β C Da,x C Da,x f (x) = C Da,x f (x) +
m+n−1 X j=n
f (j) (a)(x − a)j−m−β . Γ(1 + j − m − β)
Proof. For a suitably smooth function f (x), β m C Da,x C Da,x f (x) n d (m) =RL D−(n−β) f (x) a,x dxn =RL D−(n−β) f (m+n) (x) a,x m+β =C Da,x f (x). If we change the order of these two operators, then integration by parts yields m β C Da,x C Da,x f (x) −(n−β) (n) =RL Dm f (x) a,x RL Da,x =RL Dm−n+β f (n) (x) a,x n−1 X f (j) (a)(x − a)j−m−β m+β . = D f (x) − RL a,x Γ(1 + j − m − β) j=0
(2.134)
(2.135)
(2.136)
Define the auxiliary function φ(x) =
m+n−1 X j=0
f (j) (a)(x − a)j . Γ(j + 1)
(2.137)
Then it holds that m β C Da,x C Da,x f (x) n−1 X f (j) (a)(x − a)j−m−β =RL Dm+β φ(x) − a,x Γ(1 + j − m − β) j=0 m+n−1 X f (j) (a)(x − a)j−m−β m+β = D f (x) + . C a,x Γ(1 + j − m − β) j=n The proof is thus completed.
(2.138)
2.3. Caputo derivative
43
Remark 2.7. (I) Comparing equation (2.133) with equation (2.27), we find that composition laws of the Caputo derivative and Riemann-Liouville derivative with the integer-order derivative are different. (II) We should bear in mind that the above property is different from Property 2.5 for the RiemannLiouville derivative. The commutativity of Caputo derivatives C Dβa,x and C Dα a,x is allowed under the following conditions: f (j) (a) = 0, j = n, n + 1, . . . , m + n − 1, m = 0, 1, 2, . . . .
(2.139)
And there are no restrictions on the values of f (j) (a), j = 0, 1, . . . , n − 1. Although the Caputo derivative is different from the Riemann-Liouville one, there exist similarities between them. One of the similarities is reflected by fractional derivatives of these two types for power functions, which remain power functions of similar forms. Property 2.12. Let m − 1 < α < m ∈ Z+ and β > 0. Then we have α C Da,x (x
− a)β−1 =
Γ(β) (x − a)β−α−1 , β > m, Γ(β − α)
(2.140)
α C Dx,b (b
− x)β−1 =
Γ(β) (b − x)β−α−1 , β > m. Γ(β − α)
(2.141)
In addition, α C Da,x (x
k − a)k = 0 and C Dα x,b (b − x) = 0, k = 0, 1, . . . , m − 1.
(2.142)
In particular, α C Da,x 1
(x) = 0
and
α C Dx,b 1
(x) = 0.
(2.143)
Proof. First, we rewrite the Caputo derivative C Dα a,x f (x) in the form α C Da,x f (x)
−(m−α) (m) = RL Da,x f (x).
When β > m, it holds that m d (x − a)β−1 dxm =(β − 1)(β − 2) . . . (β − m)(x − a)β−m−1 Γ(β) (x − a)β−m−1 . = Γ(β − m) Then the relation (1.63) yields α β−1 C Da,x (x − a) Γ(β) −(m−α) = (x − a)β−m−1 RL Da,x Γ(β − m) Γ(β) Γ(β − m) = (x − a)β−α−1 Γ(β − m) Γ(β − α) Γ(β) = (x − a)β−α−1 . Γ(β − α)
(2.144)
(2.145)
(2.146)
44
Chapter 2. Fractional derivatives
When k = 0, 1, . . . , m − 1, we have dm (x − a)k = 0. dxm
(2.147)
Consequently, −(m−α) − a)k = RL Da,x 0 (x) = 0. (2.148) −(m−α) Setting k = 0 gives C Da,x 1 (x) = 0. Corresponding results for the right-sided Caputo derivative can be similarly derived and the proofs are omitted here. α C Da,x (x
Now we present formulae for the Fourier and Laplace transforms of the Caputo derivative, which are useful in analytically solving fractional differential equations with the Caputo derivative. Theorem 2.10. [139] The Fourier transform for Caputo derivative with lower terminal a = −∞ is given by αb F{C Dα (2.149) −∞,x f (x); ω} = (−iω) f (ω), α > 0, provided that f (x) and its integer-order derivatives up to order (m − 1) vanish at x = −∞. The Laplace transform for Caputo derivative with lower terminal a = 0 is L{C Dα 0,x f (x); p}
α
= p F (p) −
m−1 X
pα−k−1 f (k) (0), m − 1 < α ≤ m ∈ Z+ .
(2.150)
k=0
Here fb(ω) and F (p) are the Fourier transform and Laplace transform of f (x), respectively. α Proof. If we use C Dα −∞,x f (x) = RL D−∞,x f (x) and the corresponding result in Theorem 2.5, equation (2.149) is very briefly proved. Here we give the proof of (2.150). Rewrite the Caputo derivative C Dα 0,x f (x) in the form α C D0,x f (x)
−(m−α)
= RL D0,x
g(x)
(2.151)
with g(x) = f (m) (x), then the formulae for Laplace transforms of fractional integral and the integer-order derivative yield L{C Dα 0,x f (x); p} =p−(m−α) L{g(x); p} " # m−1 X (2.152) =p−(m−α) pm F (p) − pk f (m−k−1) (0) k=0 m−1 X α pα−k−1 f (k) (0). =p F (p) − k=0
The proof is thus completed. Based on the above results on the Caputo derivative, we conclude this section with an assessment of the fractional derivative of this type. The main advantage of the Caputo derivative is that
2.4. Some remarks on the Caputo derivative
45
the formulation of initial conditions for fractional differential equations contains the limit values of integer-order derivatives of unknown functions at the lower terminal. To see this we recall the corresponding Laplace transform formula (2.150), and then observe that the Caputo approach allows utilization of initial values of integer-order derivatives with known physical interpretations. Another advantage is elucidated by the fact that the Caputo derivative of a constant vanishes, coinciding with the case of integer-order derivatives. These two points make the Caputo derivative more commonly used than the Riemann-Liouville one in applications. Of course, there also exist drawbacks of Caputo derivatives. A function which is nondifferentiable in the classical sense is also nondifferentiable in the Caputo sense. Furthermore, the existence of the αth order Caputo derivative requires the integer-order derivative f (m) (x), whose derivative order is higher than α for m − 1 < α ≤ m.
2.4 Some remarks on the Caputo derivative Similar to the Riemann-Liouville derivative, the inequalities πα α C D0,x cos x 6= cos x + 2 and α C D0,x
πα sin x 6= sin x + 2
(2.153)
(2.154)
hold for every x > 0 and 0 < α ∈ / Z+ . The existence conditions for the Riemann-Liouville derivative and the Caputo derivative are presented as being the same (see Theorems 2.2 and 2.8), since these conditions are often used and conveniently applicable. But this does not mean that these two kinds of derivatives are the same. Differences between them are displayed in Sections 2.1 and 2.3, and the following Section 2.8. Examples below disclose their distinctions. Example 2.11. [102] Consider a discontinuous function ( 1 − x if 0 < x ≤ 1, f (x) = 2 − x if 1 < x < 2,
(2.155)
which has a jump discontinuity point at x = 1 and is differentiable at x = 0 in the classical sense. For 0 < α < 1, C Dα 0,x f (x) exits for any x ∈ (0, 2) and α C D0,x f (x)
=−
x1−α , 0 < x < 2. Γ(2 − α)
It follows that the Caputo derivative of f (x) exists on (0, 2). However, x1−α x−α − Γ(2−α) if 0 < x < 1, Γ(1−α) α D f (x) = RL 0,x x−α + (x−1)−α − x1−α if 1 < x < 2. Γ(1−α) Γ(1−α) Γ(2−α)
(2.156)
(2.157)
Obviously RL Dα 0,x f (x) does not exist at x = 0 and x = 1. Example 2.12. [102] Consider the Weierstrass function X W (x) = λ−µj sin(λj x), 0 < µ < 1, λ > 1, j≥1
(2.158)
46
Chapter 2. Fractional derivatives
which is continuous everywhere but differentiable nowhere. Its Riemann-Liouville derivative with α ∈ (0, µ) exists and is given by X α λ(1−µ)j Cx (1 − α, λj ), 0 < α < µ < 1, (2.159) RL D0,x W (x) = j≥1
where
Z x 1 tα−1 cos a(x − t)dt := Cx (α, a). (2.160) Γ(α) 0 Nevertheless, it is evident that the Caputo derivative of W (x) does not exist since the function is differentiable nowhere. −α RL D0,x cos ax =
2.5 Riesz derivative There exists a large number of papers devoted to studying anomalous diffusion, for instance, [27, 53, 127] and references therein. The study of anomalous diffusion includes proposing anomalous diffusion models like the ones established by [24, 26, 168], and also numerical algorithms that are considered in works such as [17, 73, 99, 109]. Nevertheless, there is limited research about anomalous convection. Recently, the Riesz derivative1 has been proved to be suitable for the description of anomalous convection (also called fractional convection) [97]. It is known that the classical convection-diffusion (or advection-dispersion) equation reads as ∂u ∂u ∂2u +V = D 2 , x ∈ R, t > 0, (2.161) ∂t ∂x ∂x where u(x, t) is the solute concentration, V (>0) the convection velocity, and D (>0) the diffusion coefficient. If V = 0, (2.161) is the standard diffusion equation. When V D (>0), (2.161) is called the convection-dominated diffusion equation. In this case, diffusion is not dominant but cannot be ignored. The system (2.161) gives a proper description of standard convection-diffusion phenomena. However, when the diffusion in R (or Rd ) obeys a power-law distribution rather than the Fick’s Pn ∂ 2 ∂2 law, the classical Laplacian ∂x 2 (or ∆ = i=1 ∂x2i ) cannot accurately characterize such a phenomenon which is often called anomalous diffusion. As a result, anomalous diffusion can be adequately modeled by the Riesz derivative operator [148]. In addition, if the convection process ∂ cannot perfectly model convection also obeys the power-law distribution, then the operator ∂x processes of this type either. A better choice is to apply the Riesz derivative with derivative order lying in (0, 1) to account for such an anomalous convection. In general, a Fokker-Planck equation describes the change of PDF of a particle moving in a CTRW process. It is therefore used to model the solute transport. The probability of a particle moving from position y to x during the time ∆t is denoted by p(x − y, 4t) with lim p(x − y, ∆t) = δ(x − y),
∆t→0
(2.162)
under the assumption that the initial probability of a random walk starting at the origin is a Dirac delta function [7, 179]. The above equation implies that, as the transition time vanishes, the particle at the position y is likely unmoved. For small ∆t, such a probability can be expanded as p(x − y, ∆t) (2.163) =δ(x − y) + A(y, ∆t)δ 0 (x − y) + 1 B(y, ∆t)δ 00 (x − y) + · · · . 2! 1 Here “Riesz” refers to the well-known mathematician Marcel Riesz, younger brother of another famous mathematician Frigyes Riesz, and the supervisor of Lars Hörmander.
2.5. Riesz derivative
47
Here A(y, ∆t) and B(y, ∆t) represent the behavior of the instantaneous transition probability from the first order to the second order. Denote the transition distance (x − y) by ∆y. If the stochastic process is Gaussian, then A(∆y, ∆t) and B(∆y, ∆t) are just the first and second moments, i.e., Z (x − y)p(x − y, ∆t)dx = h∆yi
(2.164)
(x − y)2 p(x − y, ∆t)dx = h∆y 2 i.
(2.165)
A(∆y, ∆t) = and
Z B(∆y, ∆t) =
Nevertheless, for a non-Gaussian process, the first moment and/or the second one may be infinite, and both-sided behaviors in space should be considered. For faster-than-Fickian diffusion in groundwater, the probability propagator is replaced by concentration [29], and the governing equation movement simplifies to ∂C D ∂C +V = p · RL Dβa,x C + q · RL Dβx,b C , (2.166) π ∂t ∂x − cos( 2 β) where p + q = 1, p, q > 0, and 1 < β < 2. In this equation, the first moment is assumed to be finite. In some situations, the interaction among particles causes blocking to particle movement. The asymptotic behavior of the convection process is much closer to a power-law distribution. In this case, the first moment is infinite and it is more appropriate to replace the convection ∂C ∂x by ∂αC 1 p˜ · RL Dα ˜ · RL Dα = a,x C + q x,b C , π α ∂|x| − cos( 2 α)
(2.167)
where p1 + q1 = 1, p˜, q˜ > 0, and 0 < α < 1, provided that the first moment does not exist but the αth moment does. This can be regarded as fractional convection. Especially, for symmetric transitions, p = q = 21 , p˜ = q˜ = 12 , then equation (2.166) with the left-hand side of equation (2.167) taking the place of ∂C ∂x becomes ∂C +V ∂t
α RZ Dx C
= D RZ Dβx C.
In this case, the above notation RZ Dα x stands for the Riesz derivative 1 γ γ γ D + D RZ Dx u = − RL RL a,x x,b u, x ∈ (a, b), 2 cos( π2 γ)
(2.168)
(2.169)
where γ ∈ (n − 1, n) with n ∈ Z+ . In particular, when γ = 0, the Riesz derivative RZ Dγx u reduces to −u if u is continuous on [a, b]. Adding the resource term f (x, t) to (2.168) gives the fractional convection-diffusion equation (0 < α < 1, 1 < β < 2) ∂u(x, t) + V α RZ Dx u(x, t) ∂t (2.170) =D RZ Dβx u(x, t) + f (x, t), (x, t) ∈ (a, b) × (0, +∞). In this situation, initial and (Dirichlet) boundary value conditions are given in the form u(x, 0) = ψ(x), a ≤ x ≤ b, α−1 (2.171) RL Da,x u(x, t)|x=a = φ1 (t), t ≥ 0, α−1 RL Dx,b u(x, t)|x=b = φ2 (t), t ≥ 0,
48
Chapter 2. Fractional derivatives
or u(x, 0) = ψ(x), a ≤ x ≤ b, β−2 RL Da,x u(x, t)|x=a = ϕ1 (t), t ≥ 0, β−2 RL Dx,b u(x, t)|x=b = ϕ2 (t), t ≥ 0.
(2.172)
Here u(x, t), V , and D have the same meanings as in (2.161). If V = 0 and D 6= 0, then (2.170) is known as the fractional diffusion equation (FDE) [179]. Furthermore, if V = 0, D 6= 0, and β = 2, it reduces to the classical diffusion equation. On the other hand, if D = 0 and V 6= 0, then (2.170) becomes the fractional convection equation (FCE). Particularly, when V D(> 0), (2.170) is the fractional convection-dominated-diffusion equation. We should also bear in mind that in equation (2.171), when the boundary value conditions are homogeneous, they are often replaced by u(a, t) = u(b, t) = 0,
(2.173)
which is adopted in most situations. After presenting the physical interpretations, we introduce the Riesz derivative (on R and/or Rd ) from the perspective of mathematics. Definition 2.13. [9, 11, 85] For a suitably smooth function f (x) defined on R, the αth order Riesz fractional derivative (or Riesz derivative for brevity) is given by α RZ Dx f (x)
= −Ψα
α RL D−∞,x
+ RL Dα x,+∞ f (x), α 6= 1, 3, 5, . . . ,
(2.174)
and for other positive values except α = 2, 4, 6, . . ., α RZ Dx f (x)
where Ψα =
=
2α−1 αΓ 1 2
π Γ 1
1+α 2 − α2
Z 0
∞
f (x + y) − 2f (x) + f (x − y) dy, y 1+α
(2.175)
1 . 2 cos( πα 2 )
Most of the properties of the Riesz derivative on R can be derived via Fourier transform. The Schwartz space ( S(R) =
) N X k N D f (x) < ∞ , f ∈ C (R) sup(1 + |x|) x∈R ∞
(2.176)
k=0
with N = 0, 1, 2, . . . , is invariant with respect to Fourier transform. We therefore present properties of the Riesz derivative in the Schwartz space settings. In the following, we always note that definitions of the Riesz derivative in equations (2.174) and (2.175) are equivalent for any f ∈ S(R) when α is noninteger. Property 2.13. Assume that f (x) ∈ S(R). Then the Fourier transform of the αth order Riesz derivative on R is given by αb F {RZ Dα / Z+ . x f (x); ω} = −|ω| f (ω), α > 0, α ∈
(2.177)
2.5. Riesz derivative
49
Proof. Equation (2.174) and the Fourier transform formulae for the left- and right-sided RiemannLiouville derivatives in (2.54) yield F {RZ Dα x f (x); ω} 1 α F RL Dα =− −∞,x + RL Dx,∞ f (x); ω πα 2 cos( 2 ) h i 1 αb αb = − (−iω) f (ω) + (iω) f (ω) 2 cos( πα 2 ) −iπα iπα (2.178) α 2 2 + e fb(ω) |ω| e =− 2 cos( πα 2 ) b 2|ω|α cos( πα 2 )f (ω) = − πα 2 cos( 2 ) = − |ω|α fb(ω), where Euler’s formula eix = cos x + i sin x is utilized. The proof is ended. Remark 2.8. The above proof indicates that some properties of the Riesz derivative can be deduced from those of the Riemann-Liouville ones. Nevertheless, we cannot take it for granted that the Riesz derivative inherits most of the properties from the Riemann-Liouville derivative. In fact, the case of the Riesz derivative is more complicated. For example, we remark that the Riesz derivative on the axis does not coincide with the integer-order one, which is different from the case of the Riemann-Liouville derivative. To see this, we compare the formula (2.177) with the following formula for Fourier transform of the integer-order derivative: F{f (m) (x); ω} = (−iω)m fb(ω), i2 = −1.
(2.179)
It immediately follows that lim
α→1
α RZ Dx f (x)
6= f 0 (x).
(2.180)
The Fourier transform formula (2.177) can also be utilized to derive the following property of the Riesz derivative. Property 2.14. The Riesz derivative on the whole real axis satisfies the relation α β RZ Dx RZ Dx f
= −RZ Dα+β f, α > 0, β > 0, x
(2.181)
for any f (x) ∈ S(R). Proof. The Fourier transform formula in (2.177) yields β F RZ Dα x RZ Dx f ; ω = − |ω|α F RZ Dβx f ; ω = −|ω|α −|ω|β fb =|ω|α+β fb = −F RZ Dxα+β f ; ω .
(2.182)
Performing the inverse Fourier transform on both sides of the equation above verifies that (2.181) is valid.
50
Chapter 2. Fractional derivatives
The above property implies that the Riesz derivative operator has no classical semigroup property. In the following, we consider the link between the Riesz derivative and the integer-order one, which also holds for the Riesz derivative of a suitably smooth function defined on an arbitrary interval (a, b) ⊆ R. Definition 2.14. For a suitably smooth function f (x) defined on (a, b) ⊆ R, the αth order Riesz derivative is given by α α α (2.183) RZ Dx f (x) = −Ψα RL Da,x + RL Dx,b f (x), α 6= 1, 3, 5, . . . , where Ψα =
1 . 2 cos( πα 2 )
Theorem 2.15. [11] Let u(x) be a suitably smooth function defined on (a, b) ⊆ R. Then the equalities (2.184) lim RZ Dα x u(x) = −u(x), α→0+
lim
α→2
α RZ Dx u(x)
= u00 (x)
(2.185)
hold for any fixed x ∈ (a, b). Proof. In view of (2.12), (2.13), and (2.183), it holds that lim+ RZ Dα x u(x) α→0 1 α RL Dα = − lim a,x + RL Dx,b u(x) πα + α→0 2 cos 2 1 α α lim RL Da,x u(x) + lim RL Dx,b u(x) = − lim+ πα α→0+ α→0 2 cos α→0+ 2 1 = − [u(x) + u(x)] 2 = − u(x).
(2.186)
Note that RZ Dα x is a continuous operator with respect to α whenever α 6= 1, 3, 5, . . . . For the suitably smooth function u(x) defined on (a, b), we have α lim RZ Dα x u(x) = [RZ Dx u(x)]α=2 α→2 (2.187) = − 1 (u00 (x) + u00 (x)) = u00 (x). 2 cos π The proof is thus completed. Remark 2.9. It is evident from the above proof that the relations for x ∈ R lim
α→4k
α RZ Dx u(x)
= −u(4k) (x), k = 0, 1, 2, . . .
(2.188)
α RZ Dx u(x)
= u(4k+2) (x), k = 0, 1, 2, . . . ,
(2.189)
and lim
α→4k+2
2.5. Riesz derivative
51
are valid for any suitably smooth u(x). For the cases with α → 4k + 1 and α → 4k + 3, k = 0, 1, 2, . . . , (2.175) yields that lim RZ Dα x u(x) α→4k+1 Z (2.190) 24k (4k + 1)Γ(2k + 1) +∞ u(x + y) − 2u(x) + u(x − y) dy = 1 4k+2 1−4k y π2Γ 2 0 and
lim
α→4k+3
α RZ Dx u(x)
4k+2
2 =
(4k + 3)Γ(2k + 2) 1 π 2 Γ − 4k+1 2
Z
+∞
0
u(x + y) − 2u(x) + u(x − y) dy, y 4k+4
(2.191)
where u(x) is a member of the Schwartz space S(R). We can see that there is no obvious link between the Riesz derivative and the integer-order one when α approaches odd numbers. Now we consider the Riesz derivative on Rd , d ∈ Z+ . Here we present its definition, existence condition, and some basic properties. Definition 2.16. [148] For α > 0, the Riesz fractional derivative RZ Dα x is realized in the form of the hypersingular integral defined by Z ∆ly f (x) 1 α (RZ Dx f )(x) = dy, l > α. (2.192) cd (l, α) Rd |y|d+α Here ∆ly f (x) denotes the centered difference of order l, l X l ∆ly f (x) = (−1)k f (x + (l/2 − k)y) , k
(2.193)
k=0
with a vector step y ∈ Rd and centered at the point x ∈ Rd , or the noncentered difference l X l ∆ly f (x) = (−1)k f (x − ky), k
(2.194)
k=0
while cd (l, α) is a constant defined by d
cd (l, α) = with Al (α) =
2−α π 1+ 2 Al (α) d+α α Γ 1 + 2 Γ 2 sin πα 2
l [P 2] 2 (−1)k−1 kl
k=0 l P
(−1)k−1 kl k=0
α
k ,
l 2
−k
α
(2.195)
, the centered case, (2.196) otherwise.
Remark 2.10. [148] The hypersingular integral (RZ Dα x f )(x) does not depend on the choice of l (l > α). It is also called the Riesz derivative in the sense that αb d F {RZ Dα x f (x); ω} = −|ω| f (ω), α > 0, ω ∈ R ,
(2.197)
52
Chapter 2. Fractional derivatives
for “sufficiently good” functions. In particular, the relation is valid for functions in the Lizorkin space Φ defined by (1.79). And it is also valid for differentiable functions f (x). For the convergence of the integral in (2.192), we have the following result, which can be viewed as the existence condition for the Riesz derivative on arbitrary dimensions. Theorem 2.17. [148] Let α > 0 and [α] be the integer part of α. Also let function f (x) be bounded together with its derivatives (Dk f )(x) (|k| = [α] + 1). Then the hypersingular integral α (RZ Dα x f )(x) is absolutely convergent. If l > 2 2 , then this integral is only conditionally convergent. Regarding basic properties of the Riesz derivative on Rd , we emphasize that the Riesz derivaα tive RZ Dα x f yields an operator inverse to the Riesz potential I f in (1.76). Property 2.15. [75] For a “sufficiently good” function f , in particular, for f belonging to the Lizorkin space Φ, the relation α α (2.198) RZ Dx I f = f, α > 0, holds with arbitrary α > 0 and x ∈ Rd . Formulations of the Riesz derivative on R and Rd are somewhat different at first glimpse. Now we show that the definition of the Riesz derivative on Rd can be viewed as a generalization of that of the Riesz derivative on R for functions in the Schwartz space. If we choose the centered difference with d = 1, l = 2, and y = y ∈ R in (2.192), then (∆2y f )(x) =
2 X k=0
(−1)k
2 f [x + (1 − k)y] = f (x + y) − 2f (x) + f (x − y), k
A2 (α) = 2
1 X
(−1)
k=0
and
k−1
2 (1 − k)α = −2, k
1 A (α) 2−α π 1+ 2 2 πα c1 (2, α) = α 1+α Γ 1 + 2 Γ 2 sin 2 1 2−α π 1+ 2 (−2) = 1+α α πα Γ Γ 1 + sin 2 2 2 1 π 2 Γ 1 − α2 , = − α−2 2 αΓ 1+α 2
(2.199)
(2.200)
(2.201)
π where two relationships, Γ(z)Γ(1 − z) = sin(πz) and Γ(z + 1) = zΓ(z), are utilized. Consequently, we obtain Z ∆ly f (x) 1 dy c1 (l, α) R |y|1+α Z ∞ 1 2α−1 αΓ 1+α f (x + y) − 2f (x) + f (x − y) l=2 2 (2.202) = − dy 1 α 2 |y|1+α 2 π Γ 1 − −∞ 2 Z ∞ 2α−1 αΓ 1+α f (x + y) − 2f (x) + f (x − y) 2 dy. =− 1 α y 1+α π2Γ 1 − 2 0
2.5. Riesz derivative
53
On the other hand, for 0 ≤ α < 1, the left- and right-sided Riemann-Liouville derivatives can be rewritten as α RL D−∞,x f (x) Z x f (t) d 1 dt = (2.203) Γ(1 − α) dx −∞ (x − t)α Z ∞ 1 f (x − y) d = dy Γ(1 − α) dx 0 yα and
α RL Dx,∞ f (x) Z ∞ 1 d f (t) =− dt Γ(1 − α) dx x (t − x)α Z ∞ 1 d f (x + y) = − dy. Γ(1 − α) dx 0 yα
(2.204)
If f (x) ∈ S(R), then f (x) and its derivatives f (k) (x) (k = 1, 2, 3, . . .) vanish at infinity. Thus, integration by parts gives Z ∆ly f (x) 1 dy c1 (l, α) R |y|1+α Z ∞ 2α−1 Γ 1+α f 0 (x + y) − f 0 (x − y) l=2 2 = − 1 dy α yα π2Γ 1 − 2 0 (2.205) Z ∞ 2α−1 Γ 1+α f (x − y) − f (x + y) d 2 = 1 dy yα π 2 Γ 1 − α2 dx 0 2α−1 Γ 1+α α 2 Γ(1 − α) RL Dα = 1 −∞,x f (x) + RL Dx,+∞ f (x) . α 2 π Γ 1− 2 Here the order of differentiation and integration can be exchanged due to the uniform convergence of the integral because f (x) ∈ S(R). Since 1+α Γ( 1−α 1 2 )Γ( 2 ) = πα 2 cos( 2 )Γ(1 − α) 2πΓ(1 − α) (2.206) 1 ) π 2 21−(1−α) 2α−1 Γ( 1+α ) Γ( 1+α 2 2 = 1 , = 2π Γ(1 − α2 ) π 2 Γ(1 − α2 ) we readily observe that the equality Z ∆ly f (x) 1 dy = −RZ Dα x f (x) c1 (l, α) R |y|1+α holds for l = 2 and 0 ≤ α < 1. When 1 < α < 2, variable substitutions yield Z ∞ 1 d2 f (x − y) α dy D f (x) = RL −∞,x Γ(2 − α) dx2 0 y α−1 and α RL Dx,∞ f (x)
1 d2 = Γ(2 − α) dx2
Z 0
∞
f (x + y) dy. y α−1
(2.207)
(2.208)
(2.209)
54
Chapter 2. Fractional derivatives
Performing integration by parts on (2.205) gives Z ∆ly f (x) 1 dy c1 (l, α) R |y|1+α Z ∞ 00 l=2 2α−1 Γ 1+α f (x + y) + f 00 (x − y) 2 = dy 1 π 2 (1 − α)Γ 1 − α 0 y α−1 2 Z ∞ 2α−1 Γ 1+α d2 f (x + y) + f (x − y) 2 = dy 1 2 α y α−1 π 2 (1 − α)Γ 1 − 2 dx 0 2α−1 Γ 1+α α 2 Γ(2 − α) RL Dα = 1 −∞,x f (x) + RL Dx,+∞ f (x) , α π 2 (1 − α)Γ 1 − 2
(2.210)
since f 0 (x) vanishes at infinity. Note that 2α−1 Γ( 1+α 1 1 2 ) = = . 1 πα πα 2 cos( 2 )Γ(2 − α) 2 cos( 2 )Γ(1 − α)(1 − α) π 2 (1 − α)Γ( 2−α 2 )
(2.211)
Hence, the equality (2.207) is also valid for l = 2 and 1 < α < 2. For the more general case m − 1 < α < m ∈ Z+ , the same result can be obtained in a similar manner. We thus conclude that the Riesz derivative on arbitrary dimensions generalizes the Riesz derivative in one dimension for functions in the Schwartz space.
2.6 The fractional Laplacian It is known that in physical sciences, the ordinary Laplacian ∆ appears as a contribution to a conservation law or evolution equation according to Fick’s law, a conductive thermal flux according to Fourier’s law, or a viscous stress according to the Newtonian constitutive equation [140]. The underlying assumption is that the rate of transport of a field of interest at a certain location depends on an appropriate field variable at that location, rather than the global structure of the transportation field. Nevertheless, in the anomalous diffusion process due to random displacements, the fractional diffusive flux at a certain location is in fact affected by the state of the field in the entire space. The generalized flux associated with the fractional Laplacian therefore emerged. Correspondingly, the generalized flux can be viewed as a generalization of the ordinary diffusive flux. The fractional Laplacian has been proved to be an adequate tool for mathematical modeling of anomalous phenomenon when traditional approaches fail. The subtlety of the underlying mathematical concept has motivated a variety of research efforts in applied mathematics and the corresponding physical sciences [108, 140]. As far as we know, there are at least ten characterizations and formulae for the fractional Laplacian on Rd , which are all equivalent in specific settings [78]. However, certain equivalences break down if we carry these definitions over to a bounded domain Ω ⊂ Rd , resulting in several distinct fractional Laplacians in a bounded domain. In this section, we introduce commonly used definitions of the fractional Laplacian in Rd . The corresponding fractional Poisson problems are also introduced. Definition 2.18. [80, 148] The fractional Laplacian (−∆)s is defined via a pseudo-differential operator with the symbol |ω|2s : (−∆)s f (x) = F −1 |ω|2s F[f ](ω); x , s > 0.
(2.212)
2.6. The fractional Laplacian
55
Although fractional differential equations with the fractional Laplacian are widely applied [12, 65, 66, 67], obtaining numerical solutions is still difficult and hot. At present, definitions of (−∆)s in numerical calculations can be mainly classified into the following categories. I. The Riesz (integral) definition Let S denote the Schwartz space defined on Rd . For any f ∈ S and s ∈ (0, 1), the fractional Laplacian (−∆)s can be defined in the following integral form [133]: (−∆)s f (x) Z f (x) − f (y) =C(d, s) P.V. dy d+2s Rd |x − y| Z f (x) − f (y) dy. =C(d, s) lim+ d+2s →0 Rd \B (x) |x − y|
(2.213)
Here P.V. is a commonly used abbreviation for “in the principle value sense,” and C(d, s) is a dimensional constant that depends on d and s, precisely given by Z C(d, s) = Rd
1 − cos ζ1 dζ |ζ|d+2s
−1 ,
(2.214)
with ζ1 denoting the first component of ζ = (ζ1 , ζ2 , . . . , ζd ) ∈ Rd . Remark 2.11. The definition (2.213) is not well defined in general due to the singularity of the kernel [133]. However, the integral in (2.213) is weakly singular near x when s ∈ (0, 21 ). Interestingly, for any f ∈ S, Z |f (x) − f (y)| dy d+2s Rd |x − y| Z Z 1 |x − y| dy + 2kf k dy ≤C ∞ d 1 L (R ) d+2s d+2s |x − y| |x − y| y∈B / R BR Z Z (2.215) 1 1 ≤C dy + dy 2 d+2s−1 d+2s BR |x − y| y∈B / R |x − y| ! Z R Z +∞ 1 1 dρ + dρ < +∞, ≤C3 2s |ρ|2s+1 R 0 |ρ| where C1 , C2 , and C3 are positive constants depending only on the dimension and on the L∞ norm of f . Such a case with s ∈ (0, 12 ) can describe fractional convection in Rd . If s ∈ ( 12 , 1), the integral in (2.213) is hypersingular, which often characterizes fractional diffusion. The case with s = 12 seems not to be very clear for physical description. In analysis, the above fractional Laplacian is often represented in a regularized form [78], by virtue of a weighted 2nd order differential quotient. Lemma 2.19. [133] Let s ∈ (0, 1) and let (−∆)s be the fractional Laplacian defined by (2.213). Then, for any f ∈ S, Z 1 f (x + y) − 2f (x) + f (x − y) s (−∆) f (x) = − C(d, s) dy, x ∈ Rd . (2.216) 2 |y|d+2s d R
56
Chapter 2. Fractional derivatives
Proof. The equivalence of (2.213) and (2.216) immediately follows by the standard changing variable formula. By choosing z = y − x, we have (−∆)s f (x) Z f (y) − f (x) dy = − C(d, s) P.V. (2.217) |x − y|d+2s d R Z f (x + z) − f (x) = − C(d, s) P.V. dz. |z|d+2s Rd ˜ = −z in the last term of the above equality, we have Moreover, by substituting z Z Z ˜) − f (x) f (x + z) − f (x) f (x − z P.V. dz = P.V. d˜ z. d+2s d+2s |z| |˜ z | d d R R ˜ as z, we obtain After relabeling z Z f (x + z) − f (x) 2 P.V. dz |z|d+2s d R Z Z f (x + z) − f (x) f (x − z) − f (x) =P.V. dz + P.V. dz d+2s |z| |z|d+2s Rd Rd Z f (x + z) + f (x − z) − 2f (x) =P.V. dz. |z|d+2s d R
(2.218)
(2.219)
Therefore, if we rename z as y in (2.217) and (2.219), we can write the fractional Laplacian in (2.213) as Z 1 f (x + y) + f (x − y) − 2f (x) (−∆)s u(x) = − C(d, s)P.V. dy. (2.220) 2 |y|d+2s Rd The above representation is useful for removing the singularity of the integral at the origin. Indeed, for any smooth function f , a 2nd order Taylor expansion yields |f (x + y) + f (x − y) − 2f (x)| kD2 f kL∞ ≤ , |y|d+2s |y|d+2s−2
(2.221)
which is integrable near 0 (for any fixed s ∈ (0, 1)). Therefore, if f ∈ S(Rd ), we can get rid of the P.V. and obtain (2.216). The integral representation in (2.213) is defined for 0 < s < 1 while the pseudo-differential definition in (2.212) is valid for any s > 0. Now we present the equivalence between these two definitions with 0 < s < 1. Theorem 2.20. [133] Let s ∈ (0, 1) and let (−∆)s : S → L2 (Rd ) be the fractional integral defined by (2.213). Then, for any f ∈ S, (−∆)s f (x) = F −1 |ω|2s F[f ]; x , ∀ω ∈ Rd . (2.222) Proof. In view of Lemma 2.19, we may use the definition via the weighted 2nd order differential quotient in (2.216). We denote by Lf the integral in (2.216), that is, Z 1 f (x − y) − 2f (x) + f (x + y) Lf (x) = − C(d, s) dy, (2.223) 2 |y|d+2s Rd with C(d, s) as in (2.214).
2.6. The fractional Laplacian
57
L is a linear operator and we are looking for its “symbol” (or “multiplier”), that is, a function L : Rd → R such that Lf = F −1 (Lfb). (2.224) We want to prove that L(ω) = |ω|2s ,
(2.225)
where we denote by ω the frequency variable. To this end, we point out that |f (x − y) − 2f (x) + f (x + y)| |y|d+2s ≤4 χB1 (y)|y|2−d−2s sup |D2 f | B1 (x) +χ d −d−2s |f (x − y) − 2f (x) + f (x + y)| ∈ L1 (R2d ). R \B1 (y) |y|
(2.226)
Consequently, by the Fubini-Tonelli Theorem, we can exchange the integral in y with the Fourier transform in x. Thus, we apply the Fourier transform in the variable x in (2.224) and obtain L(ω)(Ff )(ω) = F(Lf ) Z 1 F {f (x − y) − 2f (x) + f (x + y); ω} = − C(d, s) dy 2 |y|d+2s Rd Z e−iω·y − 2 + eiω·y 1 dyfb(ω) = − C(d, s) d+2s 2 |y| d R Z 1 − cos(ω · y) b =C(d, s) dyf (ω). |y|d+2s d R Hence, in order to obtain (2.225), it suffices to show that Z 1 − cos(ω · y) dy = C(d, s)−1 |ω|2s . d+2s |y| d R
(2.227)
(2.228)
To check this, first we observe that, if ω ∈ Rd , we have |ω1 |2 1 1 − cos ω1 ≤ ≤ d+2s d+2s d+2s−2 |ω| |ω| |ω|
(2.229)
near ω = 0. Thus, Z 0< Rd
1 − cos ω1 dω < ∞. |ω|d+2s
Now we consider the function I : Rd → R defined as Z 1 − cos(ω · y) I (ω) = dy. |y|d+2s Rd
(2.230)
(2.231)
We have that I is rotationally invariant, that is, I (ω) = I (|ω|e1 ),
(2.232)
58
Chapter 2. Fractional derivatives
where e1 denotes the first direction vector in Rd . Indeed, when d = 1, then we can deduce (2.232) by the fact that I (−ω) = I (ω). When d ≥ 2, we consider a rotation R for which ˜ = RT y, we obtain R(|ω|e1 ) = ω and we denote by RT its transpose. Then, by substituting y Z 1 − cos((R(|ω|e1 )) · y) dy I (ω) = |y|d+2s d R Z 1 − cos (|ω|e1 ) · RT y = dy |y|d+2s Rd Z ˜) 1 − cos((|ω|e1 ) · y d˜ y = I(|ω|e1 ), = d+2s |˜ y| Rd
(2.233)
which proves (2.232). In view of (2.230) and (2.232), the substitution ζ = |ω|y gives I (ω) =I (|ω|e1 ) Z 1 − cos(|ω|y1 ) = dy |y|d+2s d R Z 1 − cos ζ1 |ω|2s 1 dζ = , = d |ω| Rd |ζ/|ω||d+2s C(d, s) where we recall that
1 C(d,s)
is equal to
R
1−cos ζ1 dζ Rd |ζ|d+2s
(2.234)
by (2.214). Hence, we deduce (2.222).
It has also been shown in [133] that the fractional Laplacian defined by (2.213) coincides with integer-order derivatives in the following way. Property 2.16. [133] Let d > 1. Then for any f ∈ C0∞ (Rd ) the following statements hold for the fractional Laplacian defined by (2.213): (I) lim+ (−∆)s f = f , s→0
(II) lim− (−∆)s f = −∆f . s→1
The connection between the Riesz derivative and the fractional Laplacian in (2.213) has been considered in several works. Differently from the explanation provided by [133], the explanation given by [11] elucidates the links between the fractional Laplacian in the integral definition with the Riesz derivative on Rd with arbitrary positive integer d, by virtue of direct calculations. The following are the details. The fractional Laplacian defined by (2.213) on the whole axis R can be written as 1 (−∆)s f (x) = − C(1, s) 2
Z
∞
−∞
f (x + y) − 2f (x) + f (x − y) dy ∀x ∈ R. |y|1+2s
(2.235)
Taking d = 1 into (2.214) yields that Z
∞
C(1, s) = −∞
1 − cos ζ dζ |ζ|1+2s
−1 .
(2.236)
2.6. The fractional Laplacian
59
Integration by parts gives Z ∞ Z ∞ 1 − cos ζ 1 − cos ζ dζ = 2 dζ −∞ |ζ|1+2s ζ 1+2s 0 Z ∞ Z sin2 ζ2 1 ∞ sin ζ =4 dζ = dζ. ζ 1+2s s 0 ζ 2s 0 Performing the integration, we obtain [59] Z ∞ 1 − cos ζ 1 dζ = Γ(1 − 2s) cos(πs). 1+2s |ζ| s −∞ Using the following properties of the Gamma function that π Γ(1 − z)Γ(z) = , z 6= 0, ±1, ±2, . . . , sin(πz)
(2.237)
(2.238)
(2.239)
and
√ 1 Γ(z)Γ(z + ) = π21−2z Γ(2z), 2z 6= 0, −1, −2, . . . , 2 we obtain the alternative expression Z ∞ 1 π π 2 Γ(1 − s) 1 − cos ζ . dζ = = 1+2s Γ(1 + 2s) sin(πs) 22s sΓ 1+2s −∞ |ζ| 2
(2.240)
(2.241)
Substituting the above equation into (2.236) yields C(1, s) =
22s sΓ 1
1+2s 2
π 2 Γ(1 − s)
.
Consequently, we have that, for f (x) ∈ S(R), (−∆)s f (x) Z ∞ 2s−1 sΓ 1+2s f (x + y) − 2f (x) + f (x − y) = − 2 2 dy 1 |y|1+2s π 2 Γ(1 − s) −∞ Z ∞ 22s sΓ 1+2s f (x + y) − 2f (x) + f (x − y) 2 dy. = − 1 y 1+2s π 2 Γ(1 − s) 0 Performing integration by parts and letting s = α2 , we obtain α (−∆) 2 f (x) Z ∞ 2α−1 αΓ 1+α f (x + y) − 2f (x) + f (x − y) 2 dy = − π 1/2 Γ 2−α y α+1 0 2 Z ∞ 2α−1 Γ 1+α f 0 (x + y) − f 0 (x − y) 2 = − 1/2 dy α yα π Γ 1− 2 0 Z ∞ 2α−1 Γ 1+α f (x − y) − f (x + y) 2 d dy = 1/2 α dx yα π Γ 1 − 0 2 Z x Z ∞ 2α−1 Γ 1+α f (y) f (y) d 2 dy − dy = 1/2 α α (y − x)α x π Γ 1 − 2 dx −∞ (x − y) 2α−1 Γ 1+α α α 2 = 1/2 α Γ(1 − α) RL D−∞,x f (x) + RL Dx,+∞ f (x) π Γ 1− 2
(2.242)
(2.243)
(2.244)
60
Chapter 2. Fractional derivatives
for any f ∈ S(R). Since α−1 1+α 2 Γ 2 Γ(1 − α) π 1/2 Γ 1 − α2 Γ( 1+α )Γ(1 − α) 1/2 1−(1−α) π 2 2 = 2π Γ(1 − α2 ) 1+α Γ( 1−α 1 2 )Γ( 2 ) = = , 2π 2 cos( πα 2 )
(2.245)
we conclude that, for α ∈ (0, 1), α (−∆) 2 f (x) 1 α RL Dα = −∞,x f (x) + RL Dx,+∞ f (x) πα 2 cos 2 = − RZ Dα x f (x), f ∈ S(R).
(2.246)
In a similar manner, we can readily derive that the fractional Laplacian defined in (2.213) α also satisfies the relation (−∆) 2 f (x) = −RZ Dα x f (x) for f ∈ S(R) when α ∈ (1, 2). no Remark 2.12. (I) When α = 1, the Riesz derivative RZ Dα x f (x) in (2.174) seems to make α sense, but we use another equivalent definition in (2.175) [85]. On the other hand, (−∆) 2 f (x) exists with Z 1 1 ∞ f 0 (x + y) − f 0 (x − y) (−∆) 2 f (x) = − dy, (2.247) π 0 y in which the integral is convergent as f ∈ S(R). It is evident that the fractional Laplacian and Riesz derivative are coincident for any f ∈ S(R). (II) The Riesz derivative is essentially the fractional Laplacian defined on R. But they are not equivalent on the semiline or on a bounded domain. The partial Riesz derivatives in multiple dimensions are complicated and inconvenient to apply. So the fractional Laplacian on Rd is often adopted. Regarding the fractional Laplacian given by (2.213) on Rd , we can similarly derive that, for any f (x) ∈ S(Rd ), Z 22s−1 sΓ d+2s f (x + y) − 2f (x) + f (x − y) s 2 (−∆) f (x) = − dy. (2.248) d |y|d+2s 2 π Γ(1 − s) Rd
And it can be readily obtained that α
d (−∆) 2 f (x) = RZ Dα x f (x), f (x) ∈ S(R ), d > 1, α ∈ (0, 2).
(2.249)
Remark 2.13. In view of the above analysis and physical interpretations of the Riesz derivative in the previous section, it is believed that the fractional Laplacian (−∆)s with s ∈ (0, 21 ) (as a weak singular integral) is a suitable instrument characterizing anomalous convection (or advection), and is capable of describing anomalous diffusion when s ∈ ( 12 , 1) (as a hypersingular integral). The value s = 21 , corresponding also to a hypersingular integral, is critical. But its physical meaning seems to be unclear.
2.6. The fractional Laplacian
61
Carrying the definition (2.213) to functions on the bounded domain Ω gives the fractional Laplacian on Ω. Since (2.213) requires values of f on all of Rd , an exterior boundary condition f (x) = g(x), x ∈ Rd \ Ω
(2.250)
is required, even to define the fractional Laplacian within Ω. For functions f (x) satisfying (2.250), the fractional Laplacian defined for x ∈ Ω is given by (−∆Ω )s f (x) Z f (x) − f (y) =C(d, s) P.V. dy d+2s d (2.251) R |x − y| " # Z Z f (x) − f (y) f (x) − g(y) =C(d, s) P.V. dy + dy . d+2s d+2s |x − y| Ω Rd \Ω |x − y| In this case, the fractional Poisson problem on Ω is in the form [3, 4] ( (−∆Ω )s f = u in Ω, (2.252)
f = g in Rd \ Ω.
II. The directional definition The directional definition is believed to preserve all the original information of the fractional Laplacian. Denote by x(t) the position of a particle in d-dimensional Euclidean space Rd at time t ≥ 0, and u(x, t) the density of x(t) with the vector x = (x1 , . . . , xd ) ∈ Rd . We consider the general advection-dispersion equation ∂u(x, t) = −v∇u(x, t) + c ∇α M u(x, t), ∂t
(2.253)
∂ with v ∈ Rd being the velocity, ∇ = ( ∂x , . . . , ∂x∂ d ), c the spreading rate of the dispersion, and 1 α ∇M the general fractional derivative operator for 0 < α ≤ 2, α 6= 1. When the operator ∇α M is specified through the Fourier transform "Z # α α F {∇ f (x); ω} = (ihω, θi) M (dθ) fb(ω) (2.254) M
kθk=1
with M (dθ) being an arbitrary probability measure on the unit sphere centered at the origin, solutions to (2.253) with the initial condition u (x(0) = 0) = 1
(2.255)
yield every possible multivariable Lévy motion x(t) with 0 < α ≤ 2, α 6= 1 [124]. In the case of α = 2, the integral in (2.254) reduces to 2 Z d X − ωj θj M (dθ) = (−iω)A(−iω), (2.256) kθk=1
j=1
where the matrix A = (aij ) with aij = ∇2M f (x)
R
θ θ M (dθ). kθk=1 i j
= −∇ · ∇f (x) = −
Then it holds that
d X d X i=1 j=1
aij
∂ 2 f (x) . ∂xi ∂xj
(2.257)
62
Chapter 2. Fractional derivatives
Consequently, we obtain multivariable Brownian motion and observe that (2.253) coincides with the classical advection-dispersion equation in Rd . In one dimension, kθk = 1 implies θ = ±1 and then Z α
(ihω, θi) M (dθ) = p(iω)α + q(−iω)α
(2.258)
holds with p + q = 1. In this case, (2.253) covers the general one-dimensional fractional advection-dispersion equation introduced in [20, 147]. In addition, taking Fourier transforms on (2.253) along with the initial condition u b(ω, 0) = 1 gives " # Z α (ihω, θi) M (dθ) u b(ω, t) = exp −ihω, vti + ct kθk=1 " Z πα α = exp −ihω, vti + ct |hω, θi| cos 2 kθk=1 # n πα o × 1 + i sgn (hω, θi) tan M (dθ) , 2
(2.259)
which represents the Fourier transform of an arbitrary multivariable stable distribution with index α 6= 1 [149]. It follows that the Green function solution to (2.253) motivates the entire class of multivariable Lévy motions with α 6= 1. As a matter of fact, the asymmetric fractional derivative operator ∇α M is a mixture of the f (x). Note that the directional derivative Dθ f (x) is defined fractional directional derivative Dα θ by d X ∂f (x) dg Dθ f (x) = hθ, 4f (x)i = θj = (2.260) ∂xj ds j=0
s=0
with g(s) = f (x + sθ). To see this, we can directly check the corresponding Fourier transform of the last two terms in the above equation. Note that the fractional-order directional derivative Dα θ f (x) is defined by the scalar fractional derivative [124] Z ∞ 1 dm α D g(s) = rm−α−1 g(s − r)dr + Γ(m − α) dsm 0 Z ∞ 1 r−α−1 g(s − r)dr, m − 1 < α < m ∈ Z+ , = Γ(−α) 0
(2.261)
a convolution of generalized functions [148], in the form α Dα θ f (x) = D+ g(s) s=0 .
(2.262)
The fractional derivative ∇α M can thus be defined via convolutions of generalized functions in the real spaces, or products in Fourier space [148]. Taking a mixture over the unit sphere kθk = 1 gives the following convolution formula: ∇α M f (x) = which satisfies (2.254).
1 Γ(−α)
Z kθk=1
Z 0
∞
r−α−1 f (x − rθ)drM (dθ),
(2.263)
2.6. The fractional Laplacian
63
Let M (dθ) = m(θ)dθ and the constant cd be the volume of the unit sphere in Rd . Then we obtain Z Z ∞ 1 α r−α−1 f (x − rθ)m(θ)drdθ ∇ f (x) = M Γ(−α) kθk=1 0 Z Z ∞ 1 r−α−d m(θ)f (x − rθ)dcd rd−1 dr = dθ (2.264) dc Γ(−α) d 0 kθk=1 Z 1 y = kyk−α−d m f (x − y)dy, dcd Γ(−α) y∈Rd kyk which is a hypersingular integral with homogeneous characteristic [148]. In particular, if m(θ) is a constant function, then this integral reduces to the convolution equation for the fractional Laplacian in the form 2α Γ α+d α 2 (−∆) 2 f (x) = d (2.265) |x|−d−α ∗ f (x), α 2 π Γ −2 coinciding with the integral definition of fractional Laplacian. As a result, boundary value problems with directional fractional Laplacian are in the same form with those for the Riesz fractional Laplacian. III. The spectral definition Let Ω ⊂ Rd be a bounded domain with Lipschitz boundary ∂Ω. Suppose that the eigenpairs {λk , ϕk }∞ k=1 with λk ≥ λ0 > 0 satisfy −∆ϕk = λk ϕk ,
ϕk |∂Ω = 0,
(2.266)
1 and {ϕk }∞ k=1 form an orthogonal basis of H0 (Ω). Then for f sufficiently smooth with
f (x) =
∞ X
fk ϕk (x), (−∆)f (x) =
k=1
∞ X
fk λk ϕk (x),
(2.267)
k=1
and 0 < s < 1, the spectral fractional Laplacian is given via the spectral decomposition of the standard Laplacian [69], i.e., (−∆)s f (x) =
∞ X
fk λsk ϕk (x).
(2.268)
k=1
Such a definition has attracted increasing interest recently. However, its physical meaning seems not to be clear. It is just a simple generalization of a positive operator in finite dimensions (i.e., a symmetric and positive definite matrix in finite dimensional linear space). The spectral definition is considered as a representation which comes from the eigenvalues and eigenfunctions of −∆. In addition, the formulation of the Poisson problem with the spectral fractional Laplacian can be given as ( (−∆)s f = u in Ω, (2.269) f = 0 on ∂Ω. In other words, the boundary value condition is retained without making changes to the classical case. Besides the definition (2.213) of fractional Laplacian, the directional and spectral definitions are occasionally used. We must state that all of these definitions are generally not equivalent [45].
64
Chapter 2. Fractional derivatives
2.7 Fractional derivatives of other types In the following, we introduce fractional derivatives of other types, which are worth mentioning.
2.7.1 Grünwald-Letnikov derivative It is known that the usual differentiation of a function f (x) is in the form (∆m h f ) (x) , m ∈ Z+ , h→0 hm
f (m) (x) = lim
(2.270)
+ in which (∆m h f ) (x) is a finite difference of order m ∈ Z of the function f (x), with a step h ∈ R and at the point x ∈ R, i.e., m X k m (−1) f (x − kh). (2.271) (∆m f ) (x) = h k k=0
A natural generalization is to replace m ∈ Z+ in the above expression by α > 0. Then we have the fractional difference (∆α h f ) (x) of order α given by the following infinite series: ∞ X k α (∆α f ) (x) = (−1) f (x − kh), x, h ∈ R, α > 0, (2.272) h k k=0
where k = α(α−1)···(α−k+1) is the binomial coefficient. This series converges absolutely and k! uniformly for arbitrary α > 0 and for every bounded function f (x). In addition, the fractional difference (∆α h f ) (x) has the following properties. α
Property 2.17. [75] If α > 0 and β > 0, then the semigroup property β α+β ∆α f (x) h ∆h f (x) = ∆h
(2.273)
is valid for any bounded function f (x). Property 2.18. [75] If α > 0 and f (x) ∈ L1 (R), then the Fourier transform of ∆α h f is given by α iωh F{(∆α fb(ω), i2 = −1 (2.274) h f ) (x); ω} = 1 − e with fb(ω) being the Fourier transform of f (x). Based on the above introduced fractional difference, we can define the following GrünwaldLetnikov derivative. Definition 2.21. [172] The left- and right-sided Grünwald-Letnikov derivatives of order α > 0 are defined by x−a [X h ] 1 α k α f (x − kh), a < x < b, h > 0, (−1) GL Da,x f (x) = lim α h→0 h k
(2.275)
k=0
and α GL Dx,b f (x)
[ b−x h ] 1 X α = lim α (−1)k f (x + kh), a < x < b, h > 0, h→0 h k k=0
respectively.
(2.276)
2.7. Fractional derivatives of other types
65
The link between the Grünwald-Letnikov derivative and the Riemann-Liouville derivative was considered in [139]. It was shown that the equivalence between these two derivatives stays valid under specific conditions. Property 2.19. [139] Suppose that the function f (x) is (m − 1) times continuously differentiable on the interval [a, b] and that f (n) (x) is integrable on [a, b]. Then, for every α (0 < α < n), Riemann-Liouville derivative RL Dα a,x f (x) exists and coincides with the Grünwald-Letnikov derivative GL Dα f (x). Furthermore, if 0 ≤ m − 1 ≤ α < m ≤ n, for a < x < b, the following a,x equality holds: α RL Da,x f (x)
=
m−1 X j=0
= GL Dα a,x f (x)
1 f (j) (a)(x − a)j−α + Γ(1 + j − α) Γ(m − α)
x
Z a
f (m) (t)dt . (x − t)α−m+1
(2.277)
Remark 2.14. The above property indicates that Grünwald-Letnikov derivative can be utilized to evaluate the Riemann-Liouville derivative, resembling the case of approaching integer-order derivatives with finite differences. Interestingly, the Grünwald-Letnikov derivative is frequently viewed as a 1st order approximation to the Riemann-Liouville derivative, which is crucial for later discussions.
2.7.2 Hadamard derivative d Definition 2.22. [75, 176] Let δ = x dx be the δ derivative. Then the left- and right-sided Hadamard derivatives are defined by α m α−m H Da,x f (x) = δ (H Da,x f )(x) m Z x (2.278) d 1 x m−α−1 f (t)dt = x log dx Γ(m − α) a t t
and
α−m = (−δ)m (H Dx,b f )(x) m m−α−1 Z b 1 t f (t)dt d = −x log , dx Γ(m − α) x x t
α H Dx,b f (x)
(2.279)
α−m where 0 ≤ a < x < b ≤ +∞, m − 1 < α < m ∈ Z+ , H Da,x f , and H Dα−m x,b f are Hadamard integrals. When x > 0, a = 0, and b = ∞, we have m Z x d 1 x m−α−1 f (t)dt α log (2.280) H D0,x f (x) = x dx Γ(m − α) 0 t t
and α H Dx,∞ f (x)
=
d −x dx
m
1 Γ(m − α)
Z
∞
x
t log x
m−α−1
f (t)dt . t
(2.281)
When α = m ∈ Z+ , for 0 ≤ a < x < b ≤ ∞, m H Da,x f (x)
= (δ m f )(x) and
α H Dx,b f (x)
= (−1)m (δ m f )(x).
(2.282)
66
Chapter 2. Fractional derivatives d and m ∈ Z+ . Define the space Again, set δ = x dx ACδm ([a, b]) = g : [a, b] → C δ m−1 [g(x)] ∈ AC[a, b] .
(2.283)
Then we have the following result on the existence of Hadamard derivative. Theorem 2.23. [74] Let α ≥ 0 and m − 1 < α < m. If f (x) ∈ ACδm ([a, b]) (0 < a < b < ∞), α then the Hadamard derivatives H Dα a,x f and H Dx,b f exist almost everywhere on [a, b] and can be represented in the forms α H Da,x f (x) m−1 x k−α X (δ k f )(a) = log (2.284) Γ(1 + k − α) a k=0 Z x 1 x m−α−1 m + log (δ f )(t)dt Γ(m − α) a t and
α H Dx,b f (x) k−α m−1 k k = X (−1) (δ f )(b) log b Γ(1 + k − α) x k=0 m−α−1 Z b (−1)m t log (δ m f )(t)dt, + Γ(m − α) x x
respectively. In particular, when 0 < α < 1, then, for f (x) ∈ AC([a, b]), Z x f (a) x −α 1 x −α 0 dt α D f (x) = log + log f (t) H a,x Γ(1 − α) a Γ(1 − α) a a t
(2.285)
(2.286)
and α H Dx,b f (x) =
f (b) Γ(1 − α)
log
b x
−α −
1 Γ(1 − α)
Z
b
x
−α b dt log f 0 (t) . x t
(2.287)
It can be directly verified that the Hadamard derivative of the power function xβ yields the same power function, apart from a constant multiplication factor. Property 2.20. [75] Let α > 0 and β ∈ R. (I) If β > 0, then α β H D0,x x
= β α xβ .
(2.288)
= (−β)α xβ .
(2.289)
(II) If β < 0, then α β H Dx,∞ x
In addition, the Hadamard derivatives of the logarithmic functions [log(x/a)]β−1 and [log(b/x)]β−1 yield logarithmic functions of the same form. Property 2.21. [58, 75] If α > 0, β > 0, and 0 < a < b < ∞, then x β−1 Γ(β) x β−α−1 α = log H Da,x log a Γ(β − α) a
(2.290)
2.7. Fractional derivatives of other types
67
and α H Dx,b
β−1 β−α−1 b Γ(β) b log = log . x Γ(β − α) x
(2.291)
In particular, if β = 1 and α ≥ 0, then the Hadamard derivative of a constant, in general, is not equal to zero. For instance, when 0 < α < 1, 1 x −α log , Γ(1 − α) a
α H Da,x 1
(x) =
α H Dx,b 1
1 (x) = Γ(1 − α)
b log x
(2.292)
−α .
(2.293)
On the other hand, for j = 1, 2, . . . , [α] + 1, x α−j α =0 H Da,x log a
and
α−j b α D log = 0. H x,b x
(2.294)
Corollary 2.24. [75] Let α > 0, m = [α] + 1, and 0 < a < b < ∞. (I) The equality H Dα a,x f (x) = 0 holds if and only if m X
x α−j cj log , a j=1
f (x) =
(2.295)
where cj ∈ R (j = 1, 2, . . . , m) are arbitrary constants. In particular, when 0 < α ≤ 1, x α−1 the relation H Dα for constant c ∈ R. a,x f (x) = 0 holds if and only if f (x) = c log a (II) The equality H Dα x,b f (x) = 0 holds if and only if f (x) =
m X
dj
j=1
b log x
α−j ,
(2.296)
where dj ∈ R (j = 1, 2, . . . , m) are arbitrary constants. In particular, when 0 < α ≤ 1, b α−1 the relation H Dα for constant d ∈ R. x,b f (x) = 0 holds if and only if f (x) = d log x Instead of the Fourier and Laplace transforms, the Mellin transform is frequently adopted in solving factional differential equations with the Hadamard derivative. Definition 2.25. The Mellin transform of a function f (x) defined on R+ is given by f ∗ (s) = M {f (x); s} =
Z
∞
xs−1 f (x)dx,
s ∈ C,
(2.297)
0
and the inverse Mellin transform is given for x ∈ R+ by the formula f (x) = M−1 {f ∗ (s); x} =
1 2πi
Z
γ+i∞
γ−i∞
x−s f ∗ (s)ds,
γ = 0 and function f (x) be such that its Mellin transform f ∗ (s) = M {f (x); s} exists for s ∈ C. Then (I) When 0. When 0 < α < 1, we have α α −∞ Dx f (x) = RL D−∞,x f (x) Z x 1 d = (x − t)−α f (t)dt Γ(1 − α) dx −∞ Z ∞ d 1 t−α f (x − t)dt = Γ(1 − α) dx 0 Z ∞ (2.302) 1 = t−α f 0 (x − t)dt Γ(1 − α) 0 Z ∞ Z ∞ α dξ 0 = f (x − t)dt 1+α Γ(1 − α) 0 ξ t Z ∞ α f (x) − f (x − ξ) = dξ. Γ(1 − α) 0 ξ 1+α We can similarly derive α x D+∞ f (x)
=
α RL Dx,+∞ f (x)
α = Γ(1 − α)
Z 0
∞
f (x) − f (x + ξ) dξ. ξ 1+α
(2.303)
In this case, we introduce the Marchaud fractional derivative (or Marchaud derivative). Definition 2.27. [99, 148] For any α > 0, the left- and right-sided Marchaud derivatives are given by Z ∞ (m−1) {α} f (x) − f (m−1) (x − ξ) α dξ (2.304) M D+ f (x) = Γ(1 − {α}) 0 ξ 1+{α}
2.7. Fractional derivatives of other types
69
and α M D− f (x)
=
{α} Γ(1 − {α})
∞
Z 0
f (m−1) (x) − f (m−1) (x + ξ) dξ, ξ 1+{α}
(2.305)
respectively, where {α} = α − m + 1 with m − 1 < α < m ∈ Z+ . Remark 2.15. It is evident that the relations α M D+ f (x)
= −∞ Dα x f (x),
α M D− f (x)
= x Dα +∞ f (x)
(2.306)
hold whenever f (x), with its derivatives f (k) (x), k = 1, 2, . . . , m, is continuous and vanishes at infinity as |x|α−1− with > 0 and m − 1 < α < m. Remark 2.16. [148] In fact, the integrals (2.304) and (2.305) exist under more general assumptions for the function f (x): restrictions in the definition are needed for realizing the simple transform from Liouville derivative to Marchaud derivative. Furthermore, integrals in equations (2.304) and (2.305) exist for a wider variety of functions, for example, bounded functions satisfying the local Hölder condition of order λ > α. Remark 2.17. [148] Note that M Dα + f (x) exists for the function f (x) being a constant, and in α this case M Dα + f (x) = 0; while the Liouville derivative −∞ Dx f (x) does not exist whenever f (x) is a constant. It can be seen that the Marchaud fractional derivatives are more convenient on R than Liouville derivatives as they allow more freedom for f (x) at infinity. Apart from being related to the Liouville derivative, the Marchaud derivative M Dα ± f (x) has −α a close relation with fractional integrals RL D−α f (x) and D f (x). In fact, comparing RL x,+∞ −∞,x α the Marchaud derivative M Dα ± f (x) with fractional integrals, we see that M D± f (x) is formally −α −α obtained from fractional integrals RL D−∞,x f (x) and RL Dx,+∞ f (x) if we replace α by −α, 0 < α < 1. Note that the substraction of f (m−1) (x) in M Dα ± f (x) provides the convergence of the integral. Therefore, M Dα ± f (x) is closely connected with ideas concerning divergent integrals. Here we introduce some of the following ideas in this respect. Definition 2.28. [148] Let Φ(t) be integrable on an interval (, A) with any A > 0 and 0 < < A. The function Φ(t) is said to possess the Hadamard property at the point t = 0 if there exist constants ak , b, and λk > 0 such that Z
A
Φ(t)dt =
N X
ak −λk + b ln
k=1
1 + J0 (),
(2.307)
where lim J0 () exists and is finite. By definition →0
Z p.f.
A
Φ(t)dt = lim J0 (). 0
→0
(2.308)
RA The limit (2.308) is called a finite part (partie finie) of the divergent integral 0 Φ(t)dt in the Hadamard sense or simply an integral in the Hadamard sense. The constructive realization of RA the function J0 () is sometimes called a regularization of the integral 0 Φ(t)dt.
70
Chapter 2. Fractional derivatives
Remark 2.18. The constants ak , b, and λk > 0 in (2.307) do not depend on the choice of A. To see this, we assume that Φ(t) is integrable at infinity. By definition, it holds that Z ∞ Z A Z ∞ Φ(t)dt. (2.309) Φ(t)dt + Φ(t)dt = p.f. p.f. A
0
0
Then we can see that this definition does not depend on the choice of A. After introducing the finite part of integrals, we return to the Marchaud derivative M Dα ± f (x) R∞ and the divergent integral 0 ft(x−t) dt. First, we have the following lemma through direct 1+α verification. Lemma 2.29. [148] Let 0 < α < 1 and let f (x) be locally Hölderian of order λ > α. Then the function Φ(t) = f (x − t)t−1−α possesses the Hadamard property at the point t = 0 for each x, and if |f (t)| ≤ c|t|α− , > 0, as t → −∞, then Z ∞ Z ∞ f (x − t) − f (x) f (x − t) p.f. dt = dt. (2.310) 1+α t t1+α 0 0 It follows from Lemma 2.29 that the relations ( α α M D+ f (x) = p.f. RL D−∞,x f (x), α M D− f (x)
= p.f. RL Dα x,+∞ f (x)
(2.311)
hold whenever 0 < α < 1. Second, we can also readily prove the following regularization of the divergent integral R ∞ f (x−t) t1+α dt. 0 Lemma 2.30. [148] Let f (x) ∈ C m (R) and f (m) (x) satisfy local Hölder conditions of order λ, 0 ≤ λ < 1. Then the function Φ(t) = f (x − t)t−1−α possesses the Hadamard property at the point t = 0 for any x if α < m + λ. If |f (t)| ≤ c|t|α− holds also for t → −∞, then Z ∞ f (x − t) 1 p.f. dt Γ(−α) t1+α 0 m P k (−1)k tk! f (k) (x) Z 1 f (x − t) − 1 k=0 (2.312) = dt Γ(−α) 0 t1+α Z ∞ m X 1 f (x − t) (−1)k f (k) (x) + dt + , 1+α Γ(−α) 1 t Γ(−α)k!(k − α) k=0
where α < m + λ, α 6= 0, 1, 2, . . . . Finally, we see that the definition of the fractional derivative of order α > 0 via (2.312) agrees well with the definition (2.311) for any sufficiently “good” function f (x). Theorem 2.31. [148] Let f (x) satisfy the assumption of Lemma 2.30 with m ≥ [α] + 1. Then the Liouville fractional derivative coincides with (2.312) for any α > 0 but α 6= 1, 2, . . . . Detailed reviews of the Grünwald-Letnikov derivative, Hadamard derivative, and Marchaud derivative are given in [51, 145]. For fractional derivatives of distributions, refer to [82, 83, 84, 85, 86] for more information.
2.8. Definite conditions for fractional differential equations
71
2.8 Definite conditions for fractional differential equations In this section, we consider definite conditions for fractional differential equations with typical fractional derivatives, which is a somewhat tricky subject.
2.8.1 Caputo and Riemann-Liouville cases First, we consider fractional differential equations in the Caputo sense. As mentioned before, the formulation of initial conditions for Caputo type fractional differential equations contains the limit values of integer-order derivatives of unknown functions at the lower terminal, which can be reflected through the Laplace transform. Here we show this point via another approach. Consider Caputo type differential equation of the form α C D0,x u(x)
= f (x), m − 1 < α < m ∈ Z+ .
(2.313)
−(m−α)
Since C Dα u(m) (x), equation (2.313) can be equivalently written as 0,x u(x) = RL D0,x (m) m−α u (x) = RL D0,x f (x) Z x (2.314) 1 d = (x − t)α−m f (t)dt =: F (x). Γ(α − m + 1) dx 0 In this case, it is clear that the initial value conditions of the above equation (2.314) can be posed as (k) u(k) (0) = u0 , k = 0, 1, . . . , m − 1. (2.315) As a result, the initial and/or boundary value conditions of the Caputo type (ordinary or partial) differential equation take the same form as those of the classical (ordinary or partial) differential equation [99]. Now we consider the Riemann-Liouville type differential equation α RL D0,x u(x)
= f (x), m − 1 < α < m ∈ Z+ .
(2.316)
Recall the definition of the Riemann-Liouville derivative. It holds that α RL D0,x u(x)
=
dm −(m−α) u(x). RL D0,x dxm
In this case, the differential equation (2.316) also reads as dm −(m−α) u(x) = f (x), D RL 0,x dxm and the corresponding initial value conditions should be in the form dk −(m−α) u(x) RL D0,x dxk x=0 (k) = Dk+α−m u(x) = u0 , k = 0, 1, . . . , m − 1. RL 0,x x=0
(2.317)
(2.318)
(2.319)
It follows from the above analysis that the Riemann-Liouville derivative is more “fractional” than the Caputo derivative in this respect [99]. Note that the initial value conditions in (2.319) are nonlocal, and so are not easy to deal with. It is believed that the homogeneous case k+α−m u(x) x=0 = 0, k = 0, 1, . . . , m − 1, (2.320) RL D0,x
72
Chapter 2. Fractional derivatives
seems to be easier than the inhomogeneous one, which is still not easy to deal with since these also have nonlocal initial conditions. In some literature, the homogeneous initial value condition (2.320) is often replaced by u(k) (0) = 0, k = 0, 1, . . . , m − 1.
(2.321)
Nevertheless, these two types of initial value conditions (i.e., (2.320) and (2.321)) are not (mathematically) equivalent. We choose the definite condition (2.321) for equation (2.316) just for simplicity. We should pay particular attention to inhomogeneous initial value conditions of the type u(k) (0) = uk0 6= 0, k = 0, 1, . . . , m − 1,
(2.322)
which may cause the Riemann-Liouville type differential equation (2.316) to be ill-posed. Take the fractional differential equation ( α RL D0,x u(x) = 0, 0 < α < 1, (2.323) u(0) = u0 6= 0 as an example. According to Corollary 2.3, the first equality in the above system yields that the solution is in the form u(x) = cxα−1 (2.324) with c being a constant. In this case, limx→0 u(x) 6= 0 unless the constant c = 0. However, the initial value condition yields c 6= 0, creating a contradiction due to limx→0+ xα−1 = +∞. Therefore, initial value problem (2.323) is ill-posed. So for a Riemann-Liouville type differential equation, its initial value conditions cannot be given as the integer-order initial value conditions unless the homogeneous integer-order initial value conditions are met. Based on literature such as [75, 185], the well-posed conditions for a fractional differential equation of Riemann-Liouville type with 0 < α < 1 are given as ( α 0 < x < +∞, RL D0,x u(x) = f (x), (2.325) α−1 u(x) = u0 RL D 0,x
or
or
(
(
x=0
α RL D0,x u(x) = x1−α u(x) x=0
f (x), =
0 < x < +∞,
u0 Γ(α)
α RL D0,x u(x)
= f (x), 0 < x < +∞, u(a) = ua , a > 0.
The initial value problems (2.325) and (2.326) are equivalent due to the equality Γ(α) lim+ x1−α u(x) = lim+ RL Dα−1 0,x u(x) x→0
x→0
(2.326)
(2.327)
(2.328)
for any u(x) ∈ C1−α ([0, ∞)) provided that both limits exist. Here the space C1−α ([0, ∞)) is given by C1−α ([0, ∞)) = u(x) ∈ C ((0, ∞)) x1−α u(x) ∈ C ([0, ∞)) . (2.329) Furthermore, the solution to these two initial value problems (i.e., (2.325) and (2.326)) is given by a Volterra integral equation of the second kind, say, Z x u0 α−1 1 u(x) = x + (x − t)α−1 f (t)dt. (2.330) Γ(α) Γ(α) 0
2.8. Definite conditions for fractional differential equations
73
T Regarding the initial value problem (2.327), there exists a unique solution in the space C(R+ ) L1loc (R+ ) and it is given by α−1 Z a x 1 α−1 (a − t) f (t)dt u(x) = ua − Γ(α) aα−1 0 (2.331) Z x 1 α−1 + (x − t) f (t)dt, Γ(α) 0 T whenever f ∈ C(R+ ) L1loc (R+ ). Remark 2.19. The definite problem ( α RL D0,x u(x) = f (x), 0 < x < ∞, α ∈ (0, 1), α−1 RL D0,x u(x) x=0 = 0
(2.332)
can be simplified as α RL D0,x u(x)
(
= f (x), 0 < x < ∞, α ∈ (0, 1), (2.333)
u(0) = 0, which is also an approximation of the corresponding real-world problem. Analogously, the initial value problems with right-sided Riemann-Liouville derivative (0 < α < 1) α RL Dx,L u(x) = f (x), −∞ < x < L, (2.334) RL Dα−1 u(x) = uL , x,L x=L
α RL Dx,L u(x) = f (x), −∞ < x < L, uL (L − x)1−α u(x) = , x=L Γ(α) and
(
α RL Dx,L u(x)
u(b) = ub ,
= f (x), −∞ < x < L, −∞ < b < L
(2.335)
(2.336)
are well-posed. Furthermore, the following results given by [92] can be derived. T 1 Theorem 2.32. Let 0 < α < 1. If f (x) ∈ C((−∞, Lloc ((−∞, L)), initial value problem T L)) (2.336) admits a unique solution in C((−∞, L)) L1loc ((−∞, L)) which is given by ! α−1 Z L 1 L−x α−1 f (t)dt u(x) = ub − Γ(α) b (t − b) L−b (2.337) Z L 1 α−1 + (t − x) f (t)dt. Γ(α) x Theorem 2.33. Let 0 < α < 1. The initial value problem ( α RL Dx,L u(x) = u(x), 0 < x < L, u(b) = ub ,
0 0. Γ(αk + β)
(2.409)
2.8. Definite conditions for fractional differential equations
83
The above two theorems can be found in [120]. The general result on existence and uniqueness of the solution to the Hadamard differential equation is displayed in the following theorem. For more details, see [120]. Theorem 2.40. Let f (x, u) satisfy the conditions ( f (x, 0) = 0, |f (x, u) − f (x, v)| ≤ L|u − v|, where L is a positive constant independent of u and v. Then the initial value problem ( α H Da,x u(x) = f (x, u), 0 < a < x < b, u(x0 ) = u0 , a < x0 ≤ b,
(2.410)
(2.411)
has a unique solution u(x) ∈ C1−α,log ([a, b]). For the right-sided Hadamard case, the reader can develop the proof by consulting the above discussions and the right-sided Riemann-Liouville case.
Chapter 3
Numerical fractional integration
We introduce numerical evaluations of fractional integrals in this chapter, including methods based on polynomial interpolation, spectral approximations, the fractional linear multistep method, and diffusive approximation. We only focus on the left-sided integral. Numerical methods for the right-sided integrals are omitted here, since they can be derived in almost the same manner as those for the left-sided ones.
3.1 Numerical methods based on polynomial interpolation + Assume that f (x) is suitably smooth on (a, b). Let h = b−a N with N ∈ Z , and denote xk = a+kh with k = 0, 1, 2, . . . , N . For x = xn , the left-sided fractional integral RL D−α a,x f (x) x=xn reads as −α RL Da,x f (x) x=xn Z xn = 1 (xn − t)α−1 f (t)dt Γ(α) a (3.1) Z n−1 x k+1 1 X (xn − t)α−1 f (t)dt. = Γ(α) xk k=0
It is reasonable to approximate f (x) by a specific polynomial f˜(x) on each interval, such that R xk+1 the integrals xk (xn − t)α−1 f˜(t)dt, 0 ≤ k ≤ n − 1, can be calculated exactly. This idea likely yields quadratures in the form −α RL Da,x f (x)
x=xn
≈
n X
ωk f (xk ),
(3.2)
k=0
which can be viewed as a natural extension of those for integer-order integrals. Here ωk (k = 0, 1, . . . , n) are corresponding coefficients. Different polynomial interpolation functions lead to distinctive formulae. The frequently utilized ones in applications are given as follows. (I) Fractional rectangular formulae If f (x) is approximated by a piecewise constant function, f (x) ≈ f˜(x) = f (xk ), x ∈ [xk , xk+1 ), 87
(3.3)
88
Chapter 3. Numerical fractional integration
on each subinterval [xk , xk+1 ], k = 0, 1, . . . , n − 1, then we have −α RL Da,x f (x) x=x n Z n−1 X xk+1 = 1 (xn − t)α−1 f (t)dt Γ(α) k=0 xk n−1 Z xk+1 ≈ 1 X (xn − t)α−1 f (xk )dt. Γ(α) xk
(3.4)
k=0
Consequently, we obtain the left fractional rectangular formula −α RL Da,x f (x)
x=xn
≈
n−1 X
bn−k−1 f (xk ),
(3.5)
k=0
where the coefficients are given by bk =
hα [(k + 1)α − k α ] , 0 ≤ k ≤ n − 1. Γ(α + 1)
(3.6)
Similarly, when f (x) is approximated by the piecewise constant function f (x) ≈ f˜(x) = f (xk+1 ), x ∈ (xk , xk+1 ],
(3.7)
then we obtain the right rectangular formula −α RL Da,x f (x) x=x n
≈
n−1 X
bn−k−1 f (xk+1 ).
(3.8)
k=0
Based on the left rectangular formula (3.5) and the right rectangular formula (3.8), the weighted fractional rectangular formula gives n−1 X −α bn−k−1 [θf (xk ) + (1 − θ)f (xk+1 )] , 0 ≤ θ ≤ 1, D f (x) ≈ RL a,x x=x
n
(3.9)
k=0
or the similar formula n−1 X −α D f (x) ≈ bn−k−1 f (xk + (1 − θ)h) , 0 ≤ θ ≤ 1, RL a,x x=x
n
(3.10)
k=0
with the same accuracy O(h). Remark 3.1. Fractional rectangular formulae (3.5) and (3.8) will be recovered if the weight in the above weighted formulae is chosen as θ = 1 and θ = 0, respectively. And it was pointed out by [142] that the weighted rectangular formulae (3.9) and (3.10) are reduced to the composite trapezoidal formula and midpoint formula for the classical integral if α = 1 and θ = 21 . Remark 3.2. Apart from the above choices, other piecewise constant functions can be also utilized to construct numerical approximations to the Riemann-Liouville integral. For example, in [123], the piecewise constant approximation is chosen as f (x1 ), x0 < x < x1 , ˜ f (x) = (3.11) 1 [f (x ) + f (x )] , xk−1 < x < xk , 2 ≤ k ≤ n. k−1 k 2
3.1. Numerical methods based on polynomial interpolation
89
(II) Fractional trapezoidal formula Replacing f (x) in (3.1) with the piecewise linear function x − xk xk+1 − x f˜(x) = f (xk ) + f (xk+1 ), x ∈ [xk , xk+1 ], xk+1 − xk xk+1 − xk
(3.12)
h i −α −α ˜ D f (x) ≈ D f (x) RL a,x RL a,x x=xn x=xn Z n−1 xk+1 − t 1 X xk+1 (xn − t)α−1 f (xk ) = Γ(α) x k+1 − xk xk k=0 + t − xk f (xk+1 ) dt. xk+1 − xk
(3.13)
we have
Calculating integrals in the above equation yields the fractional trapezoidal formula −α RL Da,x f (x)
x=xn
≈
n X
ak,n f (xk ),
(3.14)
k=0
where the coefficients ak,n are given by
ak,n
(n − 1)α+1 − (n − 1 − α)nα , (n − k + 1)α+1 + (n − 1 − k)α+1 hα = −2(n − k)α+1 , Γ(α + 2) 1,
k = 0, 1 ≤ k ≤ n − 1,
(3.15)
k = n.
(III) Fractional Simpson’s formula k+1 Denote xk+ 21 = xk +x on each subinterval [xk , xk+1 ]. We approximate f (x) in (3.1) with 2 a piecewise quadratic polynomial f˜(x) =
X
lk,i (x)f (xk+i ), x ∈ [xk , xk+1 ].
(3.16)
i∈S
Here {lk,i (x)} are Lagrangian interpolation polynomials defined on the grid points {xk+j , j ∈ S}, S = {0, 21 , 1}, which are given by lk,i (x) =
Y j∈S, j6=i
x − xk+j , i ∈ S. xk+i − xk+j
(3.17)
The fractional Simpson’s formula h i −α −α ˜ RL Da,x f (x) x=x ≈ RL Da,x f (x) n x=xn n
n−1
X X ck,n f (xk ) + cˆk,n f (xk+ 12 ) = k=0
k=0
(3.18)
90
Chapter 3. Numerical fractional integration
can be obtained through direct calculation. The corresponding coefficients are given by
4 n2+α − (n − 1)2+α −(α + 2) 3n1+α + (n − 1)1+α +(α + 2)(α + 1)nα , −(α + 2) (n + 1 − k)1+α
k = 0,
hα Γ(α + 3) +6(n − k)1+α + (n − k − 1)1+α +4 (n + 1 − k)2+α −(n − 1 − k)2+α , 1 ≤ k ≤ n − 1, 2 − α, k = n,
ck,n =
(3.19)
and 4hα (α + 2) (n − k)1+α + (n − 1 − k)1+α Γ(α + 3) −2 (n − k)2+α − (n − 1 − k)2+α , 0 ≤ k ≤ n − 1.
cˆk,n =
(3.20)
(IV) Fractional Newton-Cotes formula (k) Let f (x) be approximated by a polynomial pk,r (x) of degree r on the grid points {xk = x0 , (k) (k) (k) x1 , . . . , xr−1 , xr = xk+1 }, which is given by pk,r (x) =
r X
(k)
lk,i (x)f (xi
), x ∈ [xk , xk+1 ],
(3.21)
i=0
with lk,i (x) =
(k)
r Y
x − xj (k)
j=0, j6=i
xi
(k)
− xj
, 0 ≤ i ≤ r.
(3.22)
Then we can readily derive the fractional Newton-Cotes formula n−1 r XX (k) (k) −α −α D f (x) ≈ D p (x) = Ai,n f (xi ). RL a,x RL a,x k,r x=x x=x
n
n
(3.23)
k=0 i=0
Here the coefficients are given by (k) Ai,n
1 = Γ(α)
Z
xk+1
(xn − t)α−1 lk,i (t)dt.
(3.24)
xk
Now we pause to consider truncated errors of (3.23). If f ∈ C r+1 ([a, b]), r ∈ Z+ , then the error estimate of f˜(x) = pk,r (x) on each subinterval [xk , xk+1 ] is given by r f (r+1) (ξk ) Y (k) ˜ f (x) − f (x) = x − xj , x ∈ [xk , xk+1 ], ξk ∈ [xk , xk+1 ]. (r + 1)! j=0
(3.25)
3.1. Numerical methods based on polynomial interpolation
Consequently, it holds that h i −α −α ˜ RL Da,x f (x) x=x − RL Da,x f (x) n x=x n n−1 Z 1 X xk+1 α−1 ˜ (t) − f (t) (x − t) ≤ dt f n Γ(α) xk k=0 (r+1) n−1 Z x r k+1 f Y (x) X (k) ≤ max (xn − t)α−1 t − xj dt x∈[a,xn ] (r + 1)!Γ(α) k=0 xk j=0 Z n−1 o n X xk+1 hr+1 (r+1) ≤ max (x) (xn − t)α−1 dt f (r + 1)!Γ(α) x∈[a,x ] n k=0 xk n o hr+1 = max f (r+1) (x) (xn − x0 )α , (r + 1)!Γ(α + 1) x∈[a,xn ]
91
(3.26)
indicating that the accuracy of formula (3.23) is O(hr+1 ). Remark 3.3. The error estimate for (3.23) does not coincide with that for the classical NewtonCotes formula. This is due to the nonsymmetry of the integralkernel (xn − t)α−1 , which leads Qr (k) in the integrand. to the nonsymmetry of the remainder term (xn − t)α−1 j=0 t − xj Remark 3.4. Since fractional rectangular formulae (3.5) and (3.8), the fractional trapezoidal formula (3.14), and the fractional Simpson’s formula (3.18) are special cases of the fractional Newton-Cotes formula (3.23), it is evident that their accuracies are of O(h), O(h2 ), and O(h3 ), respectively. (V) Numerical method based on cubic Hermitian interpolation When the function f (x) is approximated on each subinterval [xk , xk+1 ] by the cubic Hermitian interpolation 2 x − xk x − xk+1 ˜ f (x) = 1 − 2 f (xk ) xk − xk+1 xk − xk+1 2 x − xk+1 x − xk + 1−2 f (xk+1 ) xk+1 − xk xk+1 − xk (3.27) 2 x − xk+1 0 + (x − xk ) f (xk ) xk − xk+1 2 x − xk + (x − x ) f 0 (xk+1 ), k+1 xk+1 − xk then it holds that h i −α −α ˜ D f (x) ≈ D f (x) RL RL a,x a,x x=x n x=xn n n α X X h 0 = e f (x ) + h e ˆ f (x ) . j,n j j,n j Γ(α + 4) j=0
j=0
(3.28)
92
Chapter 3. Numerical fractional integration
Here
ej,n
− 6(n − 1)2+α (1 + 2n + α) + nα 12n3 − 6(3 + α)n2 + (1 + α)(2 + α)(3 + α) , j = 0, = 6 4(n − j)3+α + (n − j − 1)2+α (2j − 2n − 1 − α) +(1 + n − j)2+α (2j − 2n + 1 + α) , 1 ≤ j ≤ n − 1, 6(1 + α), j = n,
(3.29)
− 2(n − 1)2+α (3n + α) + n1+α 6n2 − 4(3 + α)n + (2 + α)(3 + α) , j = 0, = 2(n − j − 1)2+α (3j − 3n − α) − (n − j)2+α (8α + 24) − 2(n − j + 1)2+α (3j − 3n + α), 1 ≤ j ≤ n − 1, − 2α, j = n.
(3.30)
and
eˆj,n
In [89], the error bound for (3.28) is proved to coincide with that for the cubic interpolation, which is of O(h4 ).
3.2 Fractional linear multistep method Now we introduce the well-known fractional linear multistep method in this section, which was originally proposed by Lubich [114]. The main idea is to establish a numerical approximation preserving the following two characteristic properties of the fractional integral operator RL D−α 0,x : −α RL D0,x f (x)
= xα
and −α RL D0,x f
=
−α RL D0,t f (tx) t=1
1 α−1 t ∗ f. Γ(α)
(3.31) (3.32)
Taking the above requirements into account, Lubich considered convolution quadratures Ihα f (x) = hα
n X j=0
ωp,n−j f (jh) + hα
s X
wn,j f (jh), x = nh > 0, n ∈ Z+ ,
(3.33)
j=0
where the convolution quadrature weights ωp,n−j (p = 1, 2, . . . , j = 0, 1, 2, . . . , n) and the starting quadrature weights wn,j (n ≥ 0, j = 0, . . . , s; s fixed and s ≤ n) do not depend on h. The convolution quadratures are generally satisfying. Indeed, the factor hα keeps the homogeneity relation h i α Ihα f (x) = xα Ih/x f (tx) , (3.34) t=1
and it is easy to verify that the convolution structure is essentially preserved. It is violated only by the few correction terms of the starting quadrature which will be necessary for highorder schemes. The computation of the values Ihα f (nh) (n = 0, . . . , N − 1) needs only N evaluations of function f and only O(N log N ) additions and multiplications by using the fast Fourier transform techniques [114]. Enlightened by Dahlquist’s theorem on the classical linear multistep method [30], Lubich proved that the proposed convolution quadrature is convergent of order p if and only if it is stable
3.2. Fractional linear multistep method
93
and consistent of order p. Surprisingly, an easy way to obtain such a convolution quadrature is by using a pth order linear multistep method to the power α, which is called the fractional linear multistep method. Lubich took specific forms of the fractional Euler method, the fractional backward difference formulae, the fractional trapezoidal rule, and the generalized Newton-Gregory formulae as examples in [114], along with corresponding numerical simulations. The most widely used are the fractional backward difference formulae [188], whose implementations are as follows. Theorem 3.1. [33, 114, 158] The convolution quadrature Ihα f (x) = hα
n X
ωp,n−j f (jh) + hα
j=0
s X
wn,j f (jh), x = nh > 0, n ∈ Z+ ,
(3.35)
j=0
approximates the fractional integral RL D−α 0.x f (x) with pth order accuracy (p = 1, 2, . . . , 6). In other words, n X −α α D f (x) =h ωp,n−j f (jh) RL 0,x j=0 (3.36) s X α p +h wn,j f (jh) + O(h ), p = 1, 2, . . . , 6, α > 0. j=0
The coefficients ωp,j are respectively those of the Taylor series expansions of the corresponding generating functions [98] !α p X 1 k (α) (1 − z) (3.37) Wp (z) = k k=1
(α)
with p being the order of consistency. For p = 1, 2, 3, 4, 5, 6, functions Wp (z) are given by (α) α W1 (z) = (1 − z) , α (α) 3 1 2 W2 (z) = − 2z + z , 2 2 α 3 2 1 3 11 (α) − 3z + z − z , W3 (z) = 6 2 3 α (3.38) 25 4 3 1 4 (α) 2 W4 (z) = − 4z + 3z − z + z , 12 3 4 α 137 10 5 4 1 5 (α) 2 3 (z) = W − 5z + 5z − z + z − z , 5 60 3 4 5 α 147 15 20 15 6 1 (α) − 6z + z 2 − z 3 + z 4 − z 5 + z 6 . W6 (z) = 60 2 3 4 5 6 Technically all the coefficients ωp,j can be computed using any implementation of the fast Fourier transform. For the starting weights wn,j , if s = 0 and f (0) = 0, then the following Lubich formula holds: n X −α α ωn−j f (jh) + O(hp ), p = 1, 2, . . . , 6, α > 0. (3.39) RL D0,x f (x) = h j=0
94
Chapter 3. Numerical fractional integration
For s 6= 0, the coefficients wn,j can be constructed such that s X
wn,j j q =
j=0
n X Γ(q + 1) nq+α − ωn−j j q , q = 0, . . . , s, Γ(q + α + 1) j=0
(3.40)
exactly holds. It is easy to see that in this case it makes sense to choose s = p − 1. Remark 3.5. Apart from the choice (3.38), which corresponds to the fractional backward difference formulae, there are alternatives for the generating functions of the convolution coefficients. When we choose α 1 1+z α (3.41) W2 (z) = α 2 1−z as the generating function, we obtain the fractional trapezoidal rule, which is convergent of order 2 if α ≥ 0. Since the numerator has a zero on the unit circle, the method is not stable for α < 0 [114]. Let γi denote the coefficients of −α ∞ X ln z γi (1 − z)i = , (3.42) z−1 i=0 and choose the generating function fpα (z) = (1 − z)−α γ0 + γ1 (1 − z) + · · · + γp−1 (1 − z)p−1 , p = 1, 2, . . . . W
(3.43)
Then we obtain the generalized Newton-Gregory formula, which is convergent of order p to the α fractional integral RL D−α 0,x . In particular, direct calculation gives γ0 = 1 and γ1 = − 2 , and the corresponding generating function for the 2nd order generalized Newton-Gregory formula is given by h i fpα (z) = (1 − z)−α 1 − α (1 − z) . W (3.44) 2 In addition, the generalized Newton-Gregory formulae can be reduced to the implicit Adams method and to the (p + 1)-point backward difference quotient when α = 1 and α = −1, respectively. Recently, Ding and Li [38] proposed the 2nd order generating function α f2 (z) = 3α − 2 − 2(α − 1) z + α − 2 z 2 W , (3.45) 2α α 2α and the pth order generating function p
fp (z) = W
(α)
X λk−1,k−1 α−2 (1 − z) + (1 − z)2 + (1 − z)k 2α α
!α , p ≥ 3.
(3.46)
k=3
(α)
in which the parameters λk−1,k−1 (k = 3, 4, . . .) can be determined by the equation z
fk (e−z ) e = 1 − W zα
∞ X
(α)
λk,l z l , k = 2, 3, . . . , |z| < 1.
(3.47)
l=k
For more details on the generating functions, see [38, 114, 182, 183, 184]. Remark 3.6. [99] The starting weights {wn,j } are chosen such that the asymptotic behavior of function f (x) near the origin (x = 0) is taken into account [33]. The choice (3.40) determining
3.3. Spectral approximations
95
{wn,j } for suitably smooth function f (x) ensures that (3.36) is exact for the function f (x) = xµ , µ = 0, 1, . . . , p − 1. If the function f (x) is not suitably smooth with expression f (x) = P s σ(j) + xµ φ(x), where φ(x) is smooth and µ ≥ p − 1 ≥ σ(j), we can still construct j=0 f (jh)x the approximation as in (3.36). In such a case, we can obtain the starting weights {wn,j } by inserting f (x) = xσ(k) into (3.36) and letting (3.36) be exact, i.e., h
−α σ(k) RL D0,x x
i x=xn
= hα
n X
ωp,n−j (jh)σ(k) + hα
j=0
s X
wn,j (jh)σ(k) .
(3.48)
j=0
The above equation can be rewritten in the following equivalent form: s X j=0
wn,j j σ(k) =
n X Γ(σ(k) + 1) nσ(k)+α − ωp,n−j j σ(k) , k = 0, . . . , s. Γ(σ(k) + α + 1) j=1
(3.49)
Bounds for the starting weights are important for stability of the numerical scheme. In [33], the magnitude of the starting weights was investigated, which revealed a surprising phenomenon that the fractional backward difference formulae, which perform really well for small numbers of grid points, exhibit a poor approximate solution for a large number of grid points. Numerical examples in [33] also indicated that the truncated errors for the fractional multistep method are in general not sufficiently small when we use values of α other than α = 12 . In [150], Sanz-Serna proposed a numerical scheme for the temporal integration in a partial integro-differential equation, in which the discretized technique employed is patterned after the idea of Lubich [112, 113, 114, 115, 116]. By associating a generating function Φ(z) = φ1 z + φ2 z 2 + · · · + φn z n + · · · with a real sequence {φ0 , φ1 , . . . , φn , . . .}, which approximates the analytical solution at the grid points, the approximations to the derivative and indefinite integral at the grid points xk > 0 can be derived. Combining these approximations with the results obtained by Fourier transform of the partial integro-differential equation, the recursion for the computation of numerical approximation {φ0 , φ1 , . . . , φn , . . .} can be obtained. Error bounds for both smooth and nonsmooth initial data are also derived. Under suitable assumptions, the Lubich formula can be utilized to evaluate fractional integrals with arbitrary lower terminal a < b through suitable affine transformations and extensions. The Lubich formulae (3.39) is also valid for α < 0. In other words, it can approximate the αth order Riemann-Liouville derivative, which will be introduced later.
3.3 Spectral approximations Recalling the procedure for deriving the fractional Newton-Cotes formula, we may derive more generalized formulae in the form −α −α RL Da,x f (x) ≈ RL Da,x pN (x) =
N X
ωj,k f (xk ).
(3.50)
k=0
Here pN (x) is an approximate polynomial of f (x) of degree N ; in particular, the Gauss interpolation on the collocation points {xk }N k=0 , xk ∈ [a, b], or an orthogonal projector. This idea yields spectral approximations [174, 175]. In the following discussion, we shall present spectral approximations based on Legendre, Chebyshev, and Jacobi polynomials, separately. Since Legendre polynomials and Chebyshev polynomials are special cases of Jacobi polynomials (the Jacobi polynomials {Pju,v (x)}, x ∈ [−1, 1] can be reduced to the Legendre polynomials when u = v = 0, and to the Chebyshev polynomials if u = v = −1/2), we focus on deriving the spectral approximation based on Jacobi polynomials.
96
Chapter 3. Numerical fractional integration
We introduce here some fundamental knowledge on Jacobi polynomials as preliminaries. The Jacobi polynomials {Pju,v (x)} with u, v > −1 and x ∈ [−1, 1] are given by the three-term recurrence relation [153] u,v P (x) = 1, 0 1 1 (3.51) P1u,v (x) = (u + v + 2)x + (u − v), 2 2 u,v u,v u,v u,v (x), j ≥ 1, Pj (x) − Cju,v Pj−1 Pj+1 (x) = Au,v j x − Bj where
(2j + u + v + 1)(2j + u + v + 2) Au,v = , j 2(j + 1)(j + u + v + 1) (v 2 − u2 )(2j + u + v + 1) Bju,v = , 2(j + 1)(j + u + v + 1)(2j + u + v) (j + u)(j + v)(2j + u + v + 2) Cju,v = . (j + 1)(j + u + v + 1)(2j + u + v)
(3.52)
The Jacobi polynomials are orthogonal with respect to the weight function ω u,v (x) = (1 − x)u (1 + x)v , i.e., ( Z 1 0, m 6= n, u,v u,v u,v Pm (x)Pn (x)ω (x)dx = (3.53) u,v γn , m = n, −1 where γnu,v =
2u+v+1 Γ(n + u + 1)Γ(n + v + 1) . (2n + u + v + 1)n!Γ(n + u + v + 1)
Jacobi polynomials also have the following properties: j+u Γ(j + u + 1) Γ(j + v + 1) Pju,v (1) = = , Pju,v (−1) = (−1)j , j j!Γ(u + 1) j!Γ(v + 1) dm u,v u+m,v+m P (x) = du,v (x), j ≥ m, m ∈ Z+ , j,m Pj−m dxm j where du,v j,m =
Γ(j + m + u + v + 1) . 2m Γ(j + u + v + 2)
(3.54)
(3.55)
(3.56)
(3.57)
In addition, the following recurrence relation holds: bu,v d P u,v (x) + B b u,v d P u,v (x) + C b u,v d P u,v (x), j ≥ 1, Pju,v (x) = A j j j dx j−1 dx j dx j+1 in which
−2(j + u)(j + v) bu,v = A , j (j + u + v)(2j + u + v)(2j + u + v + 1) 2(u − v) b u,v = B , j (2j + u + v)(2j + u + v + 2) 2(j + u + v + 1) C b u,v = . j (2j + u + v + 1)(2j + u + v + 2)
bu,v is set to be zero if j = 1. Here A j
(3.58)
(3.59)
3.3. Spectral approximations
97
Now we start deriving spectral approximations based on Legendre, Chebyshev, and Jacobi polynomials. (I) Spectral approximation based on Jacobi polynomials Approximate f (x), x ∈ [−1, 1], with the function pN (x) =
N X
u,v p˜u,v j Pj (x)
(3.60)
j=0
based on the Jacobi polynomials {Pju,v } (u, v > −1), in which the coefficients p˜u,v (j = j 0, 1, . . . , N ) are given by p˜u,v = j
N 1 X
δju,v
f (xk )Pju,v (xk )ωk .
(3.61)
k=0
Here {ωk }N k=0 are the corresponding quadrature weights with respect to the Jacobi-Gauss-Lobatto (JGL) point xk [153], and ( u,v γj , j = 0, 1, . . . , N − 1, u,v δj = (3.62) u,v )γ , j = N, (2 + u+v+1 N N with γju,v being defined in (3.54). Then we have −α RL D−1,x f (x) Z x 1 −α (x − t)α−1 pN (t)dt ≈ D p (x) = RL N −1,x Γ(α) −1 Z N 1 X u,v x p ˜ (x − t)α−1 Pju,v (t)dt = j Γ(α) −1 j=0 N X u,v u,v,α (x), x ∈ [−1, 1]. p˜j Pbj =
(3.63)
j=0
In view of the recurrence formulae (3.51)–(3.52), properties (3.55) and (3.58), Pbju,v,α (x) = Rx 1 α−1 u,v Pj (t)dt can be computed by the effective recurrence formula [100] Γ(α) −1 (x − t) u,v,α (x+1)α , (x) = Γ(α+1) Pb0 α α+1 bu,v,α (x), bu,v,α (x) = u+v+2 x(x+1) − α(x+1) P + u−v 1 2 Γ(α+1) Γ(α+2) 2 P0 u,v bu,v u,v b u,v u,v bu,v P u,v (−1)) (3.64) C j j+1 bu,v,α (x) = αAj (Aj Pj−1 (−1)+Bj Pju,v(−1)+ P (x + 1)α u,v j+1 b Γ(α+1)(1+αA C ) j j b u,v u,v,α Au,v x−Bju,v −αAu,v B j j j b + P (x) j bu,v 1+αAu,v C j j u,v u,v bu,v C +αA A u,v,α − j1+αAu,vj Cbu,vj Pbj−1 (x), j ≥ 1. j
j
For f (x) defined on arbitrary interval [a, b], a simple affine transformation x ˆ= [−1, 1] yields that the fractional integral RL D−α f (x) can be evaluated by a,x α α X N b−a b−a −α −α bu,v,α (ˆ x) = p˜u,v x). RL Da,x f (x) ≈ RL D−1,ˆ j Pj x pN (ˆ 2 2 j=0
2x−a−b b−a
∈
(3.65)
98
Chapter 3. Numerical fractional integration
Enlightened by differential matrices for classical differential equations presented in [13, 153], Li and Zeng developed fractional differential matrices [100, 181] to approximate fractional integrals and derivatives. The differential matrix of a fractional integral based on the Jacobi approximation is derived as follows. Let N be a positive integer and denote x ˆk (k = 0, 1, . . . , N ) as the JGL points defined on the interval [−1, 1]. Choose x = x ˆk (k = 0, 1, . . . , N ); then we have the following formula [100]:
−α x0 ) RL D−1,ˆ x0 f (ˆ −α f (ˆ x1 ) RL D−1,ˆ x1
≈
.. .
−α xN ) RL D−1,ˆ xN f (ˆ
where RL D−α xi ) = −1,ˆ xi f (ˆ h
−α x0 ) RL D−1,ˆ x0 pN (ˆ −α p (ˆ x1 ) RL D−1,ˆ N x1
.. .
−α xN ) RL D−1,ˆ xN pN (ˆ
−α , RL D−1,x f (x) x=ˆ xi
b (u,v,−α) D −1,ˆ xN
i k,j
b (u,v,−α) e u,v = D−1,ˆxN P ,
(3.66)
T e u,v = (˜ P pu,v ˜u,v ˜u,v 0 ,p 1 ,...,p N ) , and
= Pbju,v,α (ˆ xk ), k, j = 0, 1, . . . , N.
(3.67)
e u,v can be calculated by As a matter of fact, P u,v p˜0 p˜u,v 1 .. . p˜u,v N
=
c0,0 c1,0 .. .
c0,1 c1,1 .. .
··· ··· .. .
c0,N c1,N .. .
cN,0
cN,1
···
cN,N
f (ˆ x0 ) f (ˆ x1 ) .. .
(3.68)
f (ˆ xN )
= M (f (ˆ x0 ), f (ˆ x1 ), · · · , f (ˆ xN ))T ,
where M =
c0,0 c1,0 .. .
c0,1 c1,1 .. .
··· ··· .. .
c0,N c1,N .. .
cN,0
cN,1
···
cN,N
(3.69)
and
cj,k =
Pju,v (ˆ xk )ωk , γju,v
u,v PN (ˆ xk )ωk u,v , (2+ u+v+1 )γn N
j = 0, 1, . . . , N − 1, (3.70) j = N,
with ωk being the weight with respect to the JGL point x ˆk on [−1, 1] [153, 181]. Hence, the fractional integral RL D−α ˆk can be evaluated by −1,x f (x) at x = x
−α x0 ) RL D−1,ˆ x0 f (ˆ −α f (ˆ x RL D−1,ˆ 1) x1
.. .
−α xN ) RL D−1,ˆ xN f (ˆ
b (u,v,−α) M )(f (ˆ x0 ), f (ˆ x1 ), · · · , f (ˆ xN ))T . ≈ (D −1,ˆ xN
(3.71)
3.3. Spectral approximations
99
−a−b yields For the case of arbitrary interval [a, b], an affine transformation x ˆk = 2xkb−a −α RL Da,x0 f (x0 ) −α RL Da,x1 f (x1 ) . .. −α RL Da,xN f (xN ) α b−a b (u,v,−α) M )(f (ˆ ≈ (D x0 ), f (ˆ x1 ), · · · , f (ˆ xN ))T . −1,ˆ xN 2
(3.72)
(II) Spectral approximation based on Legendre polynomials When u = v = 0, the recurrence formula (3.51) is reduced to that for Legendre polynomials {Lj (x)}, x ∈ [−1, 1], given by L0 (x) = 1, L1 (x) = x, (3.73) 2j + 1 j xLj (x) − Lj−1 (x), j ≥ 1. Lj+1 (x) = j+1 j+1 In this case, we have
−α RL D−1,x f (x) Z x N X ≈ 1 ˜lj (x − s)α−1 Lj (s)ds Γ(α) j=0 −1 N X ˜lj L bα = j (x), x ∈ [−1, 1],
(3.74)
j=0
with
b α (x) L j
=
1 Γ(α)
Rx −1
α−1
(x − s)
Lj (s)ds. It follows from the recurrence formula (3.64) that
α α b (x) = (x+1) , L 0 Γ(α+1) α α+1 b α (x) = x(x+1) − α(x+1) , L 1 Γ(α+1) Γ(α+2) n o L b α (x) = 1 b α (x) − (j − α)L b α (x) , j ≥ 1. (2j + 1)x L j+1 j j−1 j+1+α
(3.75)
The coefficients ˜lj (j = 0, 1, . . . , N ) are accordingly given by N X ˜lj = 1 f (xk )Lj (xk )ωk , γ¯j
(3.76)
k=0
2 with γ¯j = 2j+1 for 0 ≤ j ≤ N − 1, γ¯N = N2 , and {ωk }N k=0 being the corresponding quadrature weights [153]. For the case of arbitrary interval [a, b], we have
−α RL Da,x f (x)
with x ˆ=
2x−a−b b−a
∈ [−1, 1].
≈
b−a 2
α X N j=0
˜lj L bα x), j (ˆ
(3.77)
100
Chapter 3. Numerical fractional integration
(III) Spectral approximation based on Chebyshev polynomials When u = v = − 12 in (3.51), the relation − 12 ,− 12
Pj
(x) =
Γ(j + 1/2) √ Tj (x) j! π
holds, with {Tj (x)} being Chebyshev polynomials given by T0 (x) = 1, T1 (x) = x, Tj+1 (x) = 2xTj (x) − Tj−1 (x), j ≥ 1.
(3.78)
(3.79)
In this case, we obtain the Chebyshev approximation −α RL D−1,x f (x) ≈
N X
t˜j Tbjα (x), x ∈ [−1, 1].
(3.80)
j=0
Here Tbjα (x) =
1 Γ(α)
Rx −1
(x − s)α−1 Tj (s)ds can be computed by the recurrence formula
Tb0α (x) = Tb1α (x) =
Tb2α (x) =
(x+1)α Γ(α+1) , x(x+1)α Γ(α+1)
−
4x bα 2+α T1 (x)
α(x+1)α+1 Γ(α+2) ,
−
2 bα 2+α T0 (x)
+
α(x+1)α (2+α)Γ(α+1) ,
(3.81)
(j+1)(j−1−α) bα 2(j+1)x bα j+1+α Tj (x) − (j+1+α)(j−1) Tj−1 (x) 2(−1)j α(x+1)α + Γ(α+1)(j+1+α)(j−1) , j ≥ 2.
α Tbj+1 (x) =
And the coefficients t˜j are determined by N 1 X t˜j = f (xk )Tj (xk )ωk , σj
(3.82)
k=0
where {ωk }N k=0 are the corresponding quadrature weights [153] and γ − 21 ,− 12 , j = 0, 1, . . . , N − 1, j σj = 2γ − 21 ,− 12 , j = N.
(3.83)
N
For x ∈ [a, b], we readily have −α RL Da,x f (x)
≈
b−a 2
α X N
t˜j Tbjα (ˆ x),
(3.84)
j=0
by virtue of the affine transformation x ˆ = 2x−a−b ∈ [−1, 1]. b−a In [100], convergence order of spectral approximations was considered for functions in the Sobolev space H r ([a, b]). The following numerical example illustrates the proposed spectral approximations for the fractional integral.
3.4. Diffusive approximation
101
Example 3.2. [100] Let f (x) = xµ , x ∈ [0, 1]. Its fractional integral is given by −α µ RL D0,x x
=
Γ(µ + 1) xµ+α , µ > −1, α > 0. Γ(µ + 1 + α)
(3.85)
Utilizing the Jacobi approximation, Table 3.1 shows the absolute errors at JGL points when u = v = 0 and µ = 3.5. Obviously, Table 3.1 displays the spectral accuracy. Numerical results for Example 3.2 with u = v = − 21 and µ = 3.5 are shown in Table 3.2. We can see that spectral accuracy is also achieved. Table 3.1. The absolute errors for Example 3.2 with u = v = 0 and µ = 3.5. Reprinted with permission from Fract. Calc. Appl. Anal., 15 (2012), at https://www.degruyter.com/view/j/fca N
α = 0.2
α = 0.5
α = 0.8
α = 1.2
α = 1.5
α = 1.8
10
4.57E-08
3.57E-08
1.78E-08
5.18E-09
1.67E-09
6.04E-10
20
2.89E-10
1.52E-10
5.37E-11
9.88E-12
2.54E-12
1.31E-12
40
1.82E-12
6.36E-13
1.52E-13
1.74E-14
3.68E-15
2.77E-15
80
1.12E-14
2.59E-15
4.11E-16
1.67E-16
1.67E-16
1.18E-16
Table 3.2. The absolute errors for Example 3.2 with u = v = − 21 and µ = 3.5. Reprinted with permission from Fract. Calc. Appl. Anal., 15 (2012), at https://www.degruyter.com/view/j/fca N
α = 0.2
α = 0.5
α = 0.8
α = 1.2
α = 1.5
α = 1.8
10
5.49E-08
4.59E-08
2.54E-08
8.33E-09
3.09E-09
1.67E-09
20
3.08E-10
1.96E-10
7.62E-11
1.70E-11
4.97E-12
2.93E-12
40
1.81E-12
7.79E-13
2.14E-13
3.23E-14
8.09E-15
5.61E-15
80
1.06E-14
3.05E-15
5.66E-16
3.33E-16
1.80E-16
1.73E-16
3.4 Diffusive approximation There are two principle difficulties in evaluating the fractional integral Z x 1 −α (x − t)α−1 f (t)dt, 0 < α < 1. RL Da,x f (x) = Γ(α) a
(3.86)
One is that the convolution kernel xα−1 decays slowly for large x. Hence, to compute RL D−α a,x f (x) the contribution due to f (t) for t far from x cannot be neglected; i.e., systems with terms like −α RL Da,x f (x) have memory. Therefore, a naive discretization in the form
−α RL D0,x f (x) x=x j
≈
j X k=0
cj,k f (xk )
(3.87)
102
Chapter 3. Numerical fractional integration
gives rise to an algorithmic complexity which is quadratic in j, the number of subintervals, since the coefficients cj,k change as the number of subintervals varies. An additional complication comes from the fact that f may have a singularity at the end point. These two aspects result in expensive computational cost. To handle these deficiencies, concepts such as the short memory principle [139] and the logarithmic memory principle [52] emerged and have been investigated. The short memory principle simply ignores all contributions coming from points that are far away from the presently considered point xj , which leads to very cheap algorithms. However, it is shown in [52] that the corresponding approximation properties are poor. Instead, the so-called logarithmic memory principle suggests that the mesh spacing should be very fine near the current point xj and become coarser as we move away from this point. When implemented properly, this approach may lead to an arithmetic complexity only marginally higher than that in the case of integer-order operators without losing the convergence order of the method [32]. Here we introduce an alternative method dealing with the expensive computational cost for the evaluation of the fractional integral using, say, the diffusive approach. This method was proposed in [177] through a totally different path from the others. By taking advantage of the definition of the Gamma function, Z ∞ Γ(α) = e−z z α−1 dz, (3.88) 0
and the identity Γ(1 − α)Γ(α) =
π , sin(πα)
(3.89)
the fractional integral with α ∈ (0, 1) also reads −α RL D0,x f (x) Z sin(πα)Γ(1 − α) x (x − t)α−1 f (t)dt = π 0 Z Z ∞ sin(πα) x −z 1−α dz = (x − t)α−1 f (t)dt e z π z 0 0 # 1−α Z "Z ∞ dz sin(πα) x z −z f (t)dt. e = π x−t z 0 0
(3.90)
Defining the variable transformation z = (x − t)ω 2 , ω ≥ 0, we apply Fubini’s Theorem and obtain Z x Z 2 sin(πα) ∞ 1−2α −α −(x−t)ω 2 ω e f (t)dt dω (3.91) RL D0,x f (x) = π 0 0 for 0 < α < 1. Introducing the auxiliary function φ(ω, x) =
2 sin(πα) 1−2α ω π
we have −α RL D0,x f (x) =
Z
Z
x
2
e−(x−t)ω f (t)dt,
(3.92)
0
∞
φ(ω, x)dω, 0 < α < 1.
(3.93)
0
It is evident that φ(ω, x) satisfies φ(ω, 0) = 0,
(3.94)
3.4. Diffusive approximation
103
and differentiating φ(ω, x) with respect to x gives the identity ∂φ(ω, x) 2 sin(πα) 1−2α = ω f (x) − ω 2 φ(ω, x). ∂x π
(3.95)
In this case, evaluating the Riemann-Liouville integral RL D−α 0,x f (x) consists of the following two steps: solving the 1st order ordinary differential equation (ODE) (3.95) subject to the initial condition (3.94), and computing the integral (3.93) via suitable quadratures. The choices of quadrature nodes and corresponding weights have been investigated in several works; see [32, 104, 111] for more details. Apart from the idea proposed in [177], another practice of reformulating the fractional integral as a system of differential equations was adopted by [19, 155]. Instead of the utilizing properties of the Gamma function, the integral representation of the convolution kernel xα−1 , Z ∞ 1 e−zx z −α dz, (3.96) xα−1 = Γ(1 − α) 0 was utilized. In this case, Fubini’s Theorem indicates that the fractional integral can be represented as −α RL D0,x f (x) Z x Z ∞ 1 1 = e−z(x−t) z −α dzf (t)dt Γ(α) 0 Γ(1 − α) 0 Z ∞ Z x (3.97) 1 1 −z(x−t) = e f (t)dt z −α dz Γ(α) Γ(1 − α) 0 0 Z ∞ sin(πα) = g(z, x)z −α dz, π 0 where the quantity g(z, x) is defined as x
Z
e−z(x−t) f (t)dt.
g(z, x) =
(3.98)
0
In order to generate nonreflecting boundary conditions [6] and accelerate convolutions with the heat kernel [60], a well-known practice in the literature is to consider g(z, x) = e−z∆x g(z, x − ∆x) + Ψ(z, x∆x), where
Z
(3.99)
x
Ψ(z, x, ∆x) =
e−z(x−t) f (t)dt.
(3.100)
x−∆x
In other words, obtaining the value g(z, x) from the previous value g(z, x − ∆x) requires only a constant multiplication by the exponential decay term e−z∆x and the computation of Ψ(x, ∆x), which is local in time [104]. Alternatively, others works [19, 63, 128, 177] have recognized g(z, x) as the solution to the 1st order ODE dg (z, x) = −zg(z, x) + f (x), g(z, 0) = 0. dx Any ODE method can be used to obtain g(z, x), x = ∆x, 2∆x, . . . .
(3.101)
Chapter 4
Numerical Caputo differentiation
The definition of the Caputo derivative in (2.104) and (2.105) indicates that the fractional derivative of this type can be viewed as a Riemann-Liouville integral of an integer-order derivative. Naturally, basic ideas of approximating fractional integrals can be utilized to evaluate Caputo derivatives. In the present chapter, we introduce numerical approximations to Caputo derivatives.
4.1 L1, L2, and L2C methods The idea of L1, L2, and L2C methods for Caputo derivatives is analogous to that of the fractional rectangle formulae for fractional integral. Instead of approximating f (x) by piecewise constant functions, we evaluate the integer-order derivatives f (i) (x), i = 1, 2, on each subinterval, by linear combinations of values of the function at corresponding nodes.
4.1.1 L1 method As an efficient way of evaluating the Caputo derivative with 0 < α < 1, the L1 method is frequently used in discretizing time fractional differential equations since it may lead to unconditionally stable algorithms [55, 71, 72, 81, 93, 107, 144, 163, 189]. The idea of the well-known L1 method was originally introduced in [137] for numerically evaluating the Riemann-Liouville derivative with 0 < α < 1 in the equivalent form α RL Da,x f (x) =
1 (x − a)−α f (a) + Γ(1 − α) Γ(1 − α)
Z
x
(x − t)−α f 0 (t)dt.
(4.1)
a
The second term of the right-hand side of (4.1) is exactly the Caputo derivative with 0 < α < 1. That is the reason why we introduce the L1 method in detail when considering numerical approximations to the Caputo derivative. Here we present the L1 method and its modifications as follows, along with corresponding error estimates. (I) The L1 method on uniform grids b−a After setting uniform grids {xk }N k=0 with xk = a + kh, h = N , utilizing the difference quotient f (xk+1h)−f (xk ) to approximate the 1st order derivative f 0 (x) on each interval [xk , xk+1 ] 105
106
Chapter 4. Numerical Caputo differentiation
yields [81, 105, 107, 154, 163, 170] α C Da,x f (x) x=x j Z j−1 X xk+1 1 (xj − t)−α f 0 (t)dt = Γ(1 − α) k=0 xk j−1 Z xk+1 X f (xk+1 ) − f (xk ) 1 ≈ (xj − t)−α dt Γ(1 − α) h xk k=0 Z j−1 X f (xk+1 ) − f (xk ) xk+1 (xj − t)−α dt = hΓ(1 − α) x k k=0 j−1 X = bj−k−1 [f (xk+1 ) − f (xk )] , 0 < α < 1,
(4.2)
k=0
where j = 1, 2, . . . , N , and h−α (k + 1)1−α − k 1−α , k = 0, 1, 2, . . . , j − 1. Γ(2 − α)
bk =
(4.3)
We call the evaluation j−1 X α D f (x) ≈ bj−k−1 [f (xk+1 ) − f (xk )] , 0 < α < 1, C a,x x=x
j
(4.4)
k=0
the L1 method.2 Remark 4.1. The L1 method was proposed in [137] for the numerical evaluation of the RiemannLiouville derivative with order α ∈ (0, 1). The L1 method for the Caputo derivative was considered in [107] and [162]. Here we show the error estimate of the L1 method for the Caputo derivative. Theorem 4.1. Let 0 < α < 1 and f (x) ∈ C 2 ([a, b]). Then it holds that j−1 X α bj−k−1 [f (xk+1 ) − f (xk )] − C Da,x f (x) x=x ≤ Ch2−α , j
(4.5)
k=0
where C is a positive constant given by 1−α 22−α 1 −α + − (2 + 1) max |f 00 (x)|. C= x0 ≤x≤xj Γ(2 − α) 12 2−α
(4.6)
Proof. Denote Z
xj
A= x0 2 From
j−1
X f (xk+1 ) − f (xk ) f 0 (t) dt − α (xj − t) h k=0
Z
xk+1
xk
dt . (xj − t)α
[137], here “L” likely refers to the (left-sided) fractional derivative, and “1" means α ∈ (0, 1).
(4.7)
4.1. L1, L2, and L2C methods
107
Then it immediately follows that j−1 X α |A| bj−k−1 [f (xk+1 ) − f (xk )] − C Da,x f (x) x=x = . j Γ(1 − α)
(4.8)
k=0
Using the Taylor expansion with integral remainder, we have f (xk+1 ) − f (xk ) 0 f (t) − h Z t Z xk+1 1 00 00 f (s)(s − xk )ds − f (s)(xk+1 − s)ds , t ∈ [xk , xk+1 ], = h xk t
(4.9)
which yields j−1 Z xk+1 X f (xk+1 ) − f (xk ) dt 0 A = f (t) − h (x − t)α j k=0 xk Z t j−1 Z 1 X xk+1 f 00 (s)(s − xk )ds =h x x k k k=0 Z xk+1 dt 00 . − f (s)(xk+1 − s)ds α (x j − t) t
(4.10)
Exchanging the order of integration gives Z t j−1 Z 1 X xk+1 A= f 00 (s)(s − xk )ds h x x k k k=0 Z xk+1 dt 00 − f (s)(x − s)ds k+1 (x − t)α j t Z xk+1 j−1 Z xk+1 1X 00 = f (s)(s − x ) (xj − t)−α dtds k h x s k k=0 Z xk+1 Z s 00 −α − f (s)(xk+1 − s) (xj − t) dtds xk
xk
j−1 Z xk+1 X
1 s − xk = f 00 (s) [(xj − s)1−α − (xj − xk+1 )1−α ]ds 1−α h x k k=0 Z xk+1 xk+1 − s − f 00 (s) [(xj − xk )1−α − (xj − s)1−α ]ds h xk j−1 Z 1 X xk+1 s − xk = (xj − s)1−α − (xj − xk+1 )1−α 1−α h x k k=0 xk+1 − s 1−α (xj − xk ) f 00 (s)ds. + h
(4.11)
108
Chapter 4. Numerical Caputo differentiation
In the following, we show that, when 0 < α < 1, Z xk+1 s − xk 1−α (xj − xk+1 )1−α (x − s) − j x h k xk+1 − s + (xj − xk )1−α ds ≥ 0 h
(4.12)
for k = 0, 1, . . . , j − 1, and j−1 Z X k=0
xk+1
xk
s − xk (xj − xk+1 )1−α (xj − s) − h xk+1 − s 1−α + (xj − xk ) ds < +∞. h 1−α
(4.13)
Denote g(s) = (xj − s)1−α . Then it holds for any s ∈ (xk , xk+1 ) that s − xk xk+1 − s g(s) − g(xk+1 ) + g(xk ) h h 1 = g 00 (ξk )(s − xk )(s − xk+1 ) 2 = 1 (1 − α)(−α)(xj − ξk )−α−1 (s − xk )(s − xk+1 ) ≥ 0 2
(4.14)
with ξk ∈ (xk , xk+1 ). As a result, inequality (4.12) holds. For inequality (4.13), one has j−3 Z X xk+1 s − xk xk+1 − s g(x ) + g(x ) ds g(s) − k+1 k h h xk k=0 j−3 Z xk+1 X 1 = α(1 − α)(xj − ξk )−α−1 (s − xk )(xk+1 − s)ds 2 x k k=0 Z xk+1 j−3 X 1 −α−1 ≤ α(1 − α) (xj − xk+1 ) (s − xk )(xk+1 − s)ds 2 xk k=0 j−3 X h3 = α(1 − α) (xj − xk+1 )−α−1 12 k=0 j−3 Z xk+2 X h2 ≤ α(1 − α) (xj − s)−α−1 ds 12 x k+1 k=0 Z xj−1 2 h = α(1 − α) (xj − s)−α−1 ds 12 x1 h2 = (1 − α) (xj − xj−1 )−α − (xj − x1 )−α 12 ≤ 1 − α h2−α 12
(4.15)
4.1. L1, L2, and L2C methods
109
and j−1 Z xk+1 X xk+1 − s s − xk g(x ) + g(x ) ds g(s) − k+1 k h h xk k=j−2 Z xj 1 1 = g(s)ds − g(xj−2 ) + g(xj−1 ) + g(xj ) h 2 2 xj−2 Z xj 1 g(s)ds − g(xj−2 ) + g(xj−1 ) h = 2 xj−2 Z xj 1 1−α 1−α 1−α (x − x ) + (x − x ) h = (x − s) ds − j j−2 j j−1 j 2 xj−2 2−α 2 −α − (2 + 1) h2−α . = 2−α
(4.16)
The above two equations yield that inequality (4.13) holds. Combining the above analyses, one has
j−1 Z xk+1 X s − xk 1−α 0≤ (xj − s) − (xj − xk+1 )1−α h x k k=0 xk+1 − s 1−α (x − x ) ds + j k h 1−α 22−α ≤ + − (2−α + 1) h2−α . 12 2−α
(4.17)
Inserting the above estimate into (4.11) gives
|A| ≤
1 1−α 22−α + − (2−α + 1) max |f 00 (x)|h2−α . x0 ≤x≤xj 1−α 12 2−α
(4.18)
As a result, Theorem 4.1 is proved.
(II) The L1 method on nonuniform grids The existence of a weakly singular kernel (x − t)α−1 (0 < α < 1) in the fractional derivative and fractional integral makes it more difficult to get a higher-order scheme for fractional differential equations. Particularly when the solution is not smooth enough, numerical methods on uniform meshes seem to have a poor convergence rate. As a result, numerical methods on nonuniform meshes are worth considering [96]. Here we modify the classical L1 method into the more general case with nonuniform grids. Let {˜ xi } be any division of [a, b] with a = x ˜0 < x ˜1 < · · · < x ˜N −1 < x ˜N = b. Then the
110
Chapter 4. Numerical Caputo differentiation
classical L1 method is generalized into α C Da,x f (x) x=˜ xj Z x ˜j 1 (˜ xj − t)−α f 0 (t)dt = Γ(1 − α) x˜0 j−1 Z x ˜k+1 X 1 = (˜ xj − t)−α f 0 (t)dt Γ(1 − α) ˜k k=0 x j−1 Z x ˜k+1 X 1 f (˜ xk+1 ) − f (˜ xk ) ≈ (˜ xj − t)−α dt ˜k Γ(1 − α) h x ˜k k=0 j−1 X = bjk+1 [f (˜ xk+1 ) − f (˜ xk )] , j = 1, 2, . . . , N,
(4.19)
k=0
where bjk+1 =
1 (˜ xj − x ˜k )1−α − (˜ xj − x ˜k+1 )1−α ˜ Γ(2 − α)hk
(4.20)
˜k = x with h ˜k+1 − x ˜k . More precisely, we have the following result. Theorem 4.2. [186] For 0 < α < 1 and f (x) ∈ C 2 ([a, b]), it holds that Z x˜j j−1 X f 0 (t) dt = bjk+1 [f (˜ xk+1 ) − f (˜ xk )] + Rj , 1 ≤ j ≤ N, (˜ xj − t)α a
(4.21)
k=0
with j R ≤ ˜ max = Here h
˜2 ˜2 h h j−1 + max 2(1 − α) 8
! ˜ −α max |f 00 (x)| . h j−1 a≤x≤˜ xj
(4.22)
˜ j }. max {h
0≤j≤N −1
Proof. We write the integral as Z x˜j Z x˜j−1 Z x˜j f 0 (t) f 0 (t) f 0 (t) dt = dt + dt, α α (˜ xj − t) (˜ xj − t) xj − t)α a x ˜0 x ˜j−1 (˜ where x ˜0 = a. By virtue of integration by parts, we have Z x˜ j−1 f 0 (t) dt (˜ xj − t)α x ˜0 Z x˜j−1 x˜j−1 −α = (˜ x − t) f (t) − α (˜ xj − t)−α−1 f (t)dt j x ˜0 x ˜ 0 j−1 Z x ˜k X f (t)dt −α ˜ −α f (˜ = h x ) − (˜ x − x ˜ ) f (˜ x ) − α j−1 j 0 0 j−1 xj − t)α+1 ˜k−1 (˜ k=1 x ˜ −α f (˜ xj − x ˜0 )−α f (˜ x0 ) =h j−1 xj−1 ) − (˜ j−1 Z x ˜k X (˜ xk − t)f (˜ xk−1 ) + (t − x ˜k−1 )f (˜ xk ) − α dt − R1j , α+1 ˜ (˜ x − t) h x ˜ j k−1 k−1 k=1
(4.23)
(4.24)
4.1. L1, L2, and L2C methods
111
in which the linear interpolation of f (t) is utilized, and R1j
=α
j−1 Z X k=1
x ˜k
x ˜k−1
1 00 f (ξk )(t − x ˜k )(t − x ˜k−1 )(˜ xj − t)−α−1 dt, 2
(4.25)
where ξk ∈ (˜ xk−1 , x ˜k ). Noticing that x ˜k
Z α
˜ k−1 (˜ (˜ xk − t)(˜ xj − t)−α−1 dt = −h xj − x ˜k−1 )−α +
Z
x ˜k−1
x ˜k
(˜ xj − t)−α dt
(4.26)
(˜ xj − t)−α dt,
(4.27)
x ˜k−1
and Z
x ˜k
−α−1
(t − x ˜k−1 )(˜ xj − t)
α
˜ k−1 (˜ dt = h xj − x ˜k )−α −
x ˜k−1
Z
x ˜k
x ˜k−1
we can rewrite (4.24) in the following form: Z x˜j−1 f 0 (t)(˜ xj − t)−α dt x ˜0 −α −α =hj−1 f (˜ xj−1 ) − (˜ xj − x ˜0 ) f (˜ x0 ) j−1 j−1 X X −α + f (˜ x )(˜ x − x ˜ ) − f (˜ xk )(˜ xj − x ˜k )−α k−1 j k−1 k=1 k=1 Z j−1 X f (˜ xk ) − f (˜ xk−1 ) x˜k dt − R1j . + ˜ xj − t)α hk−1 x ˜k−1 (˜
(4.28)
k=1
It is easy to check that the sum of the first four terms of the right-hand side is equal to zero. Thus it follows that Z
x ˜j−1
x ˜0
and
f 0 (t)(˜ xj − t)−α dt =
Z j−1 X f (˜ xk ) − f (˜ xk−1 ) k=1
˜ k−1 h
x ˜k
(˜ xj − t)−α dt − R1j
Z x˜k j−1 X α j 00 2 ˜ max |f (x)| hk−1 (˜ xj − t)−α−1 dt R1 ≤ 8 x˜0 ≤x≤˜xj−1 x ˜ k−1 k=1 j−1 Z x ˜k X ˜2 αh max ≤ max |f 00 (x)| (˜ xj − t)−α−1 dt 8 x˜0 ≤x≤˜xj−1 x ˜ k−1 k=1 ˜2 h max 00 −α ˜ −α − (˜ = max |f (x)| h x − x ˜ ) j 0 j−1 8 x˜0 ≤x≤˜xj−1 1 ˜2 h ˜ −α ≤ max |f 00 (x)|h max j−1 . 8 x˜0 ≤x≤˜xj−1 The Taylor expansion yields that ˜ j−1 f (˜ xj ) − f (˜ xj−1 ) h 0 f (t) − ≤ ˜ 2 hj−1
max
x ˜j−1 ≤x≤˜ xj
(4.29)
x ˜k−1
|f 00 (x)|, x ˜j−1 < t < x ˜j .
(4.30)
(4.31)
112
Chapter 4. Numerical Caputo differentiation
Then the error in the interval [˜ xj−1 , x ˜j ] satisfies Z x˜j j f 0 (t)(˜ xj − t)−α dt = R 2 x˜j−1 Z f (˜ xj ) − f (˜ xj−1 ) x˜j −α − (˜ x − t) dt j ˜ j−1 h x ˜j−1 ˜ 2−α h j−1 ≤ max |f 00 (x)|. 2(1 − α) x˜j−1 ≤x≤˜xj
(4.32)
Combining (4.29), (4.30), and the above inequality, we obtain the claimed estimate. ˜ min = Remark 4.2. [99, 186] Denote h
min
˜ j }. For the quasi-uniform grids satisfying {h
0≤j≤N −1
˜ max h ≤ C0 ˜ min h
(4.33)
˜ max ≤ C0 (b − a)N −1 when N is suitably big. Then with C0 being a finite constant, one has h j the truncated error R in the above theorem is of O(N α−2 ) for any quasi-uniform grids. More precisely, the inequality j−1 X α j bk+1 [f (˜ xk+1 ) − f (˜ xk )] − C Da,x f (x) x=x j k=0 (4.34) 2−α 00 ˜ ≤C(hmax ) max |f (x)| a≤x≤b
˜
is valid for 0 < α < 1, provided that f (x) ∈ C 2 ([a, b]). Here C only depends on α and hh˜max . min ˜ max = h ˜ min , the grids reduce to the uniform one. In this case, the above theorem revisits When h the estimate of L1 approximation (4.5). A sequence of grids is not quasi-uniform if consider the nonuniform mesh with
˜ max h ˜ min h
→ +∞ as N → +∞. In particular, we
˜ k = (N − k)µ, 0 ≤ k ≤ N − 1, h
(4.35)
˜ N −1 where µ = N2(b−a) (N +1) . In this case, the stepsize {hk }k=0 is a monotonically decreasing sequence ˜ 0 = O(N −1 ), h ˜ N −1 = O(N −2 ). Theorem 4.2 implies that the truncated error for the and h nonuniform mesh (4.35) is Rj = O(N 2α−2 ). It seems that the L1 method on nonuniform meshes may lose accuracy. In fact, the estimate for R1j in the proof of Theorem 4.2 can be improved when the nonuniform mesh (4.35) is considered. Theorem 4.3. [186] For 0 < α < 1 and f (x) ∈ C 2 ([a, b]), it holds for the nonuniform mesh (4.35) that Z Z j x˜j X f (˜ xk ) − f (˜ xk−1 ) x˜k 0 −α −α f (t)(˜ x − t) dt − (˜ x − t) dt j j ˜ k−1 h x ˜0 x ˜k−1 k=1 (4.36) 1−α 2 ≤ 1 + α + max |f 00 (x)| (b − a)2−α (N + 1)α−2 1 − α x˜0 ≤x≤˜xj
4.1. L1, L2, and L2C methods
113
and
Z x˜N f 0 (t)(˜ xN − t)−α dt x˜0 Z N X f (˜ xk ) − f (˜ xk−1 )
x ˜k
−α
(˜ xN − t) − ˜ k−1 h x ˜k−1 k=1 1 + α 1−α ≤ 2 max |f 00 (x)| (b − a)2−α N −2 , a≤x≤b 1−α
dt
(4.37)
where 1 ≤ j ≤ N − 1. Proof. From the analysis in Theorem 4.2, the truncated error of the numerical integral in [˜ x0 , x ˜j−1 ] satisfies Z x˜k j−1 X Rj ≤ α max |f 00 (x)| 2 ˜ hk−1 (˜ xj − t)−α−1 dt 8 x˜0 ≤x≤˜xj−1 1 x ˜ k−1 k=1
(4.38)
j−1
X α ˜ 3 (˜ h ˜k )−α−1 . max |f 00 (x)| ≤ k−1 xj − x 8 x˜0 ≤x≤˜xj−1 k=1
It follows from the definition of the nonuniform mesh (4.35) with µ =
2(b−a) N (N +1)
that
j−1 X ˜ ˜ ˜ l = (hj−1 + hk )(j − k) = µ (j − k)(2N − j − k + 1), x ˜j − x ˜k = h 2 2 l=k ˜ k−1 = (N − k + 1)µ. h
(4.39)
Thus, we have j−1 X ˜ 3 (˜ h ˜k )−α−1 k−1 xj − x k=1 j−1 X 1+α 2−α =2 µ (j − k)−α−1 (2N − j − k + 1)−α−1 (N − k + 1)3 k=1 j−1 X 1+α 2−α ≤2 µ (j − k)−α−1 (N − k + 1)2−α k=1 j−1 X ≤21+α N 2−α µ2−α (j − k)−α−1 k=1 j−1 X 2−α α−2 (j − k)−α−1 . =8(b − a) (N + 1) k=1
(4.40)
114
Chapter 4. Numerical Caputo differentiation
In addition, it holds that j−1 j−1 j−1 X X X −α−1 −α−1 (j − k) = k = 1 + k −α−1 k=1 k=1 k=2 j−1 Z X ≤1 + k=2
and then
k
x−α−1 dx = 1 +
k−1
j R1 ≤ (1 + α)
Z 1
max
x ˜0 ≤x≤˜ xj−1
j−1
(4.41)
1 x−α−1 dx ≤ 1 + , α
|f 00 (x)| (b − a)2−α (N + 1)α−2 .
(4.42)
˜ 2−α ). Since h ˜k ≤ h ˜0 On the other hand, the truncated error in [˜ xj−1 , x ˜j ] is proved to be O(h j−1 for all k ≥ 0, it follows that ˜ 2−α h j j−1 ≤ max |f 00 (x)| R 2 2(1 − α) x˜j−1 ≤x≤˜xj ˜ 2−α h 0 (4.43) ≤ max |f 00 (x)| 2(1 − α) x˜j−1 ≤x≤˜xj 21−α = max |f 00 (x)| (b − a)2−α (N + 1)α−2 . 1 − α x˜j−1 ≤x≤˜xj Combining the above two equalities, the first inequality in Theorem 4.3 is proved. For the proof of the second estimate, the error of the numerical integral in [˜ x0 , x ˜N −1 ] satisfies N −1 X N α ˜ 3 (˜ R1 ≤ max |f 00 (x)| h ˜k )−α−1 . k−1 xN − x 8 x˜0 ≤x≤˜xN −1
(4.44)
k=1
˜ k = (N − k)µ and Noting that h x ˜N − x ˜k =
N −1 X l=k
we obtain
˜ l = 1 (h ˜ N −1 + h ˜ k )(N − k) = µ (N − k)(N − k + 1), h 2 2
N −1 X ˜ 3 (˜ h ˜k )−α−1 k−1 xN − x k=1 N −1 X 1+α 2−α =2 µ (N − k)−1−α (N − k + 1)2−α k=1
2−α N −1 X N −k+1 1+α 2−α 1−2α =2 µ (N − k) N −k k=1 N −1 X ≤8µ2−α (N − k)1−2α .
(4.45)
(4.46)
k=1
In addition, N −1 N −1 X X 1−α 1−2α 1−α µ (N − k) = µ k 1−2α k=1
k=1
(2b − 2a)1−α µ1−α 2−2α ≤ N ≤ . 2 − 2α 2 − 2α
(4.47)
4.1. L1, L2, and L2C methods
115
Therefore we obtain N −1 X k=1
1−α
(2b − 2a) ˜ 3 (˜ h ˜k )−α−1 ≤ 8µ k−1 xN − x 2 − 2α
and then
≤
24−α (b − a)2−α N −2 1−α
N 21−α α R1 ≤ max |f 00 (x)| (b − a)2−α N −2 . 1 − α x˜0 ≤x≤˜xN −1
˜ N −1 = µ implies that In addition, h N (2b − 2a)2−α [N (N + 1)]α−2 max |f 00 (x)| R2 ≤ x ˜N −1 ≤x≤˜ xN 2(1 − α) 21−α ≤ max |f 00 (x)| (b − a)2−α N 2α−4 . 1 − α x˜N −1 ≤x≤˜xN
(4.48)
(4.49)
(4.50)
Combining the above two inequalities, we obtain the required estimation. The proof is thus completed. In [96, 186], numerical simulations show that the L1 method on nonuniform grids performs better than on uniform grids. The computation efficiency of the L1 method on nonuniform grids is likely expensive. In order to overcome this drawback, the so-called sum-of-exponentials method can be applied [70]. (III) The modified L1 method x +x In the special case of nonuniform grids with x ˜0 = x0 , x ˜j = xj− 21 = j−12 j , j = 1, 2, . . ., equation (4.19) is reduced to α C Da,x f (x) x=x j+ 1 2 j (4.51) X 2−α 1 1 =b f (x ) − (b − b )f (x ) − B f (x ) + O(h ), j−k j−k+1 j 0 j+ 2 k− 2 0 k=1
with h =
b−a N .
For k = 0, 1, . . . , j, the coefficients are given by h−α (k + 1)1−α − k 1−α , b = k Γ(2 − α) " # 1−α −α 2h 1 1−α j+ −j . Bj = Γ(2 − α) 2
Replacing f (xk− 21 ) = f ( xk−12+xk ) with
f (xk )+f (xk−1 ) 2
(4.52)
yields the modified L1 method
α C Da,x f (x) x=x j+ 1 2 j 1X =− (bj−k − bj−k+1 ) [f (xk−1 ) + f (xk )] 2 k=1 + b0 [f (xj+1 ) + f (xj )] − Bj f (x0 ) + O(h2−α ). 2
(4.53)
116
Chapter 4. Numerical Caputo differentiation
The modified L1 method (4.53) is useful to obtain the Crank-Nicolson method for the timefractional subdiffusion equation, which can be regarded as a natural extension of the classical Crank-Nicolson method [22].
4.1.2 L2 and L2C methods When 1 < α < 2 and a = 0, we have the Caputo derivative defined as α C D0,x f (x) x=x j Z j−1 X xk+1 1 = (xj − t)1−α f 00 (t)dt Γ(2 − α) x k k=0 j−1 Z xk+1 X 1 = t1−α f 00 (xj − t)dt Γ(2 − α) xk
(4.54)
k=0
in the setting of uniform mesh. Approximating the 2nd order derivative f 00 (xj − t) with two specific finite difference formulae, we obtain the following L2 method and L2C method. (I) The L2 method f (xj −xk+1 )−2f (xj −xk )+f (xj −xk−1 ) Utilizing the central difference scheme to approximate h2 00 f (xj − t) on each interval [xk , xk+1 ], we have α C D0,x f (x) x=x j Z j−1 xk+1 X 1 = t1−α f 00 (xj − t)dt Γ(2 − α) x k k=0 j−1 X f (xj − xk+1 ) − 2f (xj − xk ) + f (xj − xk−1 ) Z xk+1 t1−α dt. ≈ Γ(2 − α)h2 xk
(4.55)
k=0
Precisely, the L2 method [137] is given by α C D0,x f (x) x=x j
≈
j X
Wj,k f (xj−k ),
(4.56)
k=−1
where
Wj,k
1, 22−α − 3, (k + 2)2−α − 3(k + 1)2−α h−α +3k 2−α − (k − 1)2−α , = Γ(3 − α) −2j 2−α + 3(j − 1)2−α −(j − 2)2−α , 2−α j − (j − 1)2−α ,
k = −1, k = 0, 1 ≤ k ≤ j − 2, k = j − 1, k = j.
(4.57)
4.1. L1, L2, and L2C methods
117
In the limit α = 2, the nonzero weights are Wj,−1 =
1 , h2
Wj,0 = −
2 , h2
Wj,1 =
1 , h2
which corresponds to the 2nd order accuracy discretization for the 2nd order derivative
(4.58)
d2 f dx2
x=xj
.
In the limit α = 1, the L2 scheme for the 1st order derivative is in the form of a backward two-point derivative, which indicates that the numerical approximation is of 1st order accuracy for α = 1. Consequently, we observe that the accuracy of the L2 scheme depends on the value of α. Here, “L” has the same meaning as that of L1 method, and “2” means α ∈ (1, 2). (II) The L2C method In view of the above analysis for the L2 scheme, we are aware that the L2 method may not be accurate enough for α near 1. As R xa variant of the L2 method, the L2C method was developed in [119] by evaluating the integral xkk+1 t1−α f 00 (xj − t)dt in a more symmetric form. For t ∈ [xk−1 , xk ], if we adopt the approximation 00 f (xj − t) ≈ 1 [f (x − x j k+2 ) − f (xj − xk+1 ) − f (xj − xk ) + f (xj − xk−1 )] , 2h2
(4.59)
then the L2C method is obtained in the form j+1 X α cj,k f (xj−k ). D f (x) ≈ W C 0,x x=x
j
(4.60)
k=−1
Here
−α
cj,k = W
k = −1,
1, 2−α
2
2−α
3
− 2,
k = 0, 2−α
−2×2
(k + 2)
2−α
,
− 2(k + 1)
k=1 2−α
+2(k − 1)2−α − (k − 2)2−α , 2 ≤ k ≤ j − 2,
h 2Γ(3 − α) −j 2−α − (j − 3)2−α +2(j − 2)2−α , −j 2−α + 2(j − 1)2−α −(j − 2)2−α , 2−α j − (j − 1)2−α ,
(4.61)
k = j − 1, k = j, k = j + 1.
Note that in this new scheme the value of f (x−1 ) is needed. Since we always use f 0 (x) = 0 at the lower terminal, we can set f (x−1 ) = f (x1 ). Remark 4.3. It is interesting to compare the L2 and L2C methods in the limit when α is an integer. When α = 1, the L2 and L2C methods reduce to the backward difference method and the central difference method for the 1st order derivative, respectively. If α = 2, the L2 method
118
Chapter 4. Numerical Caputo differentiation
reduces to the central difference method for the 2nd order derivative and the L2C method reduces to f (xk+1 ) − f (xk ) − f (xk−1 ) + f (xk−2 ) d2 f (xk ) ≈ dx2 2h2
(4.62)
with 1st order accuracy. Numerical experiments indicate that the L2 method is more accurate than the L2C method for 1 < α < 1.5, while the opposite result happens when 1.5 < α < 2. And these two methods behave in almost the same way near α = 1.5 [119]. Remark 4.4. Note that both the L2 and L2C methods presented here are evaluations for the Caputo derivative C Dα a,x f (x) with a = 0. For the case with arbitrary lower terminal a 6= 0, the L2 and L2C methods are still applicable after simple affine transformations. Remark 4.5. Based on the above discussions, we consider another numerical approximation with accuracy of (3 − α)th order, the idea of which was inspired by the modified L1 method. Note that the Caputo derivative C Dα a,x f (x) with 1 < α < 2 can be rewritten as α C Da,x f (x)
0 = C Dα−1 a,x f (x), 1 < α < 2.
(4.63)
Consequently, the modified L1 method given by (4.53) yields that α C Da,x f (x) x=x j+ 1 2 j X =b0 f 0 (xj+ 12 ) − (bj−k − bj−k+1 )f 0 (xj− 12 ) k=1 0 − Bj f (x0 ) + O(h3−α ),
(4.64)
where bj and Bj in this case are given by h1−α (j + 1)2−α − j 2−α , b = j Γ(3 − α) " # 2−α 1−α 2h 1 2−α j+ −j . Bj = Γ(3 − α) 2 Noting that f 0 (xk− 21 ) = following approximation:
f (xk )−f (xk−1 ) h
(4.65)
+ O(h2 ) , δx f (xk− 21 ) + O(h2 ), we can derive the
α C Da,x f (x) x=x j+ 1 2 j X =b0 δx f (xj+ 12 ) − (bj−k − bj−k+1 )δx f (xk− 21 ) k=1 − Bj f 0 (x0 ) + O(h3−α ).
(4.66)
4.2. High-order methods based on polynomial interpolation
119
4.2 High-order methods based on polynomial interpolation In view of the L1, L2, and L2C methods, it seems that some numerical approximations with higher-order accuracy can be established by utilizing higher-order interpolation if f (x) is suitably smooth. In the following, we focus on the case with α ∈ (0, 1).
4.2.1 The (3 − α)th order approximations By approximating f (x) or its derivatives using high-order interpolation, a series of high-order numerical evaluations to the Caputo derivative can be obtained, with the accuracy depending on the parameter α ∈ (0, 1). We first introduce a (3 − α)th order approximation proposed in [94]. Recall the Taylor expansions (xk+1 − x)2 00 0 f (x ) =f (x) + (x − x)f (x) + f (x) k+1 k+1 2 Z 1 xk+1 + (xk+1 − s)2 f (3) (s)ds, 2! x (xk − x)2 00 0 f (x) f (xk ) =f (x) + (xk − x)f (x) + 2 Z 1 xk + (xk − s)2 f (3) (s)ds, 2! x (xk−1 − x)2 00 f (xk−1 ) =f (x) + (xk−1 − x)f 0 (x) + f (x) 2 Z 1 xk−1 + (xk−1 − s)2 f (3) (s)ds. 2! x
(4.67)
For 0 ≤ k < j − 1 and x ∈ (xk , xk+1 ), we have f (xk+1 ) − f (xk−1 ) f 0 (x) = 2h f (x k+1 ) − 2f (xk ) + f (xk−1 ) (x − xk ) + h2 Z xk+1 1 − (xk+1 − s)2 f (3) (s)ds 2! · 2h x Z xk−1 2 (3) − (xk−1 − s) f (s)ds x Z xk+1 (x − xk ) − (xk+1 − s)2 f (3) (s)ds 2 2h x Z xk −2 (xk − s)2 f (3) (s)ds x Z xk−1 2 (3) + (xk−1 − s) f (s)ds . x
(4.68)
120
Chapter 4. Numerical Caputo differentiation
Then the Caputo derivative with 0 < α < 1 can be written as α C Da,x f (x) x=x j Z j−1 X xk+1 1 (xj − t)−α f 0 (t)dt = Γ(1 − α) k=0 xk j−1 Z xk+1 X 1 −α f (xk+1 ) − f (xk−1 ) (xj − t) = Γ(1 − α) 2h k=0
xk
(4.69)
f (xk+1 ) − 2f (xk ) + f (xk−1 ) + (t − xk ) dt + Rj h2 j−1 h−α X {ω1,j−k [f (xk+1 ) − f (xk−1 )] = Γ(3 − α) k=0 +ω2,j−k [f (xk+1 ) − 2f (xk ) + f (xk−1 )]} + Rj , where ω1,j−k =
2−α (j − k)1−α − (j − k − 1)1−α , 2
ω2,j−k = (j − k)2−α − (j − k − 1)2−α − (2 − α)(j − k − 1)1−α ,
(4.70)
(4.71)
with k = 0, 1, . . . , j − 1, j = 1, 2, . . . , N . Here Rj denotes the truncated error, and the value of f (x−1 ) is used. The computation of f (x−1 ) is to be considered in the later discussion. For the above coefficients, we have the following properties, which are necessary for analyzing numerical schemes of fractional differential equations based on the above derived approximation. Theorem 4.4. [94] Let 0 < α < 1. The coefficients ω1,j−k and ω2,j−k in (4.69) satisfy the following properties: (I) ω1,1 =
2−α 2 ,
ω2,1 = 1;
(II) 0 < ω1,j−k+1 < ω1,j−k ≤
2−α 2
< 1,
0 < ω2,j−k+1 < ω2,j−k ≤ 1; (III) ω1,j−k+1 − ω1,j−k > ω1,j−k − ω1,j−k−1 , j − k ≥ 2, ω2,j−k+1 − ω2,j−k > ω2,j−k − ω2,j−k−1 , j − k ≥ 2; ( (IV) ω1,2 + ω1,1 + ω2,2, − ω2,1
> 0,
α ∈ (0, α0 ),
≤ 0, α ∈ [α0 , 1), point of (6 − α)2−α + α − 4 = 0 with α ∈ (0, 1),
where α0 ≈ 0.68 is the unique zero
ω1,j−k + ω1,j−k−1 + ω2,j−k − ω2,j−k−1 > 0, j − k ≥ 3;
4.2. High-order methods based on polynomial interpolation
121
(V) −ω1,2 + 2ω2,1 − ω2,2 > 0, ( ω1,3 − ω1,1 + ω2,3 − 2ω2,2 + ω2,1
> 0,
α ∈ (0, α1 ),
≤ 0,
α ∈ [α1 , 1),
where α1 ≈ 0.37 is the unique
1−α zero point of −23−α +32−α −22−α + 2−α − 3(2−α) 21−α − 3α 2 3 2 2 +6 = 0 for α ∈ (0, 1),
ω1,j−k+1 − ω1,j−k−1 + ω2,j−k+1 − 2ω2,j−k + ω2,j−k−1 < 0, j − k ≥ 3.
For the truncated error in (4.69), we present the following result.
Theorem 4.5. [94] Let 0 < α < 1 and f (x) ∈ C 3 ([a, b]). For the truncated error Rj of approximation (4.69), it holds that
j R ≤ ch3−α , 1 ≤ j ≤ N,
(4.72)
with c being a positive constant and f (x−1 ) in (4.69) being used.
Proof. It is clear that the truncated error is given by
j−1 Z xk+1 X f (xk+1 ) − f (xk−1 ) 1 −α R j = (x − t) f 0 (t) − j Γ(1 − α) 2h k=0 xk f (xk+1 ) − 2f (xk ) + f (xk−1 ) − (t − xk ) dt h2 j−1 Z xk+1 X 1 = − (xj − t)−α Γ(1 − α) k=0 xk Z xk+1 1 × (xk+1 − s)2 f (3) (s)ds 2! · 2h t Z xk−1 2 (3) (xk−1 − s) f (s)ds − t Z xk+1 (t − xk ) + (xk+1 − s)2 f (3) (s)ds 2 2h t Z xk −2 (xk − s)2 f (3) (s)ds t Z xk−1 2 (3) + (xk−1 − s) f (s)ds dt. t
(4.73)
122
Chapter 4. Numerical Caputo differentiation
Interchanging the order of integration, one has j−1 Z xk+1 X 1 R j = (xk+1 − s)2 f (3) (s) 2hΓ(2 − α) x k k=0 1−α (xj − s) − (xj − xk )1−α × 2 (xj − s)1−α (s − xk ) (xj − s)2−α − (xj − xk )2−α + + ds h h(2 − α) Z 2 xk+1 2 (3) (x − s) f (s) (xj − xk+1 )1−α h + k h xk (xj − xk+1 )2−α − (xj − s)2−α 1−α −(x − s) (s − x ) + ds j k 2−α Z xk+1 (xj − xk+1 )1−α − (xj − s)1−α + (xk−1 − s)2 f (3) (s) 2 xk (xj − s)1−α (s − xk ) − (xj − xk+1 )1−α + h 2−α 2−α (xj − xk+1 ) − (xj − s) − ds h(2 − α) Z xk (xj − xk+1 )1−α − (xj − xk )1−α 2 (3) + (x − s) f (s) k−1 2 xk−1 (xj − xk+1 )2−α − (xj − xk )2−α 1−α −(x − x ) − ds j k+1 h(2 − α) j−1 X 1 Sk . = 2hΓ(2 − α)
(4.74)
k=0
For k = 0, 1, . . . , j − 1, denote
xk
(xj − xk+1 )1−α − (xj − xk )1−α 2 xk−1 (xj − xk+1 )2−α − (xj − xk )2−α −(xj − xk+1 )1−α − ds h(2 − α)
Z B = k
(xk−1 − s)2 f (3) (s)
(4.75)
and Ak , Sk − Bk ,
(4.76)
where the expression of Ak can be obtained from (4.74) and (4.75), so is omitted due to lengthiness. Let l = j − k, k = 0, 1, . . . , j − 1. The affine transformation s = xk−1 + ξh with ξ ∈ [0, 1]
4.2. High-order methods based on polynomial interpolation
123
yields Z 1 (l − 1)1−α − l1−α 2 (3) 4−α ξ f (x + ξh) B =h j−l−1 j−l 2 0 (l − 1)2−α − l2−α −(l − 1)1−α − dξ 2−α Z 1 4−α =h bl ξ 2 f (3) (xj−l−1 + ξh)dξ, l = 1, 2, . . . , j.
(4.77)
0
It is evident that b1 =
bl = l1−α
1 2−α
− 12 , and for l ≥ 2 it holds that
∞ X 1 1 1 − (−α + 1)α(α + 1) · · · (α + n − 2) ≥ 0. ln 2n! (n + 1)! n=2
(4.78)
Thus, it holds that Z 1 4−α 2 (3) |Bj−l | =h bl ξ f (xj−l−1 + ξh)dξ 0 4−α h (3) 1−α ≤ max (x) f l 3 x∈[xj−l−1 ,xj−l ] ∞ X 1 1 1 × − (−α + 1)α(α + 1) · · · (α + n − 2) ln 2n! (n + 1)! n=2 h4−α (3) 1−α max (x) ≤ f l 3 x∈[xj−l−1 ,xj−l ] ∞ X 1 1 1 1−α × − n l 2 n+1 n n=2
h4−α ≤ 3 ×
max x∈[xj−l−1 ,xj−l ]
∞ X n=2
1 ln−2
1 1 − 2 n+1
4−α
≤
h
3 4−α
(3) −1−α f (x) l
max x∈[xj−l−1 ,xj−l ]
1−α n
∞ (3) −1−α X f (x) l n=2
1 ln−2
·
1 1−α · 2 2
l (1 − α) max 12 l−1 x∈[xj−l−1 ,xj−l ] h4−α (3) 1 − α ≤ max f (x) 1+α , l ≥ 2. 6 x∈[xj−l−1 ,xj−l ] l
≤
h
(3) −1−α f (x) l
(4.79)
124
Chapter 4. Numerical Caputo differentiation
As a result, j j−1 X X B = B j−l k l=1 k=0 ( ) Z 1 X j 4−α ξ 2 f (3) (xj−2 + ξh)dξ + |Bl | ≤h b1 0
l=2
(4.80)
# " j 1 1 1 − α X −1−α (3) 1 4−α l ≤h max − + f (x) 3 2−α 2 12 x∈[x−1 ,xj−1 ] l=2 (3) 4−α ≤C2 max f (x) h x∈[x−1 ,xj−1 ]
with C2 > 0 being a constant. Note that the derivative f (3) (x) with x ∈ [x−1 , x0 ] is needed here. In this case, f (3) (xk ), k ≥ 0, can be utilized to approximate f (3) (x) when x ∈ [x−1 , x0 ] and then f (3) (x) is also bounded on [x−1 , x0 ]. Note that Ak contains all the integration terms in (4.74). Then the affine transformation s = xk + ξh, ξ ∈ [0, 1] and l = j − k, k = 0, 1, . . . , j − 1, yield Z 1 2 4−α (3) Aj−l =h f (xj−l + ξh) − (l − 1)2−α − (l − ξ)2−α 2 − α 0 (1 − ξ)2 + (l − 1)2−α − l2−α + 2 (l − ξ)1−α ξ − (l − 1)1−α 2−α (1 − ξ)2 (l − ξ)1−α − l1−α + (1 − ξ)2 (l − 1)1−α + 2 2 (ξ + 1) 1−α 1−α (l − 1) − (l − ξ) + dξ 2 Z 1 =h4−α f (3) (xj−l + ξh)al (ξ)dξ.
(4.81)
0
Rewrite al (ξ) in the form al (ξ) = l1−α
∞ X 1 a ˜n (ξ)(−α + 1)α(α + 1) · · · (α + n − 2), ln n=2
(4.82)
with 1 ξ n+1 (1 − ξ)2 1 ξ n+1 − + +2 − a ˜n (ξ) = − 2 (n + 1)! (n + 1)! (n + 1)! n! n! n 2 2 n 2 (1 − ξ) (1 − ξ) ξ (1 + ξ) ξ 1 − − + − n! 2n! 2 n! n! 2 − (1 − ξ)2 − 21 (1 + ξ)2 2ξ n+1 − 2 + (1 − ξ)2 = + , n ≥ 2. (n + 1)! n!
(4.83)
4.2. High-order methods based on polynomial interpolation
For n ≥ 2, we have a ˜n (ξ) ≥ 0 for arbitrary ξ ∈ [0, 1]. To see this, recall that 2ξ n 2(1 − ξ) 1 − 3ξ 0 − + , a ˜ (ξ) = n n! (n + 1)! n!
hold. When ξ0 =
1 2n
2ξ n−1 2 3 a ˜00n (ξ) = + − (n − 1)! (n + 1)! n! 1 n−1 1 + n+1 ∈ (0, 1), it holds that a ˜00n (ξ0 ) = 0
and
( a ˜00n (ξ)
< 0,
ξ ∈ [0, ξ0 ),
≥ 0,
ξ ∈ [ξ0 , 1].
125
(4.84)
(4.85)
(4.86)
Note that
0 ˜n (1) = 0, a (4.87) 1 2 ˜0n (0) = − > 0. a n! (n + 1)! ˜0n (0) > ˜0n (ξ1 ) = 0 since a ˜0n (1) = 0, and there exists ξ1 ∈ (0, ξ0 ) such that a One has a ˜0n (ξ0 ) < a 0. Therefore, ( > 0, ξ ∈ [0, ξ1 ), 0 a ˜n (ξ) (4.88) ≤ 0, ξ ∈ [ξ1 , 1]. Since
˜n (1) = 0, a 1 1 ˜n (0) = − > 0, a 2n! (n + 1)! it holds that a ˜n (ξ) ≥ 0 for arbitrary ξ ∈ [0, 1] when n ≥ 2. As a result, one has al (ξ) = l
1−α
∞ X 1 a ˜ (ξ)(1 − α)α(α + 1) · · · (α + n − 1) ≥ 0 n n l n=2
for l = 2, . . . , j. Furthermore, ∞ X 1 2ξ n+1 − 2 + (1 − ξ)2 1−α a (ξ) =l l ln (n + 1)! n=2 2 − (1 − ξ)2 − 12 (1 + ξ)2 (1 − α)α(α + 1) · · · (α + n − 1) + n! ∞ X 1 2ξ 3 − 2 + (1 − ξ)2 1−α ≤l ln n+1 n=2 1 1−α 2 2 + 2 − (1 − ξ) − (1 + ξ) 2 n ! √ ∞ X 1 1 1−α 35 + 13 13 2 1−α ≤l + n l n + 1 54 3 n n=2 √ 143 + 13 13 1 −1−α ≤l (1 − α) . 324 1 − 1l
(4.89)
(4.90)
(4.91)
126
Chapter 4. Numerical Caputo differentiation
Especially, when l ≥ 2,
al (ξ) ≤ l−1−α (1 − α)
√ 143 + 13 13 . 162
(4.92)
As a result, it holds that ) ( Z X j j X 1 (3) 4−α |Al | a1 (ξ)f (xj−1 + ξh)dξ + Al ≤h 0 l=2 l=1 ( ) j Z 1 Z 1 X ≤h4−α max f (3) (x) a1 (ξ)dξ + al (ξ)dξ x∈[x0 ,xj ] 0 l=2 0 " 1 1 2 (3) 4−α ≤h max f (x) − − (2 − α)(3 − α) 3(2 − α) 6 x∈[x0 ,xj ] # √ j X 31 13 + 125 −1−α + l (1 − α) 162 l=2 ≤C1 max f (3) (x) h4−α
(4.93)
x∈[x0 ,xj ]
with C1 > 0 being a constant. Consequently, the truncated error has the bound j−1 X 1 R j = (Ak + Bk ) 2hΓ(2 − α) k=0 j ) ( j X X 1 ≤ A + B j−l j−l 2hΓ(2 − α) l=1 l=1 h3−α (3) (3) C1 max f (x) + C2 max ≤ f (x) . 2Γ(2 − α) x∈[x0 ,xj ] x∈[x−1 ,xj−2 ]
(4.94)
Here f (3) (x) with x ∈ [x−1 , x0 ] can also be calculated by utilizing f (3) (xk ), k ≥ 0. Eventually, it holds that j R ≤ ch3−α , 1 ≤ j ≤ N,
(4.95)
with c > 0 being a constant. Remark 4.6. [94] In formula (4.69), if k = 0, then f (xk−1 ) = f (x−1 ) is defined outside of [a, b]. There are several choices to approach f (x−1 ). In numerical calculation, the neighboring function values are usually used to approximate f (x−1 ), that is, f (x−1 ) = f (a) − hf 0 (a) + h2 00 3 2 f (a) + O(h ).
4.2. High-order methods based on polynomial interpolation
127
(I) When f 0 (a) = f 00 (a) = 0, taking f (x−1 ) = f (a) + O(h3 ) in (4.69) gives the accuracy of O(h3−α ); (II) When f 0 (a) = 0, f 00 (a) 6= 0, taking f (x−1 ) = f (a) + accuracy of O(h2 );
h2 00 3 2 f (a) + O(h )
in (4.69) gives the
(III) When f 0 (a) 6= 0, taking f (x−1 ) = f (a) + O(h) in (4.69) gives the accuracy of O(h).
Example 4.6. [94] Consider the function f (x) = x4 , x ∈ [0, 1]. We have α 4 24 24 4−α D x = x = . C 0,x x=1 Γ(5 − α) Γ(5 − α) x=1
(4.96)
Compute the Caputo derivative numerically at x = 1 by using formula (4.69). The numerical results are shown in Table 4.1. It is obvious that the convergence order is (3 − α), which is in line with the theoretical analysis. Table 4.1. Numerical results for Example 4.6. Reprinted with permission from Commun. Appl. Ind. Math., 6: e-536 (2015), at http://caim.simai.eu/index.php/caim/article/view/536 α
0.2
0.4
0.6
0.8
h
Absolute error
Convergence order
1 10 1 20 1 40 1 80 1 160 1 10 1 20 1 40 1 80 1 160 1 10 1 20 1 40 1 80 1 160 1 10 1 20 1 40 1 80 1 160
0.0015 2.4095E-04 3.7575E-05 5.7436E-06 8.6640E-07
∗ 2.6382 2.6809 2.7097 2.7289
0.0052 9.3209E-04 1.6158E-04 2,7540E-05 4.6455E-06
∗ 2.4800 2.5282 2.5526 2.5676
0.0139 0.0028 5.4517E-04 1.0520E-04 2.0146E-05
∗ 2.3116 2.3606 2.3736 2.3846
0.0331 0.0075 0.0017 3.6991E-04 8.1011E-05
∗ 2.1419 2.1414 2.2003 2.1910
Now we revisit a (3−α)th order approximation with another approach derived by [56], which is called the L1-2 formula.
128
Chapter 4. Numerical Caputo differentiation
For the uniform grid with xj = jh, h = the difference operators
δx fj− 12 =
b−a N ,
denote xj+ 21 =
xj+1 +xj , 2
j ≥ 0, and introduce
f (xj ) − f (xj−1 ) 2 1 δx fj+ 12 − δx fj− 21 , j ≥ 1. , δx fj = h h
(4.97)
If f (x) ∈ C 1 ([a, xj ]) (j ≥ 1), the Caputo derivative with 0 < α < 1 can be represented as α C Da,x f (x) x=x j Z xj 1 = (xj − t)−α f 0 (t)dt Γ(1 − α) a j Z xk X 1 f 0 (t) = dt. α Γ(1 − α) xk−1 (xj − t)
(4.98)
k=1
On each subinterval [xk−1 , xk ], 1 ≤ k ≤ j, we denote the linear interpolation of f (x) by
P1,k (x) =
x − xk−1 x − xk f (xk ) + f (xk−1 ), x ∈ [xk−1 , xk ]. xk − xk−1 xk−1 − xk
(4.99)
It is evident that
f (x) − P1,k (x) =
f 00 (ξk ) (x − xk−1 )(x − xk ), x ∈ [xk−1 , xk ]. 2
(4.100)
Denote also the quadratic interpolation by
P2,k (x) =
2 X l=0
f (xk−l )
2 Y x − xk−i , x ∈ [xk−1 , xk ], k ≥ 2. x − xk−i i=0 k−l
(4.101)
i6=l
Then one has
f (x) − P2,k (x) =
f 000 (ϑk ) (x − xk−2 )(x − xk−1 )(x − xk ), x ∈ [xk−1 , xk ] 6
(4.102)
for some ϑk ∈ (xk−2 , xk ). And it can be directly verified that
P2,k (x) = P1,k (x) +
1 2 δx fk−1 (x − xk−1 )(x − xk ). 2
(4.103)
4.2. High-order methods based on polynomial interpolation
129
Utilize the linear interpolation P1,1 (x) to approximate f (x) on the first interval [x0 , x1 ], and approximate f (x) by the quadratic interpolation P2,k (x) on the remaining intervals [xk−1 , xk ] (k ≥ 2). Note that
f (xk ) − f (xk−1 ) = δx fk− 21 , 1 ≤ k ≤ j, h
(4.104)
1 2 0 0 P2,k (x) = P1,k (x) + δx fk−1 (2x − xk−1 − xk ) 2 = δx f 1 + δ 2 fk−1 (x − f 1 ).
(4.105)
0 P1,k (x) =
and
k− 2
x
k− 2
Then we obtain the following numerical approximation: α C Da,x f (x) x=x j Z j X xk 1 f 0 (t) = dt α Γ(1 − α) k=1 xk−1 (xj − t) "Z # j Z xk 0 x1 P 0 (t) X P (t) 1 1,1 2,k ≈ dt + dt α Γ(1 − α) x0 (xj − t)α xk−1 (xj − t) k=2 (Z x1 δx f 12 1 = dt α Γ(1 − α) x0 (xj − t) ) j Z xk X δx fk− 12 + δx2 fk−1 (t − xk− 12 ) + dt (xj − t)α xk−1 k=2 " j Z xk X dt 1 = δx fk− 12 α Γ(1 − α) (x j − t) xk−1 k=1 Z xk t − x 1 # j X k− 2 2 + δx fk−1 dt (x − t)α j xk−1 k=2 j j h1−α X (α) h2−α X (α) 2 1 + a δ f bj−k δx fk−1 . = x k− 2 j−k Γ(2 − α) Γ(2 − α) k=1
(α)
Here the coefficients ak
(α)
and bk (α)
ak
(4.106)
k=2
are given by
= (k + 1)1−α − k 1−α , 0 ≤ k ≤ j − 1,
(4.107)
130
Chapter 4. Numerical Caputo differentiation
and (α)
bk
=
(k + 1)2−α − k 2−α (k + 1)1−α + k 1−α − , 0 ≤ k ≤ j − 2. 2−α 2
(4.108)
Systematically, (4.106) can be rewritten in the form α C Da,x f (x) x=x j " j X (α) h1−α aj−k δx f (xk− 12 ) ≈ Γ(2 − α) k=1 # j X (α) + bj−k δx f (xk− 12 ) − δx f (xk− 32 ) k=2 " j 1−α X (α) h = aj−k δx f (xk− 12 ) Γ(2 − α) k=1 # j j−1 X X (α) (α) + bj−k δx f (xk− 21 ) − bj−k−1 δx f (xk− 21 ) k=2 k=1 j h1−α X (α) cj−k δx f (xk− 12 ) = Γ(2 − α) k=1 j h−α X (α) cj−k [f (xk ) − f (xk−1 )] = Γ(2 − α) k=1 " j # j−1 1−α X (α) X h (α) cj−k f (xk ) − cj−k−1 f (xk ) Γ(2 − α) k=1 k=0 " # j−1 −α X h (α) (α) (α) (α) cj−k−1 − cj−k f (xk ) − cj−1 f (x0 ) , = Γ(2 − α) c0 f (xj ) −
(4.109)
k=1
(α)
where c0
(α)
= a0
= 1 for j = 1 and for j ≥ 2,
(α) ck
=
(α) (α) a0 + b0 ,
(α) ak (α) ak
+
−
(α) (α) bk − bk−1 , (α) bk−1 ,
k = 0, 1 ≤ k ≤ j − 2,
(4.110)
k = j − 1.
Remark 4.7. Note that the first term on the last line of (4.106) happens to be the formulation of the classical L1 method (4.4). We find that the L1-2 formula (4.109) is a modification of the L1 Pj (α) 2 h2−α method (4.4), via adding a correction term Γ(2−α) k=2 bj−k δx fk−1 when j ≥ 2. Now we present the error estimate for the L1-2 formula (4.109) in the following theorem.
4.2. High-order methods based on polynomial interpolation
131
bj by the truncated error of the L1-2 formula (4.109). Suppose Theorem 4.7. [56] Denote R 3 f (x) ∈ C ([a, b]); then we have 1 α b |f 00 (x)| h2−α , R ≤ 2Γ(3 − α) x0max ≤x≤x1 1 α 1 j 00 −α−1 3 b max |f (x)| (xj − x1 ) h + R ≤ x ≤x≤x Γ(1 − α) 12 12 0 1 α 1 1 + + max |f 000 (x)| h3−α x0 ≤x≤xj 3(1 − α)(2 − α) 2 3 − α
(4.111)
with j ≥ 2. Proof. When j = 1, (4.109) is just the L1 formula and its accuracy is O(h2−α ), which can be seen from Theorem 4.1. More precisely, by (4.100), we have Z x1 1 0 b1 = R [f (t) − P1,1 (t)] (x1 − t)−α dt Γ(1 − α) x0 x1 1 = [f (t) − P1,1 (t)] (x1 − t)−α t=x 0 Γ(1 − α) Z x 1 −α [f (t) − P1,1 (t)] (x1 − t)−α−1 dt x0
x1
f 00 (ξ1 )(t − x0 ) (x1 − t)−α dt 2 x0 Z x1 α t − x0 00 = f (ξ1 ) dt α 2Γ(1 − α) (x 1 − t) x0 α = f 00 (ξ1 )h2−α , 2Γ(3 − α) α = Γ(1 − α)
Z
(4.112)
where ξ1 ∈ (x0 , x1 ). Hence the first inequality in (4.111) holds. For j ≥ 2, it holds that Z x1 0 [f (t) − P1,1 f (t)] 1 j b dt R = α Γ(1 − α) x0 (xj − t) # j Z xk 0 X [f (t) − P2,k (t)] + dt (xj − t)α xk−1 k=2 ( x Z x1 1 f (t) − P1,1 (t) 1 f (t) − P1,1 (t) = −α dt α Γ(1 − α) (x − t) (xj − t)α+1 j x0 t=x0 " #) x Z xk j X f (t) − P2,k (t) k f (t) − P2,k (t) + −α dt α+1 (xj − t)α t=xk−1 xk−1 (xj − t) k=2 "Z # j Z xk x1 X −α f (t) − P1,1 (t) f (t) − P2,k (t) dt + dt , = α+1 Γ(1 − α) x0 (xj − t)α+1 xk−1 (xj − t) k=2
(4.113)
132
Chapter 4. Numerical Caputo differentiation
where the relations x f (t) − P1,1 (t) 1 = 0, (xj − t)α t=x0
(4.114)
x f (t) − P2,k (t) k = 0, k = 2, 3, . . . , j − 1, (xj − t)α t=xk−1
(4.115)
and
x f (t) − P2,j (t) j (xj − t)α t=xj−1
xj f 000 (ϑj ) −α = (t − xj−2 )(t − xj−1 )(t − xj )(xj − t) 6 t=xj−1 xj 000 f (ϑj ) =0 (t − xj−2 )(t − xj−1 )(xj − t)1−α = − 6 t=xj−1
(4.116)
are utilized. It follows from (4.100) that Z x1 f (t) − P (t) 1,1 dt α+1 (xj − t) x0 Z x 1 f 00 (η1 ) −α−1 (t − x0 )(t − x1 )(xj − t) dt = 2 x0 00 Z x1 f (η1 ) −α−1 (t − x0 )(x1 − t)(xj − t) dt = 2 x0 ≤ 1 |f 00 (η1 )| (xj − x1 )−α−1 h3 , 12
(4.117)
where η1 ∈ (x0 , x1 ). As a result of (4.102), it holds that j−1 Z X xk f (t) − P (t) 2,k dt α+1 (x − t) j k=2 xk−1 j−1 Z X xk f 000 (ϑ )(t − x k k−2 )(t − xk−1 )(t − xk ) = dt 6(xj − t)α+1 k=2 xk−1 Z xk j−1 X (t − xk−2 )(t − xk−1 )(xk − t) 000 ≤ 1 |f (ϑk )| dt 6 (xj − t)α+1 xk−1 k=2 j−1 Z xk X 1 000 (t − xk−2 )(t − xk−1 )(xk − t) ≤ |f (η)| dt 6 (xj − t)α+1 xk−1 k=2 Z xj−1 1 000 3 ≤ |f (η)| h (xj − t)−α−1 dt 12 x 1 ≤ 1 |f 000 (η)| h3−α , 12α
(4.118)
4.2. High-order methods based on polynomial interpolation
in which |f 000 (η)| =
133
max |f 000 (ϑk )| with η ∈ (x0 , xj−1 ) and ϑk ∈ (xk−2 , xk ), 2 ≤ k ≤
2≤k≤j−1
j − 1. In addition,
Z xj f (t) − P2,j (t) dt α+1 xj−1 (xj − t) Z xj 000 f (ϑj )(t − xj−2 )(t − xj−1 )(t − xj ) dt = 6(x − t)α+1 j
xj−1
=− = −
xj
(t − xj−2 )(t − xj−1 ) dt (xj − t)α xj−1 1 1 1 1 + f 000 (ϑj )h3−α , 3 (2 − α)(1 − α) 2 3 − α
1 000 f (ϑj ) 6
Z
(4.119)
where ϑj ∈ (xj−2 , xj ). Combining the above estimates leads to the second inequality in (4.111). The proof is thus completed.
In [118], linear interpolation and quadratic interpolation are also used to get a (3 − α)th order approximation to the Caputo derivative. When j ≥ 2, instead of utilizing interpolation based on (xk−2 , f (xk−2 )), (xk−1 , f (xk−1 )), and (xk , f (xk )) on each subinterval [xk−1 , xk ], 2 ≤ k ≤ j, the interpolation polynomial
f (xk ) − f (xk−1 ) k (xk − x) H2 (x) =f (xk ) − h f (xk+1 ) − 2f (xk ) + f (xk−1 ) (xk − x)(x − xk−1 ) − h2 2
(4.120)
is chosen to interpolate f (x) on each subinterval [xk−1 , xk ], 1 ≤ k ≤ j − 1. For the last interval, [xj−1 , xj ], H2j−1 is used to interpolate f (x). For the case with j = 1, the utilized 0 linear interpolation is still based on (x0 , f (x0 )) and (x1 , f (x1 )), say, H1 (x) = x−x h f (x1 ) + x1 −x h f (x0 ). As a result, for j = 1, one has
α C Da,x f (x) x=x Z x1 1 1 (x1 − t)−α f 0 (t)dt = Γ(1 − α) x 0 Z x1 1 f (x 1 ) − f (x0 ) = (x1 − t)−α dt + Rh1 h Γ(1 − α) x0 h−α = [f (x1 ) − f (x0 )] + Rh1 , 0 < α < 1, Γ(2 − α)
(4.121)
134
Chapter 4. Numerical Caputo differentiation
with Rh1 being the truncated error on the first interval. When 2 ≤ j ≤ N , it holds that α C Da,x f (x) x=x j "j−1 Z # Z xj 0 X xk f 0 (t)dt f (t)dt 1 + = α Γ(1 − α) (xj − t)α xj−1 (xj − t) k=1 xk−1 0 Z xj H j−1 dt j−1 Z xk k 0 X 2 H2 dt 1 = + + Rhj α α Γ(1 − α) (x − t) (x − t) j j xj−1 k=1 xk−1 (j−1 −α X h = [ak f (xj−k−1 ) + bk f (xj−k ) + ck f (xj−k+1 )] Γ(3 − α) k=1 ) α 4−α + f (xj−2 ) − 2f (xj−1 ) + f (xj ) + Rhj . 2 2
(4.122)
Here Rhj denotes the truncated error and the coefficients are defined by 3 1 ak = − (2 − α)(k + 1)1−α + (2 − α)k 1−α + (k + 1)2−α − k 2−α , 2 2 1−α bk = 2(2 − α)(k + 1) − 2(k + 1)2−α + 2k 2−α , ck = − 1 (2 − α) (k + 1)1−α + k 1−α + (k + 1)2−α − k 2−α . 2
(4.123)
Arguments similar to that in Theorem 4.7 yield the following result on error estimates. Theorem 4.8. [118] For f (x) ∈ C 3 ([a, b]) and 0 < α < 1, it holds that 1 f(f )h2−α , Rh ≤ Cα M
(4.124)
j Rh ≤ Cα M (f )h3−α , j = 2, 3, . . . , N,
(4.125)
f(f ) = max |f 00 (x)|, and M (f ) = max |f (3) (x)|. where Cα depends only on α, M x∈[a,b]
x∈[a,b]
It is notable that the above mentioned numerical approximations to Caputo derivatives, which are based on polynomial interpolation, may result in different accuracy orders on the first subinterval [x0 , x1 ] and on the remaining subintervals. The overall accuracy thus deteriorates. In [5], Alikhanov proposed a new difference analogue for the Caputo derivative, which manages to modify the L1-2 formula (4.109) and to achieve the (3 − α)th order accuracy on every single subinterval. This novel approach is called the L2-1σ formula. Consider the case of a uniform grid with xj = a + jh, h = b−a N , j = 0, 1, . . . , N . Let σ = 1 − α2 with 0 < α < 1; then the Caputo derivative of f (x) ∈ C 3 ([a, b]) at the fixed point
4.2. High-order methods based on polynomial interpolation
135
xj+σ = a + (j + σ)h, 0 ≤ j ≤ N − 1, can be expressed by α C Da,x f (x) x=x j+σ Z x j+σ f 0 (t) 1 = dt Γ(1 − α) a (xj+σ − t)α " j Z # Z xj+σ 0 0 X xk f (t)dt f (t)dt 1 + . = Γ(1 − α) α (xj+σ − t)α xk−1 (xj+σ − t) xj
(4.126)
k=1
On each subinterval [xk−1 , xk ] (1 ≤ k ≤ j), the quadratic interpolation Π2,k f (x) of the function f (x) based on (xk−1 , f (xk−1 )), (xk , f (xk )), (xk+1 , f (xk+1 )), in the form (x − xk )(x − xk+1 ) Π2,k f (x) =f (xk−1 ) 2h2 (x − xk−1 )(x − xk+1 ) − f (xk ) h2 (x − xk−1 )(x − xk ) + f (xk+1 ) , 2h2
(4.127)
is used to approximate f (x). Note that the quadratic interpolation in this case is different from the one utilized in the L1-2 method, which is based on (xk−2 , f (xk−2 )), (xk−1 , f (xk−1 )), and (xk , f (xk )). Denote xk− 12 = xk−12+xk , fx,k = f (xk+1h)−f (xk ) , and fx,k = f (xk )−fh (xk−1 ) . It is clear that 0
(Π2,k f (x)) = fx,k + fxx,k (x − xk+ 12 ) = fx,k−1 + fxx,k (x − xk− 21 )
(4.128)
f 000 (ξ k ) (x − xk−1 )(x − xk )(x − xk+1 ), 6
(4.129)
and f (x) − Π2,k f (x) =
where x ∈ [xk−1 , xk+1 ] and ξ k ∈ (xk−1 , xk+1 ). Note that Z
xk
xk−1
(t − xk− 12 )(xj+σ − t)−α dt =
h2−α (α,σ) b , 1 ≤ k ≤ j, 1 − α j−k+1
(4.130)
with (k + σ)2−α − (k − 1 + σ)2−α b(α,σ) = k 2−α 1−α (k + σ) + (k − 1 + σ)1−α − , k ≥ 1. 2
(4.131)
136
Chapter 4. Numerical Caputo differentiation
Then we have α C Da,x f (x) x=x j+σ Z Z xj+σ j X xk 1 f 0 (t)dt 1 f 0 (t)dt = + Γ(1 − α) (xj+σ − t)α Γ(1 − α) xj (xj+σ − t)α k=1 xk−1 Z xj+σ j Z xk 0 X fx,j dt 1 (Π2,k f (t)) dt + ≈ α Γ(1 − α) (xj+σ − t) Γ(1 − α) xj (xj+σ − t)α k=1 xk−1 j Z xk X fx,k−1 + fxx,k (t − xk− 12 ) 1 = dt Γ(1 − α) (xj+σ − t)α k=1 xk−1 Z xj+σ fx,j dt + Γ(1 − α) xj (xj+σ − t)α ! j 1−α X h (α,σ) (α,σ) (α,σ) = fx,j aj−k+1 fx,k−1 + bj−k+1 fxx,k h + a0 Γ(2 − α) k=1 j h−α X (α,σ) cj−k [f (xk+1 ) − f (xk )] = Γ(2 − α) k=0 h−α (α,σ) = c0 f (xj+1 ) Γ(2 − α) j X (α,σ) (α,σ) (α,σ) + c − c f (x ) − c f (x ) k 0 , j j−k+1 j−k
(4.132)
k=1
which resembles the L1-2 formula (4.109). Here (α,σ)
a0 (α,σ)
c0
(α,σ)
= a0
(α,σ)
= σ 1−α , ak
= (k + σ)1−α − (k − 1 + σ)1−α , k ≥ 1,
(4.133)
for j = 0, and, for j ≥ 1,
(α,σ) ck
=
(α,σ) (α,σ) , + b1 a0
k = 0,
k = j.
(α,σ) ak (α,σ) aj
+
−
(α,σ) (α,σ) bk+1 − bk , (α,σ) bj ,
1 ≤ k ≤ j − 1,
(4.134)
Now we give the following error estimate for the L2-1σ formula (4.132). Theorem 4.9. [5] Let α ∈ (0, 1) and σ = 1 − α2 . For f (x) ∈ C 3 ([x0 , xj+1 ]), it holds that α C Da,x f (x) x=xj+σ
j h−α X (α,σ) − cj−k [f (xk+1 ) − f (xk )] = O(h3−α ). Γ(2 − α) k=0
(4.135)
4.2. High-order methods based on polynomial interpolation
137
Proof. The truncated error can be written as
α C Da,x f (x) x=x j+σ j h−α X (α,σ) cj−k [f (xk+1 ) − f (xk )] = R1j + Rjj+σ , − Γ(2 − α)
(4.136)
k=0
where
j Z xk X 1 j R = f 0 (t)(xj+σ − t)−α dt 1 Γ(1 − α) x k−1 k=1 j X Z xk 1 − (Π2,k f (t))0 (xj+σ − t)−α dt Γ(1 − α) x k−1 k=1 j X Z xk 1 0 [f (t) − Π2,k f (t)] (xj+σ − t)−α dt = Γ(1 − α) k=1 xk−1 j Z xk X α = − [f (t) − Π2,k f (t)] (xj+σ − t)−α−1 dt Γ(1 − α) x k−1 k=1 j Z xk X f 000 (ξ¯k )(t − xk−1 )(t − xk )(t − xk+1 ) α dt, = − 6Γ(1 − α) (xj+σ − t)α+1 xk−1
(4.137)
k=1
and the Taylor expansion gives
Z xj+σ 1 j+σ Rj = f 0 (t)(xj+σ − t)−α dt Γ(1 − α) xj Z f (xj+1 ) − f (xj ) xj+σ (xj+σ − t)−α dt − hΓ(1 − α) x j Z xj+σ f 0 (x 1 ) − f (xj+1 )−f (xj ) 1 j+ 2 h = dt α Γ(1 − α) (x − t) j+σ x j f 00 (xj+ 21 ) Z xj+σ (t − xj+ 12 ) + dt + O(h3−α ) α Γ(1 − α) (x − t) j+σ x j f 00 (xj+ 21 ) Z xj+σ = (t − xj+ 12 )(xj+σ − t)−α dt + O(h3−α ). Γ(1 − α) xj
(4.138)
138
Chapter 4. Numerical Caputo differentiation
We estimate R1j in a manner similar to that in [56] and obtain j Z xk α |f 000 (ξ)| X (t − xk−1 )(xk − t)(xk+1 − t) j dt ≤ R 1 6Γ(1 − α) (xj+σ − t)α+1 k=1 xk−1 j Z α |f 000 (ξ)| h3 X xk dt ≤ 3Γ(1 − α) (xj+σ − t)α+1 k=1 xk−1 Z dt α |f 000 (ξ)| h3 xj = α+1 3Γ(1 − α) (x j+σ − t) x0 |f 000 (ξ)| h3 1 1 = − 3Γ(1 − α) σ α hα (j + σ)α hα |f 000 (ξ)| ≤ α h3−α , ξ ∈ (x0 , xj ). 3σ Γ(1 − α)
(4.139)
We also have Rjj+σ = O(h3−α ) due to Z
xj+σ
xj
(t − xj+ 12 )(xj+σ − t)−α dt =
(2σ + α − 2) hx1−α σ = 0. 2(1 − α)(2 − α)
(4.140)
The proof is thus completed. Remark 4.8. (I) It is worth noticing that the superconvergent point σ is similar to those studied in [132]. (II) We can also derive the L2-1σ formula for the right-sided Caputo derivative on the basis of the above analysis. In this case, the parameter should be chosen as σ = α2 , α ∈ (0, 1). The corresponding approximation is in the form
α C Dx,b f (x) x=x j+σ
N −1 h−α X (α,σ) = c˜k−j [f (xk ) − f (xk+1 )] + O(h3−α ). Γ(2 − α)
(4.141)
k=j
Here the coefficients are given by (α,σ)
c˜0
(α,σ)
= −˜ a0
(4.142)
if j = N − 1, and, for 0 ≤ j < N − 1,
(α,σ)
c˜k
(α,σ) ˜ , k = 0, b1 (α,σ) (α,σ) ˜ ˜ = bk+1 − bk , 1 ≤ k ≤ N − j − 2, (α,σ) (α,σ) ˜ −˜ aN −j−1 − bN −j−1 , k = N − j − 1,
(4.143)
where (α,σ)
a ˜k
= (k + 1 − σ)1−α
(4.144)
4.2. High-order methods based on polynomial interpolation
139
and 1−α − (k − σ)1−α (k + 1 − σ)2−α − (k − σ)2−α ˜b(α,σ) = (k + 1 − σ) − . k 2 2−α
(4.145)
Recently, Luo et al. [117] proposed a global (3 − α)th order numerical scheme for the fractional differential equation α C D0,x f (x)
= g(x), x ∈ [0, b],
(4.146)
with g(x) being the known function. This method is also based on the piecewise linear and quadrature Lagrange interpolation. For j = 1, the only interval we have is [x0 , x1 ] with x0 = 0, for which it is not possible to utilize Lagrange interpolation. Instead, we can view the Caputo derivative as a derivative of an integral function with a parameter. Note that R x+h Rx f 0 (t)(x + h − t)1−α dt − 0 f 0 (t)(x − t)1−α dt 0 h Rx 0 1−α f (t) (x + h − t) − (x − t)1−α dt 0 = h R x+h 0 1−α dt + x f (t)(x + h − t) , x ∈ [0, b], h
(4.147)
when f (x) ∈ C 2 ([0, b]). Utilizing integration by parts gives R x+h 0 f (t)(x + h − t)1−α dt x h −1 R x+h f 0 (t)d(x + h − t)2−α x = 2−α h " # R x+h 00 f (t)(x + h − t)2−α dt −1 0 1−α x −f (x)h − , = 2−α h
(4.148)
in which the use of l’Hôpital’s rule results in
R x+h
f 00 (t)(x + h − t)2−α dt h→0 h Z x+h =(2 − α) lim f 00 (t)(x + h − t)1−α dt = 0 lim
x
h→0
(4.149)
x
and hence R x+h lim
h→0
x
f 0 (t)(x + h − t)1−α dt = 0, x ∈ [0, b]. h
(4.150)
140
Chapter 4. Numerical Caputo differentiation
Similarly, we have Rx 0 f (t) (x + h − t)1−α − (x − t)1−α dt 0 h Rx 0 −1 0 f (t)d (x + h − t)2−α − (x − t)2−α = 2 − α h ( 0 2−α 0 f (x)h − f (0) (x + h)2−α − x2−α −1 = 2−α h ) R x 00 f (t) (x + h − t)2−α − (x − t)2−α dt 0 . − h
(4.151)
Again, it follows from l’Hôpital’s rule that Rx
f 0 (t) (x + h − t)1−α − (x − t)1−α dt lim h→0 h Z x f 0 (t)(x − t)−α dt, x ∈ [0, b]. =(1 − α) 0
(4.152)
0
Consequently, we have 1 d 1−α
Rx 0
Z x f 0 (t)(x − t)1−α dt = f 0 (t)(x − t)−α dt, x ∈ [0, b], dx 0
(4.153)
and so Rx 0 1 d (1−α)Γ(1−α) f (t)(x − t)1−α dt 0 dx Z x 1 f 0 (t)(x − t)−α dt, x ∈ [0, b]. = Γ(1 − α) 0
(4.154)
Therefore, we have d
1 (1−α)Γ(1−α)
Rx 0
f 0 (t)(x − t)1−α dt
dx
= g(x), x ∈ [0, b].
(4.155)
Integrating the above equality from 0 to x, and utilizing integration by parts, it follows that 1 Γ(1 − α)
Z 0
x
f (t) f (0)x1−α dt = + α (x − t) Γ(2 − α)
Z
x
g(t)dt.
(4.156)
0
When x = x1 , substituting f (t) in the above equality with the linear Lagrange interpolation L1,1 f (t) =
x1 − t t − x0 f (x0 ) + f (x1 ), x ∈ [x0 , x1 ], h h
(4.157)
4.2. High-order methods based on polynomial interpolation
141
gives 1 (S1,1 f (x0 ) + S1,2 f (x1 )) [Dα f (x)]x=x1 , h Γ(1 − α) Z x1 f (0)x11−α = + g(t)dt + R1 , Γ(1 − α)(1 − α) 0
(4.158)
with R1 being the truncated error and Z h1−α 1 x1 1−α (x − t) dt = , S = 1,1 1 h x0 2−α Z 1 x1 t − x0 1 1 S1,2 = dt = − h1−α . h x0 (x1 − t)α 1−α 2−α
(4.159)
Then the evaluation of f (x1 ) can be obtained via the known value of f (x0 ) = f (0). For j = 2, 3, . . . , N , we write the Caputo derivative in the form α C Da,x f (x) x=x j Z xj 1 f 0 (t) = dt Γ(1 − α) 0 (xj − t)α j−1 Z xk+1 X 1 f 0 (t) = dt. Γ(1 − α) (xj − t)α xk
(4.160)
k=0
On each subinterval [xk , xk+1 ], denote by L2,k f (x) the piecewise quadratic Lagrange interpolation of f (x) taking three points (x0 , f (x0 )), (x1 , f (x1 )), (x2 , f (x2 )) when k = 0, and another three points (xk−1 , f (xk−1 )), (xk , f (xk )), (xk+1 , f (xk+1 )) when k = 1, 2, . . . , j − 1, namely, (x − x1 )(x − x2 ) f (x0 ) L2,0 f (x) = 2h2 (x − x0 )(x − x2 ) − f (x1 ) h2 (x − x0 )(x − x1 ) + f (x2 ) 2h2
(4.161)
and L2,k f (x) = (x − xk )(x − xk+1 ) f (xk−1 ) 2h2 (x − xk−1 )(x − xk+1 ) − f (xk ) h2 (x − xk−1 )(x − xk ) + f (xk+1 ), k = 1, 2, . . . , j − 1. 2h2
(4.162)
142
Chapter 4. Numerical Caputo differentiation
Then the Caputo derivative can be discretized as α C Da,x f (x) x=x j Z j−1 X xk+1 L2,k f 0 (t) 1 dt ≈ (xj − t)α Γ(1 − α) xk k=0
j X 1 = Sj,k f (xk ) 2 2Γ(1 − α)h k=0 , [Dα h f (x)]x=xj ,
(4.163)
where Z x2 2t − x1 − x2 Sj,0 = dt, (xj − t)α x0 Z x2 Z x3 2t − x0 − x2 2t − x2 − x3 Sj,1 = − 2 dt + dt, α (x − t) (xj − t)α j x0 x2 Z x2 Z x3 2t − x0 − x1 2t − x1 − x3 S = dt − 2 dt j,2 α (x − t) (xj − t)α j x0 x2 Z x4 2t − x3 − x4 dt, + (xj − t)α x3 Z xj−1 Z xj 2t − xj−3 − xj−2 2t − xj−2 − xj Sj,j−1 = dt − 2 dt α (xj − t) (xj − t)α xj−2 xj−1 Z xj 2t − xj−2 − xj−1 S = dt, j,j (xj − t)α xj−1 Z xk Z xk+1 2t − xk−2 − xk−1 2t − xk−1 − xk+1 S = dt − 2 dt j,k α (xj − t) (xj − t)α xk−1 xk Z xk+2 2t − xk+1 − xk+2 + dt, k = 3, 4, . . . , j − 2. (xj − t)α xk+1
(4.164)
Furthermore, (4.146) can be approximated by j [Dα h f (x)]x=xj = g(xj ) + R , j = 2, 3, . . . , N,
(4.165)
with Rj being the truncated error. As a result, evaluations of f (xj ), j = 2, 3, . . . , N , can be obtained. For the truncated errors of (4.158) and (4.165), we have the following result. Theorem 4.10. [117] Suppose that f (x) ∈ C 3 ([0, b]). For any α ∈ (0, 1), the truncated errors Z x1 f (t) 1 R1 = dt − [Dα (4.166) h f (x)]x=x1 Γ(1 − α) 0 (x1 − t)α and Rj =
α C D0,x f (x) x=x j
− [Dα h f (x)]x=xj , j = 2, 3, . . . , N,
(4.167)
4.2. High-order methods based on polynomial interpolation
143
satisfy 1 R ≤
1 max |f 00 (x)| h3−α , 2Γ(1 − α) x∈(x0 ,x1 )
R j ≤
1 max |f 000 (x)| h3−α , 2 ≤ j ≤ N. 3Γ(1 − α)(1 − α) x∈(x0 ,xj )
(4.168)
Proof. By the error estimates 1 00 f (ξ)(t − x0 )(t − x1 ), ξ ∈ (x0 , x1 ), 2
f (t) − L1,1 f (t) =
(4.169)
it holds that Z x1 1 |f (t) − L1,1 f (t)| 1 R ≤ dt Γ(1 − α) (x1 − t)α x0 Z x1 1 00 ≤ max f (ξ) (t − x0 )(x1 − t)1−α dt 2Γ(1 − α) x0 0.
(4.323)
For 1 < α ≤ 2 (in the present situation, two definite conditions are necessary), we can also get almost the same relation as (4.322), except that the last row of D in (4.322) is replaced by
4.5. Diffusive approximation
183
[D]N,j = 1, j = 0, 1, . . . , N . Figures 4.1 and 4.2 show the spectral radius of D for different α and N . From these two figures, we can see that the spectral radius ρ(D) of D is bounded by ρ(D) ≤ C0 N 2α , 0 < α ≤ 2, C0 > 0.
(4.324)
See [100] for more details.
Figure 4.1. The spectral radius associated with the Caputo fractional operator for 0 < α ≤ 1 [100]. Reprinted with permission from Fract. Calc. Appl. Anal., 15 (2012), at https://www.degruyter.com/view/j/fca
4.5 Diffusive approximation Considering the αth order Caputo derivative (m − 1 < α < m) as a fractional integral of order (m − α), we can also derive diffusive approximations to Caputo derivative. In this case, we have Z ∞ α D f (x) = φ(ω, x)dω, α > 0, (4.325) C 0,x 0
with φ being the auxiliary bivariate function defined by φ(ω, x) = (−1)m−1
2 sin(πα) 2α−2m+1 ω π
Z 0
x
f (m) (t) dt e(x−t)ω2
(4.326)
with m − 1 < α < m ∈ Z+ . As a matter of fact, for fixed ω > 0, the function φ(ω, ·) satisfies the differential equation ∂ 2 sin(πα) 2α−2m+1 (m) φ(ω, x) = −ω 2 φ(ω, x) + (−1)m−1 ω f (x) ∂x π
(4.327)
184
Chapter 4. Numerical Caputo differentiation
Figure 4.2. The spectral radius associated with the Caputo fractional operator for 1 < α ≤ 2 [100]. Reprinted with permission from Fract. Calc. Appl. Anal., 15 (2012), at https://www.degruyter.com/view/j/fca subject to the initial condition φ(ω, 0) = 0.
(4.328)
Consequently, evaluating C Dα 0,x involves solving the ODE (4.327) with the condition (4.328) and approximating the integral in (4.325) via quadratures. The diffusive approximation has been investigated, modified, and applied by several researchers; see [32, 104, 111] and references cited therein.
Chapter 5
Numerical Riemann-Liouville differentiation In the present chapter, we focus on numerical Riemann-Liouville differentiation. The link between the Riemann-Liouville derivative and the Caputo derivative in (2.107) gives us some hints. We present L1, L2, and L2C methods, and spectral approximations in this respect. Note that the Riemann-Liouville derivative is defined as an integer-order derivative of the Riemann-Liouville integral. We can therefore derive numerical methods via suitable compositions of those for integer-order derivatives and numerical Riemann-Liouville integrations; for instance, the approximation based on spline interpolation. In addition, other numerical evaluations such as the Grünwald-Letnikov type approximations, the fractional backward difference formulae and their modifications, the fractional average central difference method, and numerical methods based on finite-part integrals are also introduced in detail.
5.1 L1, L2, and L2C methods Recalling the relation (2.107), we can evaluate the Riemann-Liouville derivative by adding several terms relating to initial values to numerical evaluations of the Caputo derivative. The corresponding error estimates are therefore the same as those in the case of the Caputo derivative, and are thus omitted here.
5.1.1 L1 method When 0 < α < 1, the equality (2.107) reads α RL Da,x f (x)
= C Dα a,x f (x) +
f (a)(x − a)−α . Γ(1 − α)
(5.1)
Now that the L1 method for Caputo derivative has already been derived in details in the previous chapter, we can readily obtain the L1 method and its modifications for the Riemann-Liouville one as follows. (I) The L1 method on uniform grids For uniform grids, the L1 method for Riemann-Liouville derivative is given by α RL Da,x f (x) x=x j
≈
j−1 X
bj−k−1 [f (xk+1 ) − f (xk )] +
k=0
185
f (a)(xj − a)−α , Γ(1 − α)
(5.2)
186
Chapter 5. Numerical Riemann-Liouville differentiation
where j = 1, 2, . . . , N , and coefficients bk are defined by (4.3). When f (a) = 0, the above L1 method for the Riemann-Liouville derivative coincides with that for the Caputo derivative. (II) The L1 method on nonuniform grids Let {˜ xj } be any division of [a, b] with a = x ˜0 < x ˜1 < · · · < x ˜N −1 < x ˜N = b. Combining (4.19) with (5.1) gives α RL Da,x f (x) x=˜ xj
≈
j−1 X
bjk+1 [f (˜ xk+1 ) − f (˜ xk )] +
k=0
where bjk+1 =
f (a)(˜ xj − a)−α , Γ(1 − α)
1 (˜ xj − x ˜k )1−α − (˜ xj − x ˜k+1 )1−α ˜k Γ(2 − α)h
(5.3)
(5.4)
˜j = x with h ˜j+1 − x ˜j , j = 1, 2, . . . , N . (III) The modified L1 method x +x For the special case with x ˜ 0 = x0 , x ˜j = xj− 12 = j−12 j , j = 1, 2, . . ., the approximation (5.3) is reduced to the modified L1 method α RL Da,x f (x) x=x j+ 1 2 j b0 1X (5.5) = [f (x ) + f (x )] − (bj−k − bj−k+1 ) [f (xk−1 ) + f (xk )] j+1 j 2 2 k=1 2−α − Aj f (x0 ) + O(h ), j = 1, 2, . . . , N, where bk =
h−α (k + 1)1−α − k 1−α Γ(2 − α)
for k = 0, 1, . . . , j, and Aj = B j − with
−α (1 − α) j + 12 Γ(2 − α)hα
2h−α 1 1−α 1−α Bj = (j + ) . −j Γ(2 − α) 2
(5.6)
(5.7)
(5.8)
5.1.2 L2 and L2C methods When α ∈ (1, 2), the relationship (2.107) is in the form α RL Da,x f (x)
= C Dα a,x f (x) +
f 0 (a)(x − a)1−α f (a)(x − a)−α + . Γ(1 − α) Γ(2 − α)
(5.9)
Recalling (4.56) and (4.60), we obtain the following L2 and L2C methods for the RiemannLiouville derivative. (I) The L2 method α RL D0,x f (x) x=xj j X f (0)x−α f 0 (0)x1−α j j Wj,k f (xj−k ) + O(h3−α ), = Γ(1 − α) + Γ(2 − α) + k=−1
where Wj,k are defined by (4.57).
(5.10)
5.2. Approximation based on spline interpolation
187
(II) The L2C method α RL D0,x f (x) x=x j
j+1 X f (0)x−α f 0 (0)x1−α j j fj,k f (xj−k ) + O(h3−α ), = + + W Γ(1 − α) Γ(2 − α)
(5.11)
k=−1
fj,k are given by (4.61). where W Remark 5.1. Note that in both schemes (5.10) and (5.11), the 1st order derivative f 0 (0) is needed. When the value of f 0 (0) is unknown, we can utilize neighboring values of f (0) to approximate f 0 (0) with higher-order accuracy. Then the (3−α)th order accuracy is maintained. Remark 5.2. The L2 and L2C methods introduced here are for Riemann-Liouville derivatives with lower terminal a = 0. For the more general case with arbitrary lower terminal a < b, we can adopt affine transformations to modify (5.10) and (5.11).
5.2 Approximation based on spline interpolation Apart from extending numerical evaluations of the Caputo derivative to the Riemann-Liouville one by adding several terms associated with the initial values, we can approximate the RiemannLiouville derivative through other approaches; for example, the one presented and discussed in [157, 160]. Let xj = a + jh, j = 0, 1, . . . , N . When 1 < α < 2, the central difference formula for the 2nd order differentiation yields α RL Da,x f (x) x=x j 2 Z x d 1 1−α = (x − t) f (t)dt Γ(2 − α) dx2 a x=xj Z xj−1 Z xj −2 h f (t)dt f (t)dt ≈ −2 (5.12) α−1 Γ(2 − α) a (xj−1 − t) (xj − t)α−1 a Z xj+1 f (t)dt + α−1 (x j+1 − t) a −2 l = h Iα (xj−1 ) − 2Iαl (xj ) + Iαl (xj+1 ) , j = 1, 2, . . . , N − 1, Γ(2 − α) where Iαl (xj ) =
R xj a
f (t)dt (xj −t)α−1 ,
j = 1, 2, . . . , N . Utilize the linear spline slj (x) =
j X
f (xk )slj,k (x)
(5.13)
k=0
with slj,k (x) =
x−xk−1 xk −xk−1 ,
xk−1 ≤ x ≤ xk ,
xk+1 −x xk+1 −xk ,
xk ≤ x ≤ xk+1 ,
0,
otherwise,
(5.14)
188
Chapter 5. Numerical Riemann-Liouville differentiation
for 1 ≤ k ≤ j − 1, slj,0 (x)
=
x1 −x x1 −x0 ,
0,
x0 ≤ x ≤ x 1 , (5.15) otherwise,
and slj,j (x)
=
x−xj−1 xj −xj−1 ,
0,
xj−1 ≤ x ≤ xj , (5.16) otherwise,
to approximate f (x). Then we obtain an approximation to Iαl (xj ), which is given by Z xj l I (x ) = (xj − t)1−α slj (t)dt j α a j
X h2−α = alj,k f (xk ) (2 − α)(3 − α)
(5.17)
k=0
with
alj,k
(3 − α)j 2−α + (j − 1)3−α − j 3−α , k = 0, (j − k + 1)3−α − 2(j − k)3−α = +(j − k − 1)3−α , 1 ≤ k ≤ j − 1, 1, k = j.
(5.18)
Therefore, we obtain the following approximation to the left-sided Riemann-Liouville derivative: α RL Da,x f (x) x=x j "j−1 # j j+1 X X X h−α l l l ≈ aj−1,k f (xk ) − 2 aj,k f (xk ) + aj+1,k f (xk ) Γ(4 − α) k=0 k=0 k=0 "j−1 −α X h alj−l,k − 2alj,k + alj+l,k f (xk ) = Γ(4 − α) k=0 + alj+1,j − 2alj,j f (xj ) + alj+1,j+1 f (xj+1 ) .
(5.19)
In a similar manner, the right-sided Riemann-Liouville derivative can be approximated by α RL Dx,b f (x) x=x j N h−α X ≈ arj−1,k − 2arj,k + arj,k f (xk ) Γ(4 − α) k=j+1 +arj−1,j−1 f (xj−1 ) + arj−1,j − 2arj,j f (xj ) ,
(5.20)
5.2. Approximation based on spline interpolation
189
where
arj,k
(3 − α)(N − j)2−α + (N − j − 1)3−α 3−α , −(N − j) 3−α (k − j + 1) − 2(k − j)3−α = +(k − j − 1)3−α , 1,
k = N, (5.21) j + 1 ≤ k ≤ N − 1, k = j.
In the particular cases with a = −∞ and b = +∞, we have α RL D−∞,x f (x) x=x j
and α RL Dx,∞ f (x)
x=xj
∞ X h−α ql f (xj−m ) Γ(4 − α) m=−1 j,j−m
(5.22)
−1 X h−α qr f (xj−m ). Γ(4 − α) m=−∞ j,j−m
(5.23)
≈
≈
Here the coefficients are defined by l ˜lj−1,k − 2˜ alj,k + a ˜lj+1,k , qj,k = a l l l qj,j = −2˜ aj,j + a ˜j+1,j , l l qj,j+1 = a ˜j+1,j+1 and
r ˜rj−1,k − a ˜rj,k + a ˜rj+1,k , qj,k = a r r r qj,j = −2˜ aj,j + a ˜j−1,j , r r qj,j−1 = a ˜j−1,j−1
k ≤ j − 1, (5.24)
k ≥ j + 1, (5.25)
with ( a ˜lj,k
=
(j − k + 1)3−α − 2(j − k)3−α + (j − k − 1)3−α ,
k ≤ j − 1,
1,
k=j
(k − j + 1)3−α − 2(k − j)3−α + (k − j − 1)3−α ,
k ≥ j + 1,
1,
k = j.
(5.26)
and ( a ˜rj,k
=
If we define
( am =
(m + 1)3−α − 2m3−α + (m − 1)3−α ,
m ≥ 1,
1,
m=0
and qm
am−1 − 2am + am+1 , −2a0 + a1 , = a0 ,
(5.27)
(5.28)
m ≥ 1, m = 0,
(5.29)
m = −1,
then the approximations (5.22) and (5.23) are also in the forms α RL D−∞,x f (x) x=x j
≈
∞ X h−α qm f (xj−m ) Γ(4 − α) m=−1
(5.30)
190
Chapter 5. Numerical Riemann-Liouville differentiation
and
α RL Dx,+∞ f (x) x=x ≈ j
∞ X h−α qm f (xj+m ). Γ(4 − α) m=−1
(5.31)
Both series on the right-hand side of (5.30) and (5.31) converge absolutely for each 1 < α < 2 and for every bounded function f (x). This is in fact a straightforward consequence of the convergence of the series of the coefficients qm , which was discussed in [160]. Here we consider bounds for the truncated error, in which the following lemma is needed. Lemma 5.1. [160] Let f ∈ C 4 (R). For x ∈ [xk−1 , xk ] and slk (x) =
x − xk−1 xk − x f (xk−1 ) + f (xk ) h h
(5.32)
one gets f (x) − slk (x) = −
3 X 1 (r) 1 f (x)lk,r (x) − f (4) (ηk )lk,4 (x), ηk ∈ [xk−1 , xk ], r! 4! r=2
(5.33)
where |lk,r (x)| ≤ hr with r = 2, 3, 4. Proof. For x ∈ [xk−1 , xk ], it holds that xk − x x − xk−1 f (xk−1 ) − f (xk ). h h
(5.34)
3 X 1 (r) 1 f (x)lk,r (x) − f (4) (ηk )lk,4 (x), r! 4! r=2
(5.35)
f (x) − slk (x) = f (x) − The Taylor expansion yields that f (x) − slk (x) = −
where lk,r (x) are functions which depend on h and xk ; more precisely, xk − x x − xk + h lk,r (x) = (xk − x − h)r + (xk − x)r h h r−1 X r r =(x − x) + (xk − x)p+1 (−1)r−p hr−p−1 . k p p=0
(5.36)
It can be readily concluded that |lk,r (x)| ≤ hr for x ∈ [xk−1 , xk ]. In view of the above lemma, we have the following result on the approximation (5.30). Theorem 5.2. [160] Let f ∈ C 4 (R) and f (4) (x) = 0 for x ≤ a with a ∈ R being a certain constant. Denote ∞ X 1 δα f (xj ) = qm f (xj−m ) (5.37) Γ(4 − α) m=−1 with qm being defined by (5.29). Then we have α RL D−∞,x f (x) x=x j
−
1 δα f (xj ) = (xj ), hα
where |(xj )| ≤ Ch2 , and C is a constant independent of h.
(5.38)
5.2. Approximation based on spline interpolation
Proof. It is clear that 2 d l 1 α RL D−∞,x f (x) x=xj = Γ(2 − α) dx2 Iα (xj ) x=xj =
191
(5.39)
1 1 l Iα (xj−1 ) − 2Iαl (xj ) + Iαl (xj+1 ) + 1 (xj ), 2 Γ(2 − α) h
where 1 (xj ) = O(h2 ). Define the error as Es (xj ), say, Iαl (xj−1 ) − 2Iαl (xj ) + Iαl (xj+1 ) = Iαl (xj−1 ) − 2Iαl (xj ) + Iαl (xj+1 ) + Es (xj ). We have
α RL D−∞,x f (x) x=x j 1 1 l = Iα (xj−1 ) − 2Iαl (xj ) + Iαl (xj+1 ) 2 Γ(2 − α) h 1 1 + Es (xj ) + 1 (xj ), Γ(2 − α) h2
that is, α RL D−∞,x f (x) x=x j
with 2 (xj ) =
=
1 δα f (xj ) + 1 (xj ) + 2 (xj ) hα
1 1 Es (xj ). Γ(2 − α) h2
To compute the error Es (xj ), we have j−1 Z xk X Es (xj ) = f (t) − slk (t) (xj−1 − t)1−α dt k=−∞ xk−1 Z xk j X −2 f (t) − slk (t) (xj − t)1−α dt k=−∞ xk−1 j+1 Z xk X f (t) − slk (t) (xj+1 − t)1−α dt. + k=−∞
(5.40)
(5.41)
(5.42) (5.43)
(5.44)
xk−1
Corresponding to Lemma 5.1, we denote Es (xj ) = −
4 X 1 Er (xj ), r! r=2
where Er (xj ) are defined as follows. For r = 2 and r = 3, j−1 Z xk E (x ) = X lk,r (t)f (r) (t)(xj−1 − t)1−α dt r j x k−1 k=−∞ Z xk j X −2 lk,r (t)f (r) (t)(xj − t)1−α dt x k−1 k=−∞ j+1 X Z xk + lk,r (t)f (r) (t)(xj+1 − t)1−α dt, k=−∞
xk−1
(5.45)
(5.46)
192
Chapter 5. Numerical Riemann-Liouville differentiation
and, for r = 4, Z xk j−1 X (4) Er (xj ) = f (ηk ) lk,r (t)(xj−1 − t)1−α dt x k−1 k=−∞ Z xk j X (4) −2 f (ηk ) lk,r (t)(xj − t)1−α dt x k−1 k=−∞ Z xk j+1 X (4) + f (ηk ) lk,r (t)(xj+1 − t)1−α dt.
(5.47)
xk−1
k=−∞
When r = 2, 3, changing variables gives Z xk j X Er (xj ) = lk,r (t)f (r) (t − h)(xj − t)1−α dt x k−1 k=−∞ Z xk j X −2 lk,r (t)f (r) (t)(xj − t)1−α dt x k−1 k=−∞ Z xk j X + lk,r (t)f (r) (t + h)(xj − t)1−α dt,
(5.48)
xk−1
k=−∞
that is, Z j X Er (xj ) =
h lk,r (t) f (r) (t + h) − 2f (r) (t)
xk−1
k=−∞
xk
(5.49) i
+f (r) (t − h) (xj − t)1−α dt.
Let xa = Na h such that f (4) (x) = 0 for x ≤ xa . For r = 2, we have Z xk j h X lk,2 (t) f (2) (t + h) − 2f (2) (t) E (x ) = 2 j k=−∞ xk−1 i +f (2) (t − h) (xj − t)1−α dt j h2 X f (4) (ξk )cj,k,2 , = 2
(5.50)
k=Na +1
where ξk ∈ [xk−1 , xk ] and the coefficients are given by Z
xk
lk,2 (t)(xj − t)1−α dt.
cj,k,2 =
(5.51)
xk−1
Due to Lemma 5.1, it holds that |ck,j,2 | ≤ h2
Z
xk
xk−1
(xj − t)1−α dt.
(5.52)
5.2. Approximation based on spline interpolation
193
Note that Z
xj
(xj − t)1−α dt =
xa
1 (xj − xa )2−α . 2−α
(5.53)
We have |E2 (xj )| ≤
h4 max f (4) (x) (xj − xa )2−α . 2(2 − α) x∈R
(5.54)
For r = 3, j X
E3 (xj ) =
hf (4) (ξ˜k )cj,k,3 , ξ˜k ∈ [xk−1 , xk ]
(5.55)
k=Na +1
and |cj,k,3 | ≤ h3
Z
xk
(xj − t)1−α dt.
(5.56)
xk−1
We have |E3 (xj )| ≤
2h4 max f (4) (x) (xj − xa )2−α . 2 − α x∈R
(5.57)
Finally for r = 4, the first term in the right-hand side on (5.47) can be estimated as
j−1 X
j−1 X
f
(4)
Z
xk 1−α
lk,4 (t)(xj−1 − t)
(ηk ) xk−1
k=Na +1
dt
j−1 Z xk (4) X 4 ≤h max (x) (xj−1 − t)1−α dt f x∈R x k−1 k=Na +1 4 h = max f (4) (x) (xj−1 − xa )2−α . 2 − α x∈R
(5.58)
Therefore we have
k=Na +1
f (4) (ηk ) lk,4 (t)(xj−1 − t)1−α dt xk−1 Z
xk
(5.59)
4 ≤ h max f (4) (x) (xj − xa )2−α + h2−α , 2 − α x∈R due to the inequality |x + y|p ≤ |x|p + |y|p for 0 < p ≤ 1. In a similar manner, we have for the second term on the right-hand side of (5.47)
j X
k=Na +1
f
(4)
Z
xk 1−α
lk,4 (t)(xj − t)
(ηk ) xk−1
4 ≤ h max f (4) (x) (xj − xa )2−α , 2 − α x∈R
dt
(5.60)
194
Chapter 5. Numerical Riemann-Liouville differentiation
and for the third term
j X
k=Na +1
f (4) (ηk ) lk,4 (t)(xj+1 − t)1−α dt xk−1 Z
xk
(5.61)
4 ≤ h max f (4) (x) (xj − xa )2−α + h2−α . 2 − α x∈R Finally, we have |E4 (xj )| ≤
3h4 2h6−α max f (4) (x) (xj − xa )2−α + max f (4) (x) . 2 − α x∈R 2 − α x∈R
(5.62)
From the previous three inequalities, the error Es (xj ) is of O(h4 ) and therefore 2 (xj ) is of O(h2 ). Remark 5.3. When α = 1, schemes (5.30) and (5.31) reduce to the 2nd order finite difference formula for the 1st order derivative. When α = 2, (5.30) and (5.31) are consistent with the central difference formula for the 2nd order derivative.
5.3 Grünwald-Letnikov type approximations As mentioned in Chapter 2, the Grünwald-Letnikov derivative can be viewed as a numerical approximation to the Riemann-Liouville derivative. In this section, we introduce numerical evaluations of this type in detail, as well as some variants.
5.3.1 Classical Grünwald-Letnikov formulae For the suitably smooth function f (x), the Grünwald-Letnikov derivative approximates the Riemann-Liouville derivative with 1st order accuracy, i.e., α RL D−∞,x f (x)
=
∞ 1 X (α) ωl f (x − lh) + O(h). hα
(5.63)
l=0
(α) Here the coefficients are given by ωl = (−1)l αl . This can be verified through the Fourier P∞ transform. Since (1 + z)α = l=0 αl z l , we have ) ( ∞ 1 X (α) ωl f (x − lh); ω F hα l=0 ∞ 1 X (α) ihlω b ωl e f (ω) = α h (5.64) l=0 1 α = α 1 − eihω fb(ω) h α 1 − eihω αb . =(−iω) f (ω) −ihω Note that the Taylor expansion gives 1 − eihω α − 1 ≤ ch|ω| −ihω
(5.65)
5.3. Grünwald-Letnikov type approximations
with c being a positive constant. The inverse Fourier transform yields ∞ 1 X (α) α ω f (x − lh) − D f (x) RL −∞,x l α h l=0 α Z ∞ 1 − eihω 1 e−iωx (−iω)α fb(ω) − 1 dω = 2πi −∞ −ihω Z ∞ ch ≤ |ω|1+α |fb(ω)|dω 2π −∞ ≤Ch,
195
(5.66)
provided that f (x) ∈ C [α]+2 (R) and all the derivatives of f (x) up to order [α] + 3 belong to L1 (R). Consequently, equation (5.63) is valid. For a suitably smooth function defined on bounded interval [a, b] with f (a) = 0, zero extension yields that the following equality also holds: α RL Da,x f (x)
j 1 X (α) = α ωl f (x − lh) + O(h), jh = x − a. h
(5.67)
l=0
Remark 5.4. In general, the classical Grünwald-Letnikov formula (5.67) is applicable for approximating the Riemann-Liouville derivative with 0 < α < 1 in fractional differential equations [151]. However, it may not be suitable for the discretization of fractional differential equations with α ∈ (1, 2) [88]. And it has been shown in [126] that both explicit and implicit Euler methods for the space-fractional advection-dispersion equation are unstable when the classical Grünwald-Letnikov formula is used.
5.3.2 Shifted Grünwald-Letnikov formulae In order to construct stable schemes for fractional differential equations, we often adopt shifts [8, 141]; in other words, we replace f (x − lh) in (5.67) by f (x − (l − p)h) with p ∈ Z+ , which makes the coefficient matrices diagonally dominated. In this case, we obtain the shifted Grünwald-Letnikov formula α RL Da,x f (x)
[j+p] 1 X (α) = α ωl f (x − (l − p)h) + O(h), jh = x − a, h
(5.68)
l=0
under certain assumptions. Remark 5.5. It is believed that minimizing |p − α2 | yields the best performance of the above approximation. In addition, equation (5.68) coincides with the 2nd order central difference of the classical 2nd order differentiation [126, 137]. If the shift is chosen to be noninteger, numerical method (5.68) may have superconvergence behaviors [54, 132, 187]. To show the difference between the classical Grünwald-Letnikov formula and the shifted one, we consider the leading terms of the truncated errors for those two approaches and clarify that they are slightly different, although they are both of O(h) accuracy. Lemma 5.3. [164] Let 1 < α < 2, n ≥ 2, and f ∈ C n+3 (R) such that all derivatives of f up to order n + 3 belong to L1 (R). For any integer p ≥ 0, define the shifted Grünwald difference
196
Chapter 5. Numerical Riemann-Liouville differentiation
operator ∆α h,p f (x)
=
∞ X l=0
α (−1) f (x − (l − p)h). l l
(5.69)
Then we have for some constant ak independent of h, f , and x that n−1 X ∆α h,p f (x) α+k α = D f (x) + ak RL D−∞,x f (x) hk + O(hn ) RL −∞,x α h
(5.70)
k=1
holds uniformly in x ∈ R. Proof. By the Riemann-Lebesgue lemma, our assumptions on f imply that for some constant C1 > 0 we have b (5.71) f (ω) ≤ C1 (1 + ω)−n−3 for all ω ∈ R, where F{f (x); ω} = fb(ω) is the Fourier transform of f . In view of the Fourier transform formula (2.54) and the equalities F{f (x − y); ω} = eiyω fb(ω), i2 = −1, and (1 + z)α =
∞ X α l z , |z| < 1, l
(5.72)
(5.73)
l=0
1 we have ∆α h,p f ∈ L (R). Therefore it holds that F h−α ∆α h,p f (x); ω ∞ X −α −iωph l α =h e (−1) eilωh fb(ω) l l=0 α =h−α e−iωph 1 − eiωh fb(ω) α 1 − eiωh α =(−iω) e−iωhp fb(ω) −iωh =(−iω)α Wα,p (−iωh)fb(ω),
(5.74)
−z
Wα,p (z) is analytic in some neighborhood of the origin, where Wα,p (z) = ( 1−ez )α epz . SinceP ∞ we have the power series Wα,p (z) = k=0 ak z k , which converges absolutely for all |z| < R with some R > 0. It is evident that a0 = 1 and a1 = − α2 + p. Next we show that there exists a constant C2 > 0 such that n−1 X k ak (−ix) ≤ C2 |x|n (5.75) Wα,p (−ix) − k=0
holds uniformly for x ∈ R. For |x| ≤ R we have ∞ n−1 X X k k W (−ix) − a (−ix) = a (−ix) α,p k k k=0
∞ X n ≤|x| |ak ||x|k−n ≤ C3 |x|n , k=n
k=n
(5.76)
5.3. Grünwald-Letnikov type approximations
where C3 = R−n
P∞
k=0
197
|ak |Rk < ∞. When |x| > R, we have
α 1 − eix α −ipx ≤ 2 ≤ C4 |x|n , |Wα,p (−ix)| = e −ix Rα where C4 =
2α Rα+n
(5.77)
< ∞, and n−1 n−1 X X n k ≤ |x| |ak ||x|k−n ≤ C5 |x|n , a (−ix) k
(5.78)
k=0
k=0
Pn−1 in which C5 = k=0 |ak |Rk−n < ∞. Now if we set C2 = max{C3 , C4 + C5 }, then it follows that (5.75) holds for all x ∈ R. In view of (5.74), it holds that F h−α ∆α f (x); ω h,p n−1 X = ak (−iω)α+k hk fb(ω) + ϕ(ω, b h) (5.79) k=0 n−1 X αb =(−iω) ak (−iω)α+k fb(ω)hk + ϕ(ω, f (ω) + b h), k=1
where α
ϕ(ω, b h) = (−iω)
Wα,p (−iωh) −
n−1 X
! ak (−iωh)
k
fb(ω).
(5.80)
k=0
Note that we have (−iω)α+k fb(ω) ∈ L1 (R) for 0 ≤ k ≤ n − 1. Moreover, ϕ(ω, b h) ∈ L1 (R) α+n n with |ϕ(ω, b h)| ≤ C|ω| h for all ω ∈ R, with C = C1 C2 . By the inverse Fourier transform we get α h−α ∆α h,p f (x) = RL D−∞,x f (x) +
n−1 X
α+k ak RL D−∞,x f (x)hk + ϕ(x, h),
(5.81)
Z Z −iωx |ϕ(x, h)| = C e ϕ(ω, b h)dω ≤ C |ϕ(ω, b h)| dω ≤ Chn
(5.82)
k=1
where
R
R
holds uniformly for x ∈ R. This concludes the proof. Regarding the above Taylor expansion for the error in the shifted Grünwald finite difference formula, we have the following result based on the above lemma. Theorem 5.4. [99, 158] If 1 < α < 2, f (x) = (x − a)µ , µ is a non-negative integer, and x = xj = a + jh, then the following relations hold: j 1 X (α) α D f (x) = ωl f (xj−l ) RL a,x x=xj hα l=0 (−α)(xj − a)−1−α − (1 − α) h + O(h2 ), µ = 0, 2Γ(1 − α)
(5.83)
198
Chapter 5. Numerical Riemann-Liouville differentiation
j 1 X (α) α ωl f (xj−l ) = D f (x) RL a,x x=xj hα l=0 Γ(µ + 1)(xj − a)µ−1−α − (−α) h + O(h2 ), µ > 0, 2Γ(µ − α)
(5.84)
j+1 1 X (α) α D f (x) = ωl f (xj−l+1 ) RL a,x x=xj hα l=0 (−α)(xj − a)−1−α h + O(h2 ), µ = 0, − (3 − α) 2Γ(1 − α)
(5.85)
j+1 1 X (α) α ωl f (xj−l+1 ) RL Da,x f (x) x=xj = hα l=0 Γ(µ + 1)(xj − a)µ−1−α − (2 − α) h + O(h2 ), µ > 0. 2Γ(µ − α)
(5.86)
and
Remark 5.6. It follows from the above theorem that the shifted Grünwald-Letnikov formula (5.68) does not reach 1st order accuracy for the smooth function f (x) if f (a) 6= 0. The remedy may be α RL Da,x f (x) x=x j f (a)(xj − a)−α = RL Dα (f (x) − f (a)) + a,x x=xj Γ(1 − α) (5.87) j+p X f (a)(xj − a)−α (α) ≈ 1 ωl [f (xj−l+p ) − f (a)] + . hα Γ(1 − α) l=0
Such a remedy guarantees that the above formula is exact whenever f (x) is a constant.
5.3.3 Further modification Another modification of the Grünwald-Letnikov formulae introduced here is the weighted and shifted Grünwald-Letnikov formulae. For f (x) = (x − a)µ , µ > 0, Theorem 5.4 implies that (2 − α) RL Dα a,x f (x) x=xj j 2 − α X (α) = α ωl f (xj−l ) h l=0 Γ(µ + 1)(xj − a)µ−1−α h + O(h2 ) − (2 − α)(−α) 2Γ(µ − α)
(5.88)
5.3. Grünwald-Letnikov type approximations
199
and α RL Dα a,x f (x) x=xj j+1 α X (α) = α ωl f (xj−l+1 ) h l=0 Γ(µ + 1)(xj − a)µ−1−α h + O(h2 ). − α(2 − α) 2Γ(µ − α) Eliminating the term α(2 − α) order approximation
Γ(µ+1)(xj −a)µ−1−α h 2Γ(µ−α)
from the above two equations yields the 2nd
α RL Da,x f (x) x=x j " # j j+1 1 2 − α X (α) α X (α) ωl f (xj−l ) + ωl f (xj−l+1 ) + O(h2 ). = hα 2 2 l=0
(5.89)
(5.90)
l=0
This approach is applicable for any suitably smooth function with f (a) = 0. For other suitably smooth functions not satisfying f (a) = 0, the remedy introduced in Remark 5.6 yields the following 2nd order method: α RL Da,x f (x) x=x j " j 1 2 − α X (α) = ωl (f (xj−l ) − f (a)) hα 2 l=0 # j+1 X α (α) + ωl (f (xj−l+1 ) − f (a)) 2 l=0 f (a)(xj − a)−α + + O(h2 ). Γ(1 − α)
(5.91)
If we choose arbitrary integer shifts p and q, the 2nd order weighted and shifted GrünwaldLetnikov approximation can be obtained by choosing appropriate weights. Theorem 5.5. [165] Let f (x) ∈ L1 (R), L1 (R). Then it holds that
α+2 RL Da,x f (x),
and its Fourier transform belong to
α RL D−∞,x f (x) x=x j " ∞ X (α) =h−α α − 2q ωl f (xj − (l − p)h) 2(p − q) l=0 # ∞ 2p − α X (α) ωl f (xj − (l − q)h) + O(h2 ) + 2(p − q) l=0
(5.92)
200
Chapter 5. Numerical Riemann-Liouville differentiation
for arbitrary integers p, q. If f (a) = 0, then we have α D f (x) RL a,x x=xj " j+p X (α) =h−α α − 2q ωl f (xj − (l − p)h) 2(p − q) l=0 # j+q 2p − α X (α) ωl f (xj − (l − q)h) + O(h2 ). + 2(p − q)
(5.93)
l=0
Accordingly, if f (b) = 0, it holds that α D f (x) RL x,b x=xj " NX −j+p (α) =h−α α − 2q ωl f (xj + (l − p)h) 2(p − q) l=0 # N −j+q 2p − α X (α) ωl f (xj + (l − q)h) + O(h2 ). + 2(p − q)
(5.94)
l=0
Proof. Denote by −α Aα h,p f (x) = h
∞ X
(α)
ωl f (x − (l − p)h)
(5.95)
l=0
and α α Bh,p,q f (x) = a1 Aα h,p f (x) + b1 Ah,q f (x)
(5.96)
the shifted Grünwald-Letnikov operator and the weighted and shifted Grünwald-Letnikov operα−2q 2p−α and b1 = 2p−2q . Then the Fourier transform gives ator, respectively. Here a1 = 2p−2q ∞ X (α) −α F Aα ωl ei(l−p)ω fb(ω) h,p f (x); ω = h
(5.97)
l=0
and
α F Bh,p,q f (x); ω ∞ h i X (α) =h−α ωl a1 ei(l−p)hω + b1 ei(l−q)hω fb(ω) l=0
(5.98)
=h−α a1 (1 − eihω )α e−iphω + b1 (1 − eihω )α e−iqhω fb(ω) α −iphω 1 − eihω a1 e + b1 e−iqhω (−iω)α fb(ω). = −ihω Observe that
1 − e−z z
α
α ezr = 1 + r − z + O(|z|2 ). 2
(5.99)
Since a1 and b1 satisfy (
a1 + b1 = 1, a1 (p − α2 ) + b1 (q − α2 ) = 0,
(5.100)
5.3. Grünwald-Letnikov type approximations
201
we have α ( f (x); ω − F RL Dα |ϕ(ω, b h)| = F Bh,p,q −∞,x f (x); ω ≤ Ch2 |iω|α+2 |fb(ω)|. With the condition F
α+2 RL D−∞,x f (x); ω
(5.101)
∈ L1 (R), the inverse Fourier transform yields
α Bh,p,q f (x) − RL Dα f (x) −∞,x Z 1 |ϕ(ω, b h)|dω =|ϕ| ≤ 2π R
1 . ≤Ch2 F RL Dα+2 −∞,x f (x); ω L (R)
(5.102)
In other words, (5.92) holds uniformly for x ∈ R. Equations (5.93) and (5.94) immediately follow from (5.92) and zero extension due to homogeneous boundary conditions. This ends the proof. Remark 5.7. To apply the weighted and shifted formulae (5.93) and (5.94), additional points on the right/left hand of the node xj are needed. The amount of additional points depends on the shifts p and q. It has been pointed out in [165] that we should choose p, q such that |p| ≤ 1, |q| ≤ 1 when adopting these difference schemes to numerically calculate the RiemannLiouville derivative of nonperiodic functions on a bounded interval, to ensure that the values of the required nodes are within the bounded interval. Remark 5.8. [165] When (p, q) = (0, −1), the approximation formula is proved to be unstable for time dependent problems. In effect, there are two applicable pairings (p, q) = (1, 0) and 2+α 2−α (p, q) = (1, −1), and the corresponding weights are ( α2 , 2−α 2 ) and ( 4 , 4 ). In addition, the weighted and shifted Grünwald-Letnikov formula (5.93) is reduced to the centered difference approximation of the 2nd order derivative if the shifts are chosen as (p, q) = (0, 1) or (p, q) = (1, −1) when α = 2. And the centered difference scheme for the 1st order derivative is recovered when (p, q) = (1, 0) and α = 1. The following 3rd order weighted and shifted Grünwald-Letnikov formula can be also obtained in a manner similar to the 2nd order one. 1 Theorem 5.6. [165] Let f ∈ L1 (R), RL Dα+3 −∞,x f (x), and its Fourier transform belong to L (R). Then it holds that
α RL D−∞,x f (x) x=x j ∞ ∞ X λ2 X (α) (α) = λ1 ω f (x − (l − p)h) + ωl f (xj − (l − q)h) j l hα hα l=0 l=0 ∞ λ3 X (α) ωl f (xj − (l − r)h) + O(h3 ). + hα l=0
(5.103)
202
Chapter 5. Numerical Riemann-Liouville differentiation
Here p, q, r are integers and mutually nonequal. The weights are given by 12qr − (6q + 6r + 1)α + 3α2 λ1 = , 12(qr − pq − pr + p2 ) 12pr − (6p + 6r + 1)α + 3α2 , λ2 = 12(pr − pq − qr + q 2 ) 12pq − (6p + 6q + 1)α + 3α2 λ3 = . 12(pq − pr − qr + r2 ) Especially, the equalities α RL Da,x f (x) x=x j j+p j+q X λ2 X (α) (α) = λ1 ω f (x − (l − p)h) + ωl f (xj − (l − q)h) j l hα hα l=0 l=0 j+r λ3 X (α) ωl f (xj − (l − r)h) + O(h3 ) + hα
(5.104)
(5.105)
l=0
and α RL Dx,b f (x) x=x j NX −j+p N −j+q λ2 X (α) (α) = λ 1 ω f (x + (l − p)h) + ωl f (xj + (l − q)h) j l hα hα l=0 l=0 N −j+r λ3 X (α) ωl f (xj + (l − r)h) + O(h3 ) + hα
(5.106)
l=0
are valid for f (a) = 0 and f (b) = 0, respectively. Example 5.7. [165] Consider the fractional differential problem α RL Da,x f (x)
=
Γ(3 + α) 2 x , x ∈ (0, 1), α ∈ (1, 2), 2
(5.107)
with f (0) = 0, f (1) = 1. The exact solution is f (x) = x2+α . Utilizing the 3rd order weighted and shifted Grünwald-Letnikov formula (5.103) with (p, q, r) = (1, 0, −1) to evaluate the solution gives the numerical results in Table 5.1. It follows from Table 5.1 that the 3rd order accuracy of (5.103) is verified. In effect, linear combinations of shifted Grünwald-Letnikov formulae with different shifts and appropriate weights are able to evaluate the Riemann-Liouville derivative with higher-order accuracy. However, more complex weights and a large number of shifted Grünwald-Letnikov operators are needed. There are two ways to improve this situation. One is to introduce compact operators; for instance, the method proposed in [190]. Another is to construct suitable linear combinations of shifted Lubich formulae with higher accuracy. For example, a class of numerical algorithms with 4th order accuracy based on the 2nd order Lubich formula was considered in [23], and will be introduced later.
5.4. Fractional backward difference formulae with modifications
203
Table 5.1. The maximum errors, L2 errors, and their converc Copyright 2015 gence rates for Example 5.7 with α = 1.1. American Mathematical Society α = 1.1 rate kf j − F j kL2
N
kf j − F j kL∞
8
9.48629E-04
∗
5.92003E-04
∗
16
1.19530E-04
2.99
7.51799E-05
2.98
32
1.50130E-05
2.99
9.47995E-06
2.99
64
1.88094E-06
3.00
1.18999E-06
2.99
128
2.35382E-07
3.00
1.49052E-07
3.00
256
2.94392E-08
3.00
1.86501E-08
3.00
rate
5.4 Fractional backward difference formulae with modifications The 1st order Lubich formula (3.39) with α replaced by −α happens to be the classical GrünwaldLetnikov approximation (5.67). Based on this observation, we can derive a series of numerical approximations to the Riemann-Liouville derivative by replacing α > 0 in fractional backward difference formulae with −α. The corresponding accuracy can be verified through the Fourier transform. In this section, we shall introduce fractional backward difference formulae with some of the variants.
5.4.1 Classical Lubich formulae In [35], the Lubich formulae for Riemann-Liouville derivative are studied in detail. Here we present some of the important results on the classical Lubich formulae. Theorem 5.8. [35] If a suitably smooth function f (x) satisfies f (k) (a+) = 0 (k = 0, 1, . . . , p − 1), then the left-sided Riemann-Liouville derivative has the following approximation:
α RL Da,x f (x)
[ x−a h ] 1 X (α) = α $p,l f (x − lh) + O(hp ), h
(5.108)
l=0
(α)
in which h is the step size. For p = 1, 2, 3, 4, 5, 6, the convolution coefficients $p,l are defined by ∞ X (α) Wp(α) (z) = $p,l z l , |z| < 1, (5.109) l=0 (α)
where the corresponding generating functions Wp (z) are defined by (3.37). Remark 5.9. (I) The proof of Theorem 5.8 can be obtained via Fourier transform, so it is omitted here.
204
Chapter 5. Numerical Riemann-Liouville differentiation
(II) The classical Lubich formulae for the right-sided Riemann-Liouville can be similarly obtained as b−x [X h ] 1 (α) α $p,l f (x + lh) + O(hp ), (5.110) RL Dx,b f (x) = α h l=0
provided that f
(k)
(b−) = 0 for k = 0, 1, . . . , p − 1.
Now we list properties of the coefficients (5.108). Theorem 5.9. [91, 99, 110] Let 0 < α < 1 and n ∈ Z+ be suitably large. Then the 1st order (α) coefficients $1,l (l = 0, 1, . . .) satisfy 1+α (α) (α) (α) (α) $1,l−1 , $1,0 = 1, $1,l < 0, l ≥ 1, (5.111) $1,l = 1 − l ∞ X
(α)
(α)
$1,l = 0, lim $1,l = 0,
l=0
and
n−1 X
(α)
$1,l =
l=0
(5.112)
l→+∞
Γ(n − α) n−α = + O(n−1−α ). Γ(1 − α)Γ(n) Γ(1 − α)
(5.113)
(α)
Theorem 5.10. [35] Let 0 < α < 1. The 1st order coefficients $1,l satisfy (α)
B1L (α, l) < |$1,l | < B1R (α, l), l ≥ 3, and S1L (α, l)
$2,l+1 for l ≥ 5. (α)
(1+α)
Theorem 5.13. [35] Let 0 < α < 1. The 2nd order coefficients $2,l and $2,l (α)
B2L (α, l) < |$2,l | < B2R (α, l), l ≥ 4,
satisfy (5.126)
and L
(1+α)
B 2 (1 + α, l) < |$2,l
R
| < B 2 (1 + α, l), l ≥ 4.
(5.127)
Here l ! 2(1+α) α " 1 α(1 − α) 2 3 L 1 + B (α, l) = 2 2 3 2 l ! # l−1 1 α2 22α+1 − 1 − , 3 1 + (α + 1)l α " l ! 3 1 α2α+1 1+ B2R (α, l) = 2 3 (l + 1)α+1 l−1 ! 4(α+1) # α2 (1 − α)2 42α+1 1 2 − 1− , 2 3 l
(5.128)
206
Chapter 5. Numerical Riemann-Liouville differentiation
and 1+α " 2(2+α) l ! 3 1 (1 − α)α(1 + α) 3 L B (1 + α, l) = 1 + 2 2 3 6 l ! l−3 4(2+α) (1 − α)2 α2 (1 + α)2 1 6 + 1− 216 3 l l−1 ! 2+α # 1 3 α(1 + α)2 1 + , − 2 3 3 l 1+α " l ! 3 1 α(α + 1)3α+2 R (1 + α, l) = B 1 + 2 2 3 2(l + 1)α+2 l−3 ! 2 1 α (1 + α)2 32(2+α) + 1− 3 24 (1 + (2 + α)l) l−1 ! 2(2+α) # 2 (1 − α)α(1 + α) 1 1 3 − + . 6 3 3 l−1
(5.129)
Theorem 5.14. [35] Let 0 < α < 1. Then the following inequality holds for the coefficients (α) $p,l : ∞ X
(α)
$p,l cos(lθ) ≥ 0, θ ∈ [−π, π], p = 2, 3, 5, 6.
(5.130)
l=0
For the 4th order coefficients, it holds that ∞ X
(α)
$4,l cos(lθ) ≥ 0, θ ∈ [−π, π],
(5.131)
l=0
if 0 < α ≤
π
π−arccos
√ 191 6 1 5 +2 arctan 317
≈ 0.8439.
Remark 5.10. The coefficients of high-order schemes (till 10th order) for Riemann-Liouville derivatives were computed in [171]. A conjecture on coefficients of the 3rd, 4th, and 5th order schemes was proposed by Li and Ding and was rephrased on page 80 of [99]: (α)
(α)
(α)
(α)
(α)
(α)
(α)
(α)
(α)
(α)
(α)
(α)
(I) If 0 < α < 1, then $3,l ≤ $3,l+1 for l ≥ 4, $4,l ≤ $4,l+1 for l ≥ 7, and $5,l ≤ $5,l+1 for l ≥ 12. (II) If 1 < α < 2, then $3,l ≥ $3,l+1 for l ≥ 7, $4,l ≥ $4,l+1 for l ≥ 12, and $5,l ≥ $5,l+1 for l ≥ 16. (α)
Recently, the above assertion on $3,l with α ∈ (0, 1) was proved in [95], as presented in the following theorem.
5.4. Fractional backward difference formulae with modifications
207 (α)
Theorem 5.15. Let 0 < α < 1. Then the 3rd order coefficients $3,l satisfy the following relationships. (I) α 11 (α) $3,0 = > 0, 6 α 18α 11 (α) $3,1 = − < 0, 11 6 α 9α(18α − 7) 11 7 (α) $ = ≤ 0 for α ∈ 0, , 3,2 121 6 18 7 (α) , 1 , $ > 0 for α ∈ 3,2 18 α 2α(486α2 − 567α + 202) 11 (α) $ = − < 0, 3,3 1331 6 α 9α(α − 1)(972α2 − 1296α + 761) 11 (α) < 0, $3,4 = 29282 6 $(α) < 0. 3,l
(5.132)
(II) (α)
(α)
$3,l ≤ $3,l+1 for l ≥ 4.
(5.133)
Proof. (I) can be readily obtained by direct calculations. The proof of part (II) is presented as follows. Denote 3 1 (2α − l + 2), c3,l = − (3α − l + 3), 2 3
c1,l = −3(α − l + 1), c2,l =
(5.134)
and (α)
Bl =
$3,l−1 (α) $3,l
, m(l) =
l+2 . l − 3α − 1
(5.135)
It follows from (5.132) that 6 (α) (α) (α) (α) $ = c $ + c $ + c $ 1,l 2,l 3,l 3,l 3,l−1 3,l−2 3,l−3 , 11l 1 6 = (c1,l + c2,l Bl−1 + c3,l Bl−1 Bl−2 ) . Bl 11l
(5.136)
We aim to prove that for l ≥ 5, 1 ≤ Bl ≤ m(l) by mathematical induction. For l = 5, 6, . . . , 17, we display the sketch maps of Bl and m(l) in Figures 5.1–5.4. One (α) can find from this computer-assisted proof that Bl ’s satisfy 1 ≤ Bl ≤ m(l), and $3,l < 0 and (α)
(α)
$3,l ≤ $3,l+1 when l = 5, 6, . . . , 17.
208
Chapter 5. Numerical Riemann-Liouville differentiation
Figure 5.1. Plots of Bl and m(l) with l = 5, 6, 7, 8. Reprinted by permission from Springer Nature: [95].
Figure 5.2. Plots of Bl and m(l) with l = 8, 9, 10, 11. Reprinted by permission from Springer Nature: [95]. Suppose that 1 ≤ Bl ≤ m(l) for l = k − 2, k − 1, k ≥ 18. When l = k, Bk is computed as 6 1 = (c1,k + c2,k Bk−1 + c3,k Bk−1 Bk−2 ) B 11k k 6 = [c1,k + Bk−1 (c2,k + c3,k Bk−2 )] 11k 6 (5.137) ≤ [c1,k + Bk−1 (c2,k + c3,k m(k − 2))] 11k 6 ≤ [c1,k + c2,k + c3,k m(k − 2)] 11k ≤1,
5.4. Fractional backward difference formulae with modifications
209
Figure 5.3. Plots of Bl and m(l) with l = 11, 12, 13, 14. Reprinted by permission from Springer Nature: [95].
Figure 5.4. Plots of Bl and m(l) with l = 14, 15, 16, 17. Reprinted by permission from Springer Nature: [95]. k since m(k − 2) = k−3α−3 < 2 for k ≥ 18, and c2,k + 2c3,k = α − 65 k + 1 < 0. On the other hand, 6 1 = [c1,k + Bk−1 (c2,k + c3,k Bk−2 )] Bk 11k 6 ≥ [c1,k + Bk−1 (c2,k + c3,k )] 11k 6 ≥ [c1,k + m(k − 1)(c2,k + c3,k )] , 11k
(5.138)
due to c2,k + c3,k = 2α − 67 k + 2 < 0 for k ≥ 18. Multiplying m(k) on both sides of the above
210
Chapter 5. Numerical Riemann-Liouville differentiation
inequality gives 6 m(k) ≥ [c1,k + m(k − 1)(c2,k + c3,k )] m(k) Bk 11k 6 = I1 , 11k
(5.139)
where I1 = [c1,k + m(k − 1)(c2,k + c3,k )] m(k). Substitution the expressions of c1,k , c2,k , c3,k and m(k), m(k − 1) into I1 yields (k + 2)(3k − 3α − 3) (k + 1)(k + 2)(2α − 76 k + 2) + I = 1 k − 3α − 1 (k − 3α − 1)(k − 3α − 2)
=
11 3 6 k
2 − 10αk 2 − 92 k 2 + 9α2 k − 3αk − 25 3 k + 18α + 34α + 16 . (k − 3α − 1)(k − 3α − 2)
(5.140)
Therefore, 39 2 2 (1 + α)k 2 − 15 11k 2 α + 2 α + 12 k + 18α + 34α + 16 = . I1 − 6 (k − 3α − 1)(k − 3α − 2)
(5.141)
Define f (k) as 2
f (k) = (1 + α)k −
15 2 39 α + α + 12 k + 18α2 + 34α + 16. 2 2
(5.142)
By simple calculation, we have 15 2 39 0 f (k) =2(1 + α)k − α + α + 12 2 2 15 2 39 ≥2(1 + α) · 18 − α + α + 12 2 2 1 = (−15α2 + 33α + 48) > 0 2
(5.143)
f (k) ≥ f (18) = −117α2 + 7α + 124 > 0
(5.144)
for any α ∈ (0, 1). Hence,
for arbitrary α ∈ (0, 1). In view of equations (5.139) and (5.141), and m(k) > 0, we have m(k) > 1, Bk
(5.145)
1 1 > . Bk m(k)
(5.146)
i.e.,
5.4. Fractional backward difference formulae with modifications
211
Combining the above equation with equation (5.137), we obtain 1 ≤ Bl ≤ m(l) for l ≥ 5, ∀α ∈ (0, 1), (α)
(5.147)
(α)
which yields $3,l−1 ≤ $3,l for l ≥ 5. (α)
(α)
(α)
It is also evident that the sign of $3,l is same as that of $3,l−1 . Since $3,4 < 0, then (α)
$3,l < 0 for l ≥ 4. This completes the proof. The classical Lubich formulae have attracted numerous researchers since they emerged. There is a variety of literature studying, developing, and adopting Lubich formulae, such as [28, 34, 178]. As mentioned before, the classical 1st order Lubich formula coincides with the classical Grünwald-Letnikov formula (5.67), which is unstable for spatial fractional differential equations with 1 < α < 2. Thus, the approximation in Theorem 5.8 may not be suitable for the discretization of fractional differential equations with α ∈ (1, 2), either.
5.4.2 Shifted fractional backward difference formulae To offset the (potential) instability of the classical Lubich formula (5.108), we often adopt the following shifted formula:
α RL Da,x f (x)
1 hα
=
n+1 P
$p,l f (xn−l+1 ) + O(h),
l=0
(5.148)
nh = x − a, p = 1, 2, . . . , 6.
The improvement on the stability is at the cost of convergence, and the shifted Lubich formula has only 1st order accuracy [38], which can be verified through the Fourier transform if certain assumptions are satisfied. In this case, the modified generating functions are needed to maintain high-order accuracies. Now we present shifted fractional backward difference formulae with higher-order accuracy by virtue of modifying the generating functions. Theorem 5.16. [38] Suppose that f (x) ∈ C [α]+3 (R) and all derivatives of f (x) up to order [α] + 4 belong to L1 (R). Let L
B2α f (x) =
∞ 1 X (α) k2,l f (x − (l − 1)h) ; hα
(5.149)
l=0
then α RL D−∞,x f (x)
= L B2α f (x) + O(h2 ), h → 0,
(5.150)
(α)
holds uniformly for x ∈ R. Here k2,l (l = 0, 1, . . .) are the coefficients of the generating α f2 (z) = 3α−2 − 2(α−1) z + α−2 z 2 , i.e., function W 2α α 2α f2 (z) = W
3α − 2 2(α − 1) α−2 2 − z+ z 2α α 2α
α =
∞ X l=0
(α)
k2,l z l , |z| < 1.
(5.151)
212
Chapter 5. Numerical Riemann-Liouville differentiation
Proof. Taking the Fourier transform yields F L B2α f (x); ω ∞ 1 X (α) i(l−1)hω b k e f (ω) = α h l=0 2,l ∞ 1 −ihω b X (α) ilhω e f (ω) k2,l e = hα l=0 =(−iω)α χ(−ihω)fb(ω),
(5.152)
where χ(z) =
ez f −z 2α2 − 6α + 3 2 z + O(|z|3 ). W (e ) = 1 − 2 zα 6α
(5.153)
So there exists a constant c1 > 0 satisfying |χ(−ihω) − 1| ≤ c1 |ω|2 h2 .
(5.154)
F L B2α f (x); ω =(−iω)α fb(ω) + (−iω)α [χ(−ihω) − 1] fb(ω) =F RL Dα b ω), −∞,x f (x); ω + ϕ(h,
(5.155)
Furthermore,
where ϕ(h, b ω) = (−iω)α [χ(−ihω) − 1] fb(ω). It follows from (5.154) that |ϕ(h, b ω)| ≤ c1 |ω|α+2 h2 |fb(ω)|.
(5.156)
Note that f (x) ∈ C [α]+3 (R) and all the derivatives of f (x) up to order [α] + 4 belong to L (R). There exists a positive constant c2 such that 1
b f (ω) ≤ c2 (1 + |ω|)−([α]+4) .
(5.157)
Taking the inverse Fourier transform on ϕ(h, b ω) yields |ϕ(h, x)| Z ∞ Z ∞ 1 1 −iωx e ϕ(h, b ω)dω ≤ |ϕ(h, b ω)| dω = 2πi −∞ 2π −∞ Z ∞ c c ≤ 1 2 (1 + |ω|)α−[α]−2 dω h2 = ch2 , 2π −∞ in which c =
c1 c2 π([α]+1−α) .
(5.158)
Using again the inverse Fourier transform leads to α RL D−∞,x f (x)
The proof is thus completed.
= L B2α f (x) + O(h2 ).
(5.159)
5.4. Fractional backward difference formulae with modifications
213
Remark 5.11. For the right-sided Riemann-Liouville derivative, the approximation α RL Dx,+∞ f (x)
= R B2α f (x) + O(h2 )
(5.160)
holds under the conditions of Theorem 5.16. Here the right difference operator R B2α is given by R
B2α f (x) =
∞ 1 X (α) k2,l f (x + (l − 1)h) . hα
(5.161)
l=0
Remark 5.12. Apart from using the fast Fourier transform technique to calculate the coefficients, we can also compute them by (α) k2,l
(α)
with $1,l = (−1)l
=
α l
3α − 2 2α
α X l i=0
α−2 3α − 2
i
(α)
(α)
$1,i $1,l−i , l = 0, 1, . . . ,
(5.162)
. This follows from the relation
f2 (z) W α α 3α − 2 α−2 α (1 − z) 1 − z = 2α 3α − 2 " #" ∞ s # ∞ α X X 3α − 2 α α − 2 α s = (−1)k zk − z k s 2α 3α − 2 s=0 k=0 α X s ∞ X ∞ α−2 3α − 2 α α k+s k = (−1) − z 2α 3α − 2 k s k=0 s=0 " l α X l−k # ∞ X α − 2 α α 3α − 2 k = (−1) − zl k l − k 2α 3α − 2 l=0 k=0 " α X k # ∞ l X 3α − 2 α−2 α α l = (−1) zl 2α 3α − 2 k l−k l=0 k=0 " # k ∞ l α X X 3α − 2 α − 2 (α) (α) $1,k $1,l−k z l , = 2α 3α − 2 l=0
(5.163)
k=0
which also gives the recursive relations (α) k2,0 = (α) k2,1 =
3α−2 α , 2α 4α(1−α) (α) 3α−2 k2,0 ,
h 1 (α) (α) k2,l = 4(1 − α)(α − l + 1)k2,l−1 l(3α − 2) i (α) +(α − 2)(2α − l + 2)k2,l−2 , l ≥ 2, with the help of automatic differentiation techniques [143].
(5.164)
214
Chapter 5. Numerical Riemann-Liouville differentiation
The coefficients in (5.164) have the following properties. (α)
Theorem 5.17. [38] Let 1 < α < 2. The coefficients k2,l (l = 0, 1, . . .) satisfy the following equalities: α 3α − 2 (α) > 0, k = 2,0 2α 4α(1 − α) (α) (α) k2,1 = k < 0, 3α − 2 2,0 (5.165) (α) α(8α3 − 21α2 + 16α − 4) (α) k2,0 , k2,2 = (3α − 2)2 (α) (α) k2,2 < 0 if α ∈ (1, α∗ ), k2,2 ≥ 0 if α ∈ [α∗ , 2), with 7 α = + 8 ∗
p 3
√ 621 + 48 87 19 + p √ ≈ 1.5333. 3 24 621 + 48 87
(5.166)
In addition, (α) k2,l ≥ 0, if l ≥ 3, sin(πα)Γ(α + 1) −α−1 (α) k2,l ∼ − l , l → ∞, π (α) k2,l → 0, l → ∞, and
∞ X
(α)
k2,l = 0.
(5.167)
(5.168)
l=0
By virtue of the Fourier transform, we have the following asymptotic expansion for the operator L B2α f (x). Theorem 5.18. Suppose that f (x) ∈ C [α]+n+1 (R) and all the derivatives of f (x) up to order [α] + n + 2 belong to L1 (R). Then L
B2α f (x) = RL Dα −∞,x f (x) +
n−1 X
l n (γlα RL Dα+l −∞,x f (x))h + O(h ), n ≥ 2.
(5.169)
l=1 z f2 (e−z ) = 1 + P∞ γ α z l , in which the Here the coefficients γlα (l = 1, 2, . . .) satisfy zeα W l=1 l 2 3 2 +12α−4 coefficients of the first three terms are γ1α = 0, γ2α = − 2α −6α+3 , and γ3α = 3α −11α . 6α 12α2
For shifted fractional backward difference formulae with arbitrary accuracy, we give the following two assertions that can be obtained by the same reasoning as in Theorem 5.16. Theorem 5.19. Let p ≥ 3, f (x) ∈ C [α]+p+1 (R), and all the derivatives of f (x) up to order [α] + p + 2 belong to L1 (R). Set L
Bpα f (x) =
∞ 1 X (α) kp,l f (x − (l − 1)h) hα l=0
(5.170)
5.4. Fractional backward difference formulae with modifications
and R
Bpα f (x) =
215
∞ 1 X (α) kp,l f (x + (l − 1)h) . hα
(5.171)
l=0
Then α RL D−∞,x f (x)
= L Bpα f (x) + O(hp ), p ≥ 3,
(5.172)
α RL Dx,+∞ f (x)
= R Bpα f (x) + O(hp ), p ≥ 3.
(5.173)
and (α)
Here the generating functions of coefficients kp,l (l = 0, 1, . . .) with p ≥ 3 are p
fp (z) = W
(α)
X λk−1,k−1 α−2 (1 − z)2 + (1 − z)k (1 − z) + 2α α
!α ,
(5.174)
k=3
i.e., fp (z) = W
∞ X
(α)
kp,l z l , |z| < 1, p ≥ 3,
(5.175)
l=0 (α)
in which the parameters λk−1,k−1 (k = 3, 4, . . .) can be determined by the following equation: z
fk (e−z ) e = 1 − W zα
∞ X
(α)
λk,l z l ,
k = 2, 3, . . . .
(5.176)
l=k (α)
In the following, we derive recursive formulae for the coefficients kp,l with p = 3 and p = 4. The cases with p ≥ 5 can be similarly obtained so are omitted here. (α) (I) When p = 3, the generating function for coefficients k3,l (l = 0, 1, 2, . . .) reads as f3 (z) = a31 + a32 z + a33 z 2 + a34 z 3 W
α
=
∞ X
(α)
k3,l z l ,
(5.177)
l=0
where
11α2 − 12α + 3 −6α2 + 10α − 3 , a = , a31 = 32 6α2 2α2 2 2 a33 = 3α − 8α + 3 , a34 = −2α + 6α − 3 . 2α2 6α2 This generating function can be also written as f3 (z) = a31 + a32 z + a33 z 2 + a34 z 3 α W α α = aα 1 + b31 z + b32 z 2 31 (1 − z) s ∞ X α b32 s α α z = a31 (1 − z) (b31 z) 1 + b31 s s=0
=
aα 31 (1
− z)
α
∞ X α s=0
t s X s b32 (b31 z) z s t b31 t=0 s
[ 2s ] ∞ l X s−2t t s+t X X (−1) (s − t)!b31 b32 (α) (α) = aα $1,l−s $1,s−t z l , 31 t!(s − 2t)! s=0 t=0 l=0
(5.178)
(5.179)
216
Chapter 5. Numerical Riemann-Liouville differentiation
in which b31 =
−7α2 + 18α − 6 2α2 − 6α + 3 , b = . 32 11α2 − 12α + 3 11α2 − 12α + 3
(5.180)
Therefore, we have (α) k3,l
= aα 31
[ 2s ] l X X
(α)
(α)
P (α, s, t)$1,l−s $1,s−t , l = 0, 1, 2, . . . ,
(5.181)
s=0 t=0
where P (α, s, t) =
(−1)s+t (s − t)! s−2t t b31 b32 . t!(s − 2t)!
(5.182) (α)
In addition, we can get the following recursion relations by using the expressions of k3,l in (5.181) and automatic differentiation techniques: α 11α2 − 12α + 3 (α) k = , 3,0 6α2 (α) 3α(6α2 − 10α + 3) (α) k = − k , 3,1 11α2 − 12α + 3 3,0 3α(108α5 − 402α2 + 520α3 − 312α2 + 87α − 9) (α) (5.183) (α) k = k3,0 , 3,2 2(11α2 − 12α + 3)2 1 h (α) (α) (α) a32 (α − l + 1)k3,l−1 + a33 (2α − l + 2)k3,l−2 k3,l = a l 31 i (α) , l ≥ 3. +a (3α − l + 3)k 34
3,l−3
(α)
(II) When p = 4, the generating function for coefficients k4,l (l = 0, 1, 2, . . .) reads as f4 (z) = a41 + a42 z + a43 z 2 + a44 z 3 + a45 z 4 W
α
=
∞ X
(α)
k4,l z l ,
(5.184)
l=0
in which −24α3 + 52α2 − 27α + 4 25α3 − 35α2 + 15α − 2 a = , a = , 41 42 12α3 6α3 6α3 − 19α2 + 12α − 2 −8α3 + 28α2 − 21α + 4 a = , a = , 43 44 2α3 6α3 3 2 a45 = 3α − 11α + 9α − 2 . 12α3
(5.185)
Similarly, we have (α)
k4,l = aα 41
2s l [X 3 ] X
[ 2t ] X
(α)
(α)
P (α, s, t, r)$1,l−s $1,s−t ,
(5.186)
s=0 t=0 r=max{0,2t−s}
where P (α, s, t, r) =
(−1)s+t (s − t)! t−3r r bs+r−2t b42 b43 r!(t − 2r)!(s + r − 2t)! 41
(5.187)
5.4. Fractional backward difference formulae with modifications
217
and
−23α3 + 69α2 − 39α + 6 , b = 41 25α3 − 35α2 + 15α − 2 13α3 − 45α2 + 33α − 6 , b42 = 25α3 − 35α2 + 15α − 2 3 2 b43 = −3α + 11α − 9α + 2 . 25α3 − 35α2 + 15α − 2 The corresponding recursion formula is given as α 25α3 − 35α2 + 15α − 2 (α) , k4,0 = 12α3 2α(24α3 − 52α2 + 27α − 4) (α) (α) k = − k4,0 , 4,1 25α3 − 35α2 + 15α − 2 (α) 2α k4,2 = (576α7 − 2622α6 + 4441α5 3 − 35α2 + 15α − 2)2 (25α (α) − 3835α4 + 1844α3 − 497α2 + 70α − 4)k4,0 , 2α (α) k4,3 = − 27648α11 − 197856α10 3 − 35α2 + 15α − 2)3 3(25α + 591000α9 − 995240α8 + 1067901α7 − 775354α6 + 390051α5 (α) −135738α4 + 31923α3 − 4820α2 + 420α − 16 k4,0 , 1 h (α) (α) (α) k = a42 (α − l + 1)k4,l−1 + a43 (2α − l + 2)k4,l−2 4,l a l 41 i (α) (α) +a44 (3α − l + 3)k4,l−3 + a45 (4α − l + 4)k4,l−4 , l ≥ 4.
(5.188)
(5.189)
(α)
Properties of the coefficients kp,l , l ≥ 0 with p ≥ 3, are not available yet. Remark 5.13. Let f (x) be defined on [a, b] satisfying the homogeneous conditions f (n) (a) = f (n) (b) = 0, (n = 0, 1, . . . , p − 1). If f (x) ∈ C [α]+p+1 ([a, b]) and all its derivatives up to order [α] + p + 2 belong to L1 ([a, b]), then we have, through zero extension, 1 α RL Da,x f (x) = α h
[ x−a h ]+1 X
(α)
(5.190)
(α)
(5.191)
kp,l f (x − (l − 1)h) + O(hp ), p ≥ 2,
l=0
and α RL Dx,b f (x)
1 = α h
[ b−x h ]+1 X
kp,l f (x + (l − 1)h) + O(hp ), p ≥ 2.
l=0
(α)
Here the coefficients kp,l are generated by the corresponding generating functions in (5.151) when p = 2 and (5.174) when p ≥ 3. Apart from the case with the Riemann-Liouville derivative, the shifted fractional backward difference formulae are also modified for the evaluation of Riemann-Liouville tempered fractional derivatives in [39]. The above discussion on the shifted fractional backward difference
218
Chapter 5. Numerical Riemann-Liouville differentiation
formulae are all with integer shifts. In effect, we can also choose noninteger shifts in applications. For example, the 2nd order approximation to the Riemann-Liouville derivative proposed in [42] can be viewed as a shifted Lubich formula with noninteger shifts. Here we introduce this approximation in detail. Theorem 5.20. [42] Define the space Z Ln+α (R) = f ∈ L1 (R) : (1 + |ω|)n+α |fb(ω)|dω < ∞ .
(5.192)
R
If f ∈ L2+α (R), then one has α RL D−∞,x+ h f
x+
2
h 2
= h−α
∞ X
(α)
$l f (x − lh) + O(h2 )
(5.193)
l=0
(α)
as h → 0. Here the coefficients $l are the expansion coefficients of the generating function α 3α + 1 2α + 1 α+1 2 − z+ z . (5.194) G(z) = 2α α 2α Proof. The Fourier transform yields that iωh h α F RL D−∞,x+ h f x + ; ω = (−iω)α e− 2 fb(ω) 2 2 and
( F
∞ X
) (α) $l f (x
− lh); ω
=
l=0
∞ X
(α) $l eiωlh fb(ω) = G(eiωh )fb(ω).
(5.195)
(5.196)
l=0
It follows that ( ) ∞ X h (α) α −α F RL D−∞,x+ h f x + −h $l f (x − lh); ω 2 2 l=0 =(−iω)α e− iωh b(ω) − h−α G(eiωh )fb(ω) 2 f iωh iωh G(e ) αb − iωh 2 2 =(−iω) f (ω)e 1 − e (−iωh)α iωh =(−iω)α fb(ω)e− 2 [1 − S(iωh)] ,
(5.197)
z
where S(z) = Note that
e 2 G(ez ) (−z)α . z
α
G(e ) = (−z)
8α2 + 9α + 3 2 1 1− z− z + ··· 2 24α
(5.198)
and z S(z) = (−z)−α e 2 G(ez ) z z2 1 8α2 + 9α + 3 2 = 1+ + + ··· 1− z− z + ··· 2 8 2 24α 8α2 + 12α + 3 2 =1− z + O(|z|3 ). 24α
(5.199)
5.4. Fractional backward difference formulae with modifications
219
There exists a constant c1 > 0 such that |1 − S(iωh)| ≤ c1 |ωh|2 . Therefore, it holds that (∞ ) X (α) h α F RL D−∞,x+ h f x + ;ω = F $l f (x − lh); ω + ϕ(h, b ω) (5.200) 2 2 l=0
with ϕ(h, b ω) = (−iω)α fb(ω)e−
iωh 2
[1 − S(iωh)] .
The inverse Fourier transform yields |ϕ(h, ω)| Z 1 = e−iωh ϕ(h, b ω)dω 2πi R Z c h2 ≤ 1 (1 + |ω|)2+α |fb(ω)|dω. 2π R
(5.201)
(5.202)
This ends the proof. Remark 5.14. If f (x) is suitably smooth and has compact support for x ∈ (a, b), then the approximation (5.193) is reduced to α RL Da,x f (x) x=x j+ 1 2 j (5.203) X (α) −α 2 $l f (xj − lh) + O(h ), j = 0, 1, . . . , N − 1. =h l=0
Here h =
b−a N
and xj = a + jh. (α)
Remark 5.15. The coefficients $l (α) $l
with (α) gm
=
in (5.193) can be expressed as
3α + 1 2α
α X m l α+1 (α) (α) gm gl−m 3α + 1 m=0
α Γ(1 + α) . = (−1) = (−1)m m Γ(m + 1)Γ(1 + α − m) m
Furthermore, for α ∈ (0, 1), one has the following recursive formula: 1−α 4 − 3α (1−α) $ = , 0 2(1 − α) 2(α − 1)(3 − 2α) (1−α) $1(1−α) = , $0 4 − 3α h 1 (1−α) (1−α) $ = 2(l + α − 2)(3 − 2α)$l−1 l (4 − 3α)l i (1−α) +(2 − α)(4 − 2α − l)$l−2 , l ≥ 2. For properties of the coefficients in (5.193), we present the following result.
(5.204)
(5.205)
(5.206)
220
Chapter 5. Numerical Riemann-Liouville differentiation
Theorem 5.21. [42] Let 0 < α < 1. (1−α)
are nonpositive if l ≥ 5;
(1−α)
are increasing with respect to l ≥ 7.
(I) The coefficients $l (II) The coefficient $l
5.4.3 Weighted and shifted Lubich formulae Enlightened by the idea of the weighted and shifted Grünwald-Letnikov approximations, the weighted and shifted Lubich formulae were proposed in [23]. Note that the generating functions for the coefficients of the `th order (` ∈ Z+ , ` ≤ 6) fractional backward difference formulae are also in the form !α ` X 1 k α (1 − z) . (5.207) W (z) = k k=1
When ` = 2, it holds that α 3 1 2 − 2z + z 2 2 α α 3 1 α = (1 − z) 1 − z 2 3 m ∞ ∞ α X X 3 α m 1 n α n (−1) = z z − n 2 3 m n=0 m=0 " ∞ α X m # ∞ X 3 α 1 α = − (−1)n z m+n n 2 3 m n=0 m=0 " k α X m # ∞ X 3 α 1 α k (−1) − zk = 2 k − m 3 m k=0 m=0 " # α k ∞ X X α α 3 (−1)k 3−m zk = k − m m 2 m=0 k=0 ∞ X (α) $2,k z k , =
(5.208)
k=0
with
α X k 3 α α (α) k −m $2,k =(−1) 3 2 k − m m m=0 k α X 3 α α = 3−m $1,m $1,k−m 2 m=0
α k X k 3 1 α α = 3m $1,m $1,k−m 2 3 m=0 α k 3 1 α = 2 F1 (−k, −α; −k + α + 1; 3). 2 3 k
(5.209)
5.4. Fractional backward difference formulae with modifications α k
Here $1,k = (−1)k by
221
, and 2 F1 (a, b; c; z) is the Gauss hypergeometric function [156] defined 2 F1 (a, b; c; z)
=
∞ X (a)k (b)k z k (c)k k!
(5.210)
k=0
with (z)n =
Γ(z+n) Γ(z) .
Define the operator α L Ap
=
∞ 1 X (α) $2,k f (x − (k − p)h), hα
(5.211)
k=0
(α)
where $2,k is defined by (5.209) and the shift p is an integer. Then we have the following assertion, which can be verified through the Fourier transform. α+2 Lemma 5.22. [23] Let f (x), RL Dα+1 −∞,x f (x) (or RL D−∞,x f (x)) with α ∈ (1, 2), and their 1 Fourier transforms belong to L (R) when p 6= 0 (or p = 0). Then α RL D−∞,x f (x)
= L Aα p f (x) + O(h), p 6= 0,
α RL D−∞,x f (x)
2 = L Aα p f (x) + O(h ), p = 0,
(
(5.212)
holds uniformly for x ∈ R. Under reasonable regularity assumptions posed on f (x), the higher-order schemes proposed in [23] can be derived, which are given as follows. Theorem 5.23. Let f (x), RL Dα+2 −∞,x f (x) with α ∈ (1, 2), and their Fourier transformations belong to L1 (R). Then the 2nd order approximation is given by (
α RL D−∞,x f (x) α 2L Ap,q f (x)
2 = 2L Aα p,q f (x) + O(h ),
(5.213)
α = Wp L Aα p f (x) + Wq L Aq f (x),
α where L Aα p , L Aq are defined in (5.211), Wp =
q q−p ,
Wq =
p p−q ,
p 6= q, and p, q are integers.
Theorem 5.24. Let f (x), RL Dα+3 −∞,x f (x) with α ∈ (1, 2), and their Fourier transformations belong to L1 (R). Then the 3rd order approximation is in the form (
α RL D−∞,x f (x)
3 = 3L Aα p,q,r,s f (x) + O(h ),
α 3L Ap,q,r,s f (x)
α = Wp,q 2L Aα p,q f (x) + Wr,s 2L Ar,s f (x),
α where 2L Aα p,q and 2L Ar,s are defined in (5.213), Wp,q = and p, q, r, s are integers.
3rs+2α 3(rs−pq) ,
Wr,s =
3pq+2α 3(pq−rs) ,
(5.214) rs 6= pq,
Furthermore, we have the 4th order weighted and shifted Lubich formula in the following theorem, which can be readily verified through the Fourier transform. Theorem 5.25. Let f (x), RL Dα+4 −∞,x f (x) with α ∈ (1, 2) and their Fourier transformations belong to L1 (R). Denote that α 4L Ap,q,r,s,p,¯ ¯ q ,¯ r ,¯ s f (x)
α = Wp,q,r,s 3L Aα ¯ q ,¯ r ,¯ s 3L Ap,¯ p,q,r,s f (x) + Wp,¯ ¯ q ,¯ r ,¯ s f (x),
(5.215)
222
Chapter 5. Numerical Riemann-Liouville differentiation
α where 3L Aα p,q,r,s and 3L Ap,¯ ¯ q ,¯ r ,¯ s are defined in (5.214). And
with
Wp,q,r,s =
ap,q,r,s¯bp,¯ ¯ q ,¯ r ,¯ s , ¯ ap,q,r,s bp,¯ ¯p,¯ ¯ q ,¯ r ,¯ s−a ¯ q ,¯ r ,¯ s bp,q,r,s
(5.216)
Wp,¯ ¯ q ,¯ r ,¯ s =
a ¯p,¯ ¯ q ,¯ r ,¯ s bp,q,r,s , a ¯p,¯ b − ap,q,r,s¯bp,¯ ¯ q ,¯ r ,¯ s p,q,r,s ¯ q ,¯ r ,¯ s
(5.217)
ap,q,r,s =rs − pq, a ¯p,¯ rs¯ − p¯q¯, ¯ q ,¯ r ,¯ s =¯ bp,q,r,s =6pqrs(r + s − p − q) + 4α [rs(r + s) − pq(p + q)] + 9α(rs − pq), ¯bp,¯ pq¯r¯s¯(¯ r + s¯ − p¯ − q¯) ¯ q ,¯ r ,¯ s =6¯ + 4α [¯ rs¯(¯ r + s¯) − p¯q¯(¯ p + q¯)] + 9α(¯ rs¯ − p¯q¯), ap,q,r,s¯bp,¯ ¯p,¯ ¯ q ,¯ r ,¯ s 6= a ¯ q ,¯ r ,¯ s bp,q,r,s ,
(5.218)
and p, q, r, s; p¯, q¯, r¯, s¯ are integers. Then α RL D−∞,x f (x)
4 = 4L Aα p,q,r,s,p,¯ ¯ q ,¯ r ,¯ s f (x) + O(h ).
(5.219)
All the above schemes are applicable to the case with bounded domain (a, b), by virtue of performing zero extensions, provided that f (x) satisfies suitably homogeneous boundary conditions. Let f˜(x) be the zero extended function of f (x) from (a, b). Whenever f˜(x) satisfies the requirements of the above corresponding theorems, we denote 1 eα L Ap f (x) = α h
[ x−a h ]+p X
(α) $2,k f˜(x − (k − p)h).
Then it follows from the above analysis that eα RL Dα a,x f (x) = L Ap f (x) + O(h), p 6= 0,
α RL Da,x f (x)
(5.220)
k=0
2 = L Aeα p f (x) + O(h ), p = 0.
(5.221)
Furthermore, α RL Da,x f (x) α RL Da,x f (x)
2 = 2L Aeα p,q f (x) + O(h ),
(5.222)
3 = 3L Aeα p,q,r,s f (x) + O(h ),
(5.223)
and α RL D−∞,x f (x)
with
4 = 4L Aα p,q,r,s,p,¯ ¯ q ,¯ r ,¯ s f (x) + O(h ),
eα 2L Ap,q f (x) eα 3L Ap,q,r,s f (x)
(5.224)
eα =Wp L Aeα p f (x) + Wq L Aq f (x), eα =Wp,q 2L Aeα p,q f (x) + Wr,s 2L Ar,s f (x),
α α 4L Ap,q,r,s,p,¯ ¯ q ,¯ r ,¯ s f (x) =Wp,q,r,s 3L Ap,q,r,s f (x) α + Wp,¯ ¯ q ,¯ r ,¯ s 3L Ap,¯ ¯ q ,¯ r ,¯ s f (x).
(5.225)
5.4. Fractional backward difference formulae with modifications
In particular, with M = max{|p|, |q|, |r|, |s|, |¯ p|, |¯ q |, |¯ r|, |¯ s|} and N = a + jh, j = −M, . . . , 0, 1, . . . , N − 1, N, . . . , N + M . Then we have
223 b−a h ,
we denote xj =
f (xj ) = 0 for j = −M, −M + 1, . . . , 0, and j = N, N + 1, . . . , N + M.
(5.226)
In this case, it holds that
j+p 1 X (α) α e $2,k f (xj−k+p ) A f (x ) = L p j hα k=0
1 = α h
j+M X k=M −p
j+M 1 X (α) (α) $2,k+p−M f (xj−k+M ) = α $2,k+p−M f (xj−k+M ), h
(5.227)
k=0
(α)
in which $2,k+p−M = 0 as k + p − M < 0, and p is an integer. Hence, i h α eα + O(h) A f (x) D f (x) = L RL p a,x x=xj x=xj j+M 1 X (α) = $ f (xj−k+M ) + O(h), p 6= 0, hα k=0 2,k+p−M h i α eα + O(h2 ) RL Da,x f (x) x=x = L Ap f (x) j x=x j j+M 1 X (α) = $2,k+p−M f (xj−k+M ) + O(h2 ), p = 0, hα
(5.228)
k=0
i h α eα + O(h2 ) RL Da,x f (x) x=x = 2L Ap,q f (x) j x=xj j+M 1 X (α) (α) = Wp $2,k+p−M + Wq $2,k+q−M f (xj−k+M ) + O(h2 ), α h
(5.229)
h i α eα + O(h3 ) RL Da,x f (x) x=x = 3L Ap,q,r,s f (x) j x=x j j+M 1 X (α) (α) Wp,q Wp $2,k+p−M + Wp,q Wq $2,k+q−M = α h k=0 (α) 3 + Wr,s Wr $(α) + W W $ r,s s 2,k+r−M 2,k+s−M f (xj−k+M ) + O(h ),
(5.230)
k=0
and h i α α e D f (x) = A f (x) + O(h4 ) RL 4L a,x p,q,r,s,p,¯ ¯ q ,¯ r ,¯ s x=xj x=xj j+M 1 X (α) ϕk f (xj−k+M ) + O(h4 ), = hα k=0
(5.231)
224
Chapter 5. Numerical Riemann-Liouville differentiation
where (α) (α) ϕα k =Wp,q,r,s Wp,q Wp $2,k+p−M + Wp,q,r,s Wp,q Wq $2,k+q−M (α) (α) + Wp,q,r,s Wr,s Wr $2,k+r−M + Wp,q,r,s Wr,s Ws $2,k+s−M (α)
(α)
(α)
(α)
(5.232)
+ Wp,¯ + Wp,¯ ¯ q ,¯ r ,¯ s Wp,¯ ¯ q Wp¯ $2,k+p−M ¯ q ,¯ r ,¯ s Wp,¯ ¯ q Wq¯ $2,k+¯ ¯ q −M
+ Wp,¯ ¯ q ,¯ r ,¯ s Wr¯,¯ s Wr¯ $2,k+¯ ¯ q ,¯ r ,¯ s Wr¯,¯ s Wq¯ $2,k+¯ r −M + Wp,¯ s−M . T
Taking F = [f (x0 ), f (x1 ), . . . , f (xN )] , we have the matrix form for (5.227) as eα L Ap F
=
1 α A F, hα p
(5.233)
where p is an integer,
(ω)
(ω)
$2,p−1
$2,p
(ω) $2,p+1 (ω) $2,p+2 .. . α Ap = (ω) $2,N .. . $(ω) 2,p+N −1 (ω)
$2,p+N
(ω)
(ω)
...
$2,0
(ω)
...
$2,0
(ω)
$2,p
$2,p−1
$2,p+1
(ω)
$2,p
$2,p−1
(ω)
...
$2,0
..
.
..
.
..
..
.
···
···
..
.
$2,p+1
(ω)
$2,p
(ω)
$2,p−1
(ω)
···
..
.
..
.
..
..
···
..
.
$2,p+1
(ω)
$2,p
···
$2,p+2
(ω)
$2,p+1
..
(ω)
.
··· ..
··· (ω)
.
(ω)
···
$2,p+N −1
.
$2,N
(ω)
.
..
.
.
(ω)
(ω)
(5.234) (ω) $2,0 .. . (ω) $2,p−1 (ω)
$2,p
($)
and $2,k = 0 as k < 0. Similarly, we can readily obtain the matrix forms 1 α α α A F, Aα p,q = Wp Ap + Wq Aq , hα p,q
(5.235)
1 α α α A F, Aα p,q,r,s = Wp,q Ap,q + Wr,s Ar,s , hα p,q,r,s
(5.236)
1 α A ¯ q ,¯ r ,¯ s F, hα p,q,r,s,p,¯
(5.237)
eα 2L Ap F
eα 3L Ap F
=
=
and eα 4L Ap F
=
with α α Aα ¯ q ,¯ r ,¯ s Ap,¯ p,q,r,s,p,¯ ¯ q ,¯ r ,¯ s = Wp,q,r,s Ap,q,r,s + Wp,¯ ¯ q ,¯ r ,¯ s.
(5.238)
Remark 5.16. When p = 0, the matrix Aα triangular matrix, and all of its p reduces to the lower α α eigenvalues are greater than 1. In this case, we have λ Ap = 23 with α ∈ (1, 2). This is the reason that the scheme for time dependent problems is unstable when directly using the 2nd order Lubich formula with α ∈ (1, 2) to discretize a fractional derivative. When applying the above weighted and shifted Lubich formulae to proposing numerical schemes for space fractional differential equations, it is of significance to choose suitable pa-
5.4. Fractional backward difference formulae with modifications
225
α rameters p, q, r, s, p¯, q¯, r¯, s¯, such that all the eigenvalues of the matrix Aα p,q (or Ap,q,r,s , or α Ap,q,r,s,p,¯ ¯ q ,¯ r ,¯ s ) possess negative real parts. We have the following assertions in this respect. α Theorem 5.26. Let Aα p,q be given by (5.235) and 1 < α < 2. Then any eigenvalue λ of Ap,q satisfies < λ Aα < 0 for (p, q) = (1, q), |q| ≥ 2. (5.239) p,q α T Furthermore, the matrices Aα are negative definite. p,q and Ap,q
Theorem 5.27. Let Aα p,q,r,s with 1 < α < 2 be given by (5.236). Then any eigenvalue λ of Aα satisfies p,q,r,s < λ Aα p,q,r,s
< 0 for (p, q, r, s) = (1, q, 1, s), |q| ≥ 2, |s| ≥ 2, and qs < 0.
α Moreover, the matrices Aα p,q,r,s and Ap,q,r,s
T
(5.240)
are negative definite.
Theorem 5.28. Let Aα ¯, q¯, r¯, s¯) p,q,r,s,p,¯ ¯ q ,¯ r ,¯ s with 1 < α < 2 be given by (5.237), where (p, q, r, s, p = (1, 2, 1, −2, 1, q¯, 1, s¯), |¯ q | ≥ 2, |¯ s| ≥ 2, (¯ q , s¯) 6= (2, −1), and q¯s¯ < 0. Then any eigenvalue λ of Aα p,q,r,s,p,¯ ¯ q ,¯ r ,¯ s satisfies < λ Aα < 0, (5.241) p,q,r,s,p,¯ ¯ q ,¯ r ,¯ s T α and the matrices Aα are negative definite. Moreover, if (p, q, r, s, p,q,r,s,p,¯ ¯ q ,¯ r ,¯ s and Ap,q,r,s,p,¯ ¯ q ,¯ r ,¯ s p¯, q¯, r¯, s¯) takes the values (p, q, r, s, p¯, q¯, r¯, s¯) = (1, 2, 1, 0, 1, 2, 1, −2), (p, q, r, s, p¯, q¯, r¯, s¯) = (1, 2, 1, 0, 1, −1, 1, −2), (p, q, r, s, p¯, q¯, r¯, s¯) = (1, 2, 1, −1, 1, 2, 1, −2), (p, q, r, s, p¯, q¯, r¯, s¯) = (1, 2, 1, −1, 1, −1, 1, −2), (5.242) (p, q, r, s, p¯, q¯, r¯, s¯) = (1, 0, 1, −1, 1, 2, 1, −2), (p, q, r, s, p¯, q¯, r¯, s¯) = (1, 0, 1, −2, 1, 2, 1, −2), (p, q, r, s, p¯, q¯, r¯, s¯) = (1, −1, 1, −2, 1, 2, 1, −2), T α then < λ Aα < 0, and the matrices Aα are negp,q,r,s,p,¯ ¯ q ,¯ r ,¯ s p,q,r,s,p,¯ ¯ q ,¯ r ,¯ s and Ap,q,r,s,p,¯ ¯ q ,¯ r ,¯ s ative definite. Remark 5.17. To relax the regularity assumptions on f (x) in Theorem 5.25 while keeping the high-order convergence at the same time, we can introduce the starting weights [114]. In this case,
α RL Da,x f (x) x=x j
j+M s 1 X α 1 X α ϕk f (xj−k+M ) + α ϕj+M,k f (xk ), = α h h k=0
(5.243)
k=0
α where ϕα k is defined by (5.232). And the starting weights ϕj+M,k can be derived based on (4.1) of [114]. However, the approximation (5.243) does not always work well for time dependent space fractional partial differential equations due to the eigenvalue issue.
226
Chapter 5. Numerical Riemann-Liouville differentiation
Example 5.29. [23] We simulate the fractional equation −RL Dα 0,x f (x) = −
Γ( 5 ) 3 Γ(7) x 2 −α x6−α − 5 2 Γ(7 − α) Γ( 2 − α)
(5.244) 3
with f (0) = 0, f (1) = 2, and 1 < α < 2. The exact solution is f (x) = x6 + x 2 . Note that f (x) fails to meet the regularity assumptions in Theorem 5.25 in this case. Numerical results given by approximations (5.231) and (5.243) are presented in Table 5.2. It is evident from the numerical results that (5.243) works in this case.
Table 5.2. The maximum errors and convergence orders for schemes (5.231) and (5.243), respectively. The parameters α = c Copy1.2 and (p, q, r, s, p¯, q¯, r¯, s¯) = (1, 2, 1, 0, 1, 2, 1, −2). right 2014 Society for Industrial and Applied Mathematics h
scheme (5.243)
rate
scheme (5.231)
rate
1 100 1 200 1 400 1 800
1.7530E-06
∗
1.8242E-01
∗
1.0994E-07
3.9950
1.4817E-01
0.3000
6.8857E-09
3.9970
1.2035E-01
0.3000
4.3504E-10
3.9844
9.7758E-02
0.3000
5.4.4 Fractional-compact formulae Now we present another variant of the fractional backward difference formulae slightly different from the shifted one, called the fractional-compact formulae given by [36]. (I) The 3rd order fractional-compact formula I Define the two difference operators, L
B2α f (x + sh) =
∞ 1 X (α) k2,l f (x − (l − s − 1)h) hα
(5.245)
∞ 1 X (α) k2,l f (x + (l − s − 1)h) , hα
(5.246)
l=0
and R
B2α f (x + sh) =
l=0
(α)
where s ∈ R, and the coefficients k2,l can be obtained by the generating function f2 (z) = W
3α − 2 2(α − 1) α−2 2 − z+ z 2α α 2α
α ,
(5.247)
|z| < 1.
(5.248)
i.e.,
3α − 2 2(α − 1) α−2 2 − z+ z 2α α 2α
α =
∞ X l=0
(α)
k2,l z l ,
5.4. Fractional backward difference formulae with modifications
227
It can be verified through the Fourier transform that the above two difference operators have the following asymptotic behaviors. Theorem 5.30. Let f (x) ∈ C [α]+n+1 (R) and all derivatives of f (x) up to order [α] + n + 2 belong to L1 (R). Then, for any s ∈ R,
L
B2α f (x + sh)
α =RL D−∞,x f (x) +
n−1 X
(α)
α+l D f (x) hl + O(hn ), n ≥ 2, RL −∞,x
(α)
α+l RL Dx,+∞ f (x)
σl,s
(5.249)
l=1
and
R
B2α f (x + sh)
α =RL Dx,+∞ f (x) +
n−1 X
σl,s
hl + O(hn ), n ≥ 2,
(5.250)
l=1 (α)
hold uniformly on R. Here the coefficients σl,s (l = 1, 2, . . .) satisfy the equation ∞ P (α) =1+ σl,s z l . Especially, the first three coefficients are
e(1−s)z −z ) z α W2 (e
l=1
(α) σ1,s = −s, (α) 2α2 − 3α(s2 + 2) + 3 σ2,s = − , 6α 3 2 3 σ (α) = α (4s + 3) − α (2s + 12s + 11) + 6α(s + 2) − 4 . 3,s 12α2 If we define the fractional-compact difference operator L as (α) Lf (x) = 1 + σ2,0 δx2 f (x),
(5.251)
(5.252)
with δx2 f (x) = f (x − h) − 2f (x) + f (x + h) being a 2nd order central difference operator, then the following assertions can be obtained. Theorem 5.31. Let f (x) ∈ C [α]+4 (R) and all derivatives of f (x) up to order [α] + 5 belong to L1 (R). Then it holds that L
3 B2α f (x) = L RL Dα −∞,x f (x) + O(h )
(5.253)
and R
3 B2α f (x) = L RL Dα x,∞ f (x) + O(h )
(5.254)
uniformly for x ∈ R. Proof. Taking n = 3, s = 0 in equation (5.249), and noticing (α)
σ1,0 = 0,
α+2 RL D−∞,x f (x)
=
d2 dx2
α RL D−∞,x f (x)
,
(5.255)
228
Chapter 5. Numerical Riemann-Liouville differentiation
it holds that L α B2 f (x) 2 2 (α) d α 3 =RL Dα RL D−∞,x f (x) h + O(h ) −∞,x f (x) + σ2,0 dx2 (α) 2 2 3 =RL Dα δx2 RL Dα −∞,x f (x) + σ2,0 h −∞,x f (x) + O(h ) + O(h ) 3 =L RL Dα −∞,x f (x) + O(h ).
(5.256)
Using the same method, we can prove (5.254). For the function f (x) defined on the bounded interval [a, b] that satisfies f (n) (a) = f (n) (b) = 0, n = 0, 1, 2, we can apply zero extension to obtain the following results. Theorem 5.32. Suppose f (x) ∈ C [α]+4 ([a, b]), f (n) (a) = f (n) (b) = 0, n = 0, 1, 2, and all derivatives of f (x) up to order [α] + 5 belong to L1 ([a, b]). Then, for any x ∈ [a, b], L
α 3 Aα 2 f (x) = L RL Da,x f (x) + O(h )
(5.257)
α 3 Aα 2 f (x) = L RL Dx,b f (x) + O(h ).
(5.258)
and R
R α Here, the operators L Aα 2 and A2 are defined as follows:
1 L α A2 f (x) = α h
[ x−a h ]+1 X
(α)
k2,l f (x − (l − 1)h)
(5.259)
l=0
and 1 R α A2 f (x) = α h
[ b−x h ]+1 X
(α)
k2,l f (x + (l − 1)h) .
(5.260)
l=0
Remark 5.18. The fractional-compact formulae (5.253) and (5.254) share the same generating f2 (z) with the shifted fractional backward difference formulae (5.150) and (5.160), function W (α) indicating that the coefficients can be obtained in the same way. And the properties of k2,l have been presented in Theorem 5.17. However, it should be noted that the fractional-compact formulae achieve higher-order accuracy. (II) The 3rd order fractional-compact formula II Choose the generating function as α X ∞ f (α) e f 2 (z) = 3α + 2 − 2(α + 1) z + α + 2 z 2 W = k2,l z l , |z| ≤ 1, 2α α 2α
(5.261)
l=0
and define the difference operators ∞ 1 X e(α) Be2α f (x + sh) = α k2,l f (x − (l − s + 1)h) h
(5.262)
∞ 1 X e(α) Be2α f (x + sh) = α k2,l f (x + (l − s + 1)h) . h
(5.263)
L
l=0
and R
l=0
5.4. Fractional backward difference formulae with modifications
229
In this case, the coefficients are given by (α) e k2,l =
3α + 2 2α
α X l k=0
α+2 3α + 2
k
(α)
(α)
$1,k $1,l−k , l = 0, 1, 2, . . . ,
from which follows the recursion formula α 3α + 2 (α) e , k2,0 = 2α e(α) −4α(1 + α) e(α) k2,1 = 3α + 2 k2,0 , −4α(1 + α)(α − l + 1) e(α) (α) e k2,l−1 k2,l = l(3α + 2) (α + 2)(2α − l + 2) e(α) k2,l−2 , l ≥ 2. + l(3α − 2)
(5.264)
(5.265)
We can also obtain the following asymptotic behaviors similar to those of the previous 3rd order compact difference operators. Theorem 5.33. Let f (x) ∈ C [α]+n+1 (R) and all derivatives of f (x) up to order [α] + n + 2 belong to L1 (R). Then, for any s ∈ R, L α e B2 f (x + sh) n−1 (5.266) X (α) l n α σ el,s RL Dα+l =RL D−∞,x f (x) + −∞,x f (x) h + O(h ), n ≥ 2, l=1
and
R
Be2α f (x + sh)
α =RL Dx,+∞ f (x) +
n−1 X
(α)
σ el,s
α+l RL Dx,+∞ f (x)
hl + O(hn ), n ≥ 2,
(5.267)
l=1 (α)
hold uniformly on R. Here the coefficients σ el,s (l = 1, 2, . . .) satisfy the equation ∞
X (α) e−(1+s)z f f 2 (e−z ) = 1 + W σ el,s z l . α z
(5.268)
l=1
Especially, the first three coefficients are 2α2 − 3α(s2 − 2) + 3 (α) (α) e1,s = −s, σ e2,s = − , σ 6α α3 (4s + 3) − α2 (2s3 − 12s − 11) + 6α(s + 2) + 4 (α) σ . e3,s = 12α2 If we define the fractional-compact difference operator Le (α) e (x) = 1 + σ Lf e2,0 δx2 f (x),
(5.269)
(5.270)
230
Chapter 5. Numerical Riemann-Liouville differentiation
then we get the following results, which can be verified by virtue of the results in the previous theorem. Theorem 5.34. Let f (x) ∈ C [α]+4 (R) and all derivatives of f (x) up to order [α] + 5 belong to L1 (R). Then it holds that L
3 Be2α f (x) = Le RL Dα −∞,x f (x) + O(h )
(5.271)
and R
3 Be2α f (x) = Le RL Dα x,∞ f (x) + O(h )
(5.272)
uniformly for x ∈ R. Remark 5.19. Let f (x) ∈ C [α]+4 ([a, b]) , f (n) (a) = f (n) (b) = 0 (n = 0, 1), and all derivatives of f (x) up to order [α] + 5 belong to L1 ([a, b]). Then, for any x ∈ [a, b], L
α 3 e Aeα 2 f (x) = L RL Da,x f (x) + O(h )
(5.273)
α 3 e Aeα 2 f (x) = L RL Dx,b f (x) + O(h ).
(5.274)
and R
R eα Here, the operators L Aeα 2 and A2 are defined as follows:
L
Aeα 2
1 = α h
[ x−a h ]−1 X
(α) e k2,l f (x − (l + 1)h)
(5.275)
(α) e k2,l f (x + (l + 1)h) .
(5.276)
l=0
and R
Aeα 2
1 = α h
[ b−x h ]−1 X l=0
(III) The 4th order fractional-compact formula In view of Theorems 5.30 and 5.33, we can further deduce the following 4th order fractionalcompact formula for the Riemann-Liouville derivative. Theorem 5.35. Let f (x) ∈ C [α]+5 (R) and all derivatives of f (x) up to order [α] + 6 belong to L1 (R). Define the fractional-compact operator i (α) (α) (α) (α) + σ2,0 σ e3,0 − σ e2,0 σ3,0 δx2 f (x);
(5.277)
(α) (α) H RL Dα e3,0 L B2α f (x) − σ3,0 L Be2α f (x) + O(h4 ) −∞,x f (x) = σ
(5.278)
Hf (x) =
h
(α)
(α)
σ e3,0 − σ3,0
then the equalities
and (α) R
H RL Dα e3,0 x,∞ f (x) = σ hold uniformly on R.
(α) R
B2α f (x) − σ3,0
Be2α f (x) + O(h4 )
(5.279)
5.5. Fractional average central difference method
231
5.5 Fractional average central difference method In [167], the left-shifted difference operator α C ∆−h f (x)
=
∞ X k=0
α (α) h , h > 0, ωk f x − k − 2
(5.280)
(α) with ωk = (−1)k αk , was proposed to construct approximation to Riemann-Liouville derivative. It happens that this operator can be reduced to the standard central difference operators if α is a positive integer [167]. Furthermore, it possesses the following asymptotic behavior. Theorem 5.36. Let α be a positive number, f ∈ C [α]+2n+1 (R), and all derivatives of f up to order [α] + 2n + 2 exist and belong to L1 (R). Then the following equality holds uniformly on R: n−1
X 1 α+2k α α f (x)h2k + O(h2n ). b2k RL D−∞,x C ∆−h f (x) = RL D−∞,x f (x) + α h
(5.281)
k=1
Here the coefficients b2k are those of the Taylor expansion
z
z
e 2 − e− 2 z
α =
∞ X
b2k z 2k , |z| < 1.
(5.282)
k=0
Proof. For the function f ∈ C [α]+2n+1 (R), where all its derivatives up to order [α] + 2n + 2 exist and belong to L1 (R), the Fourier transform of the left-shifted operator (5.280) exists and has the form F C ∆α −h f (x); ω ∞ X α (α) ωk eiω(k− 2 )h fb(ω) = k=0 (5.283) − iωαh iωh α b 2 =e 1−e f (ω) iωh α = e− 2 − e iωh 2 fb(ω). It follows from the above relation that 1 α F = (−iω)α fb(ω) C ∆−h f (x); ω hα Let the function
z
z
e 2 −e− 2 z
α
e
−iωh 2
−e −iωh
iωh 2
!α .
(5.284)
have the Taylor expansion
z
z
e 2 − e− 2 z
α =
∞ X
bk z k , |z| < 1,
(5.285)
k=0
that converges in the vicinity of the origin. It is easy to see that b0 = 1 and bk = 0 whenever k is an odd integer. Therefore, it holds that
z
z
e 2 − e− 2 z
α =1+
∞ X k=1
b2k z 2k , |z| < 1.
(5.286)
232
Chapter 5. Numerical Riemann-Liouville differentiation
As a result, 1 α ∆ f (x); ω F C −h hα α =(−iω)
1+
n−1 X
(5.287)
! 2k
b2k (−iωh)
fb(ω) + ϕ(ω, b h),
k=1
where " ϕ(ω, b h) =
e
−iωh 2
−e −iωh
iωh 2
!α −
n−1 X
# b2k (−iωh)
2k
(−iω)α fb(ω).
(5.288)
k=0
Since there exists a universal constant C such that −iωh !α n−1 e 2 − e iωh X 2 − b2k (−iωh)2k ≤ C|ωh|2n , −iωh
(5.289)
k=0
it holds that |ϕ(ω, b h)| ≤ Ch2n |ω|α+2n |fb(ω)|.
(5.290)
From the condition imposed on f , we conclude that (1 + |x|)[α]+2n+2 fb(ω) is bounded on R. Hence |ω|α+2n |fb(ω)| ∈ L1 (R). Consequently, |ϕ(x, h)| ≤ Ch2n , where ϕ(x, h) is the inverse Fourier transform of the function ϕ(ω, b h). Since (−iω)α+2k fb(ω) = F RL Dα+2k −∞,x f (x); ω , k = 0, 1, . . . , n − 1,
(5.291)
(5.292)
we obtain equation (5.281). The following theorem exhibits a special case of the above asymptotic behavior, which implies that the left-shifted difference operator can be utilized to construct a 2nd order approximation to the Riemann-Liouville derivative. Theorem 5.37. [167] Let f ∈ C [α]+3 (R) and all derivatives of f up to order [α] + 4 exist and belong to L1 (R). Then 1 α α 2 C ∆−h f (x) = RL D−∞,x f (x) + O(h ) hα
(5.293)
holds uniformly on R. Remark 5.20. [167] Instead of utilizing the infinite sum in the left-shifted operator, we can take a finite sum with the same accuracy. Define the truncated operator α C ∆−h,m f (x) =
m X k=0
α (α) ωk f x − k − h , h > 0. 2
(5.294)
In fact, f is uniformly bounded on R due to the conditions imposed on f . Consequently, ∞ X k=m
α (α) ωk f x − k − h = O(m−α ), 2
(5.295)
5.5. Fractional average central difference method
233
since, by Stirling’s formula α = O(k −α−1 ). k
(5.296)
1 ∆α h,m f (x) α h m α 1 X (α) = α h ωk f x − k − h 2 k=0 n−1 X = Dα 2k b2k RL Dα+2k + O(h2n ), RL −∞,x f (x) + −∞,x f (x)h
(5.297)
It follows that
k=1
2n
whenever m ∈ Z+ is sufficiently big and satisfies m ≥ ch− α with an arbitrary positive constant c. In this case, we have 1 α α 2 C ∆−h,m f (x) = RL D−∞,x f (x) + O(h ), hα
(5.298)
2
if m is such a natural number that m ≥ ch− α with an arbitrary positive number c. We can also define the right-shifted operator α C ∆+h f (x)
=
∞ X k=0
α (α) ωk f x + k − h , h > 0. 2
(5.299)
And assertions for the right-sided Riemann-Liouville derivative analogous to Theorems 5.36 and 5.37 can be obtained. Recently, Ding and Li modified the above shifted operators and got higher-order approximations in [40]. Define fractional average operators that are analogous to the integer-order finite difference formula in the forms µα −h f (x − sh) =
f (x − (s − α2 )h) + f (x − (s + α2 )h) , 2
(5.300)
µα +h f (x + sh) =
f (x + (s − α2 )h) + f (x + (s + α2 )h) . 2
(5.301)
And define also the left and right fractional average central difference operators α AC ∆−h f (x) α =µα −h C ∆−h f (x) ∞ X α α α (−1)k h = µ−h f x − k − 2 k k=0 ∞ 1X k α (−1) [f (x − kh) + f (x − (k − α)h)] = 2 k k=0
(5.302)
234
Chapter 5. Numerical Riemann-Liouville differentiation
and α AC ∆+h f (x) α =µα +h C ∆+h f (x) ∞ X α α α h = (−1)k µ+h f x + k − 2 k k=0 ∞ 1X k α (−1) [f (x + kh) + f (x + (k − α)h)] . = 2 k
(5.303)
k=0
Then we obtain the following high-order approximations to Riemann-Liouville derivatives, which can be verified through Fourier analysis. Theorem 5.38. [40] Suppose that f (x) and the Fourier transforms of α+2 1 RL Dx,+∞ f (x) are in L (R). Then =
α AC ∆−h f (x) hα
RL Dα x,+∞ f (x) =
α AC ∆+h f (x) hα
RL Dα
−∞,x f (x)
α+2 RL D−∞,x f (x)
and
+ O(h2 ), (5.304) 2
+ O(h )
hold uniformly on R. Proof. Here we only prove the first equality. Since f (x) belongs to L1 (R), the Fourier transform of the fractional average central difference operator (5.302) exists and has the following form: α AC ∆−h f (x) F ; ω hα ∞ 1 X iωkh iω(k−α)h b k α e + e f (ω) (−1) = 2hα k k=0 ! ∞ X 1 α 1 + e−iωαh b k ikωh = α f (ω) (−1) e h k 2 k=0 iωh α 1 + e−iωαh b =(−iω)α 1 − e f (ω). −iωh 2
(5.305)
α 1 + e−iωαh 1 − eiωh −iωh 2 =1 + α(3α + 1) (−iωh)2 + O(| − iωh|4 ). 24
(5.306)
Note that
If we denote b h) = F φ(ω,
α AC ∆−h f (x) ;ω hα
−F
α RL D−∞,x f (x); ω
,
(5.307)
5.5. Fractional average central difference method
235
then we have b h)| ≤ C1 h2 |(iω)α+2 fb(ω)|. |φ(ω, In view of the condition F
Z
α+2 RL D−∞,x f (x); ω
(5.308)
∈ L1 (R), i.e.,
F{RL Dα+2 −∞,x f (x); ω} dω < C2
(5.309)
R
with C2 being a finite constant, we obtain α AC ∆−h f (x) α − RL D−∞,x f (x) α h Z 1 −iωh b φ(ω, h)dω =|φ(ω, h)| = 2π R e Z b h)|dω ≤ 1 |φ(ω, 2π R Z C1 α+2 b f (ω) dω h2 (iω) ≤ 2π R Z C1 α+2 = F{RL D−∞,x f (x); ω} dω h2 2π R ≤Ch2 , where C =
C1 C2 2π .
(5.310)
This completes the proof.
By virtue of a similar proof, we have the following assertion. α+4 Theorem 5.39. [40] Let f (x) and the Fourier transforms of RL Dα+4 −∞,x f (x) and RL Dx,+∞ f (x) 1 be in L (R). Then
α(3α + 1) 2 α δ 1 + x RL D−∞,x f (x) = 24 α(3α + 1) 2 1+ δx RL Dα x,+∞ f (x) = 24
1 α 4 AC ∆−h f (x) + O(h ), hα 1 α 4 AC ∆+h f (x) + O(h ) hα
(5.311)
hold uniformly on R, with δx2 denoting the 2nd order central difference operator defined by 2 hδx f (xj ) = f (xij+1 ) − 2f (xj ) + f (xj−1 ). Here (5.311) means that when the operator
1 + α(3α+1) δx2 is nonsingular, the left- and right-sided Riemann-Liouville derivatives can be 24 approximated by 1 α RL D−∞,x f (x) = hα 1 RL Dα x,+∞ f (x) = α h
−1 α(3α + 1) 2 α 4 1+ δx AC ∆−h f (x) + O(h ), 24 −1 α(3α + 1) 2 α 4 1+ δx AC ∆+h f (x) + O(h ). 24
(5.312)
236
Chapter 5. Numerical Riemann-Liouville differentiation
When f (x) defined on [a, b] satisfies the homogeneous conditions f (n) (a) = f (n) (b) = 0, 0 ≤ n ≤ 3, numerical approximations given by (5.304) and (5.312) can be truncated to the domain [a, b] by using zero extension.
5.6 Spectral approximations Spectral approximations to the Riemann-Liouville derivative can be readily obtained based on those to the Caputo derivative. For m − 1 < α < m ∈ Z+ , one has α RL D−1,x f (x) x=x j
=
α C D−1,x f (x) x=x j
+
m−1 X i=0
f (i) (−1)(xj + 1)i−α . Γ(i − α + 1)
(5.313)
In this case, the corresponding differential matrix for the Riemann-Liouville derivative at JGL points xk ∈ [−1, 1] (k = 0, 1, . . . , N ) is given by [100, 181]
α RL D−1,x0 pN (x0 ) α RL D−1,x1 pN (x1 )
.. .
α RL D−1,xN pN (xN )
(u,v,α) T = RL D−1,xN (f (x0 ), f (x1 ), . . . , f (xN )) ,
(5.314)
where i h (u,v,α) D RL −1,xN k,l m−1 i−α h i X D(i) 0,l (xk + 1) (u,v,α) + , = C D−1,xN Γ(i + 1 − α) k,l i=0
(5.315) k, l = 0, 1, . . . , N
with D(i) being the ith order classical differential matrix proposed in [153], and (u,v,α) C D−1,xN
(u,v,α)
b = CD −1,xN M.
(5.316)
b (u,v,α) is defined by (4.317). Some applications of spectral Here M is defined by (3.69), and C D −1,xN approximation for the Riemann-Liouville derivative can be found in [180]. Remark 5.21. Spectral approximations and the corresponding differential matrices for the Riemann-Liouville derivative of functions on arbitrary intervals can be obtained via affine transforms. In addition, differential matrices for the right-sided Riemann-Liouville derivative can be derived similarly.
5.7 Numerical method based on finite-part integrals In [31], a numerical approximation to the Riemann-Liouville derivative was proposed. Instead of utilizing the relationship (2.107), Diethelm rewrote the definition of the Riemann-Liouville derivative into a Hadamard finite-part integral, and then established a numerical approximation to this integral via a first-degree compound quadrature formula.
5.7. Numerical method based on finite-part integrals
237
Consider the case 0 < α < 1 and x ∈ [a, b] with a = 0. Based on the observation in [46], it is reasonable to change the order of the differentiation and integration in (2.5) and obtain Z x f (t) 1 α dt, (5.317) RL D0,x f (x) = Γ(−α) 0 (x − t)α+1 where the integral now must be interpreted as a Hadamard finite-part integral [106]. Then for a given N ∈ Z+ and an equispaced grid xj = jh with h = Nb , the Riemann-Liouville derivative of f (x) at nodes xj is given by α RL D0,x f (x) x=xj Z xj 1 f (t) = dt (5.318) Γ(−α) 0 (xj − t)α+1 Z 1 x−α f (xj − xj t) j = dt, j = 1, 2, . . . , N. Γ(−α) 0 tα+1 The integral in the right-hand side of the above equation can be approximated based on a firstdegree compound quadrature formula with the equispaced nodes 0, 1j , 2j , . . . , 1, Z
1 −α−1
g(t)t 0
dt ≈
j X k=0
k ak,j g . j
Here the explicit expressions for the weights αk,j with j ≥ 0 are given by the relation k = 0, −1, α j 2k 1−α − (k − 1)1−α − (k + 1)1−α , k = 1, 2, . . . , j − 1, ak,j = α(1 − α) (α − 1)k −α − (k − 1)1−α + k 1−α , k = j.
(5.319)
(5.320)
In this case, g(t) = f (xj − xj t) for each fixed xj . Consequently, the corresponding numerical approximation to Riemann-Liouville derivative is given by
j X x−α j α D f (x) ≈ ak,j f ((j − k)h) . RL 0,x x=xj Γ(−α)
(5.321)
k=0
Remark 5.22. In the above discussion, the Riemann-Liouville derivative is rewritten in the form of a singular integral in (5.317), which must be interpreted as a Hadamard finite-part integral, which is formally written as Z x f (t) 1 α p.f. dt. (5.322) RL D0,x f (x) = α+1 Γ(−α) 0 (x − t) In effect, other fractional derivatives can be rewritten as finite-part integrals. The finite-part integrals associated with the fractional derivatives are presented in Table 5.3 [152]. Here dαe denotes the smallest integer greater than α and Cα = − cos1πα . The left- and right-sided Hadamard 2 derivatives are given by α H Da,x,µ f (x) Z x µ (5.323) 1 t x dαe−α−1 dt −µ dαe µ = x δ x log f (t) Γ(dαe − α) x t t a
238
Chapter 5. Numerical Riemann-Liouville differentiation
Table 5.3. Fractional derivatives associated with finite-part integrals. Factional derivatives
The finite-part integrals
α RL Da,x f (x), dαe
1 Γ(−α)
p.f.
Rx
α RL Dx,b f (x), dαe
1 Γ(−α)
p.f.
Rb
α RZ Dx f (x), α dαe
Cα Γ(−α)
h
α H Da,x f (x), α dαe
1 Γ(−α)
p.f.
Rx
α H Dx,b f (x), α dαe
1 Γ(−α)
p.f.
Rb
f (t)
x
(log xt )
α H Da,x,µ f (x), dαe
x−µ Γ(−α)
p.f.
Rx
α H Dx,b,µ f (x), dαe
xµ Γ(−α)
p.f.
Rb
f (x) ∈ C f (x) ∈ C f (x) ∈ C
f (x) ∈ C
f (x) ∈ C f (x) ∈ C
f (x) ∈ C
and = with δ =
α>0 (a, b] α>0 [a, b)
>0 [a, b] >0 (a, b], x > a > 0 >0 [a, b), b > x > 0
α>0 (a, b], x > a > 0 α>0 [a, b), b > x > 0
f (t)dt a (x−t)α+1 f (t)dt x (t−x)α+1
p.f.
Rx
f (t)dt a (x−t)α+1
a
a
x
f (t) α+1
dt t
α+1
dt t
(log xt )
tµ f (t)
(log
x t
+ p.f.
)
α+1
t−µ f (t) α+1
(log xt )
Rb
f (t)dt x (t−x)α+1
i
dt t dt t
α H Dx,b,µ f (x)
1 x−µ (−δ)dαe xµ Γ(dαe − α)
d x dx ,
a > 0, and 0 < x < b.
Z
b
x
dαe−α−1 x µ t dt log f (t) t x t
(5.324)
Chapter 6
Numerical Riesz differentiation
As mentioned in Chapter 2, the Riesz derivative frequently appears when characterizing anomalous diffusion and anomalous convection processes. In effect, mathematical descriptions of the fractional diffusion genuinely involve a Riesz derivative with order α ∈ (1, 2), while those of the fractional convection utilize a Riesz derivative with order α ∈ (0, 1). Before introducing numerical approximations to the Riesz derivative, we show this issue through continuous time random walk (CTRW) processes. Consider the CTRW process depicted via a Poissonian waiting time and a Lévy distribution. In this case, the jump length variance Σ2 diverges, while the finite characteristic waiting time T stays finite. Furthermore, the jump length probability density function (PDF) possesses the asymptotic behavior λ(x) ∼ Aα σ −α |x|−1−α , |x| σ > 0, 1 < α < 2,
(6.1)
and its Fourier transform takes the form α α b λ(ω) = e−σ |ω| ∼ 1 − σ α |ω|α , 1 < α < 2.
(6.2)
Combining the asymptotic expansion (6.2) with the relation (1.9) gives f (ω, u) = W
1 , 1 < α < 2. u + Kα |ω|α
(6.3)
Consequently, it follows from the inverse Fourier and Laplace transforms that ∂ W (x, t) = Kα RZ Dα x W (x, t), 1 < α < 2. ∂t
(6.4)
α
Here the generalized diffusion constant is given by Kα = στ . The notation RZ Dα x refers to the Riesz derivative in one dimension, which can be defined via the Fourier transformation αb F {RZ Dα x f (x); ω} = −|ω| f (ω),
(6.5)
with fb(ω) being the Fourier transform of f (x). In the literature, an explicit expression for the Riesz derivative defined via (6.5) is usually given by (2.174), a linear combination of the leftand right-sided Riemann-Liouville derivatives. From the above description, it is evident that a Riesz derivative with order α ∈ (1, 2) can be utilized to characterize the anomalous diffusion. Now we derive the fractional convection 239
240
Chapter 6. Numerical Riesz differentiation
equation (FCE) through a CTRW process with nonlocal jump length distribution. Let the density of particles at position x and at time t be u(x, t). Denote the jump length on a one-dimensional lattice by kh, k = 0, 1, 2, . . . , where h is the one-step jump length. For a right-sided nonlocal jump, the particles only jump to the right of their current positions. The density of particles at position x after a waiting time ∆t, u(x, t + ∆t), can be written as u(x, t + ∆t) =
∞ X
p(k)u(x − kh, t).
(6.6)
k=0
Here p(k) the probability density for jump step k, which is independent of waiting time and Pis ∞ satisfies k=0 p(k) = 1. Then the process described by (6.6) is determined by p(k). Furthermore, it holds that u(x − kh, t) − u(x, t) u(x, t + ∆t) − u(x, t) X = p(k) . ∆t ∆t
(6.7)
k
Consider the power-law distribution of p(k), namely, p(y) = Cα |y|−(α+1) with α ∈ (0, 1) and Cα > 0. Then it follows from (6.7) that u(x, t + ∆t) − u(x, t) ∆t α X h u(x − kh, t) − u(x, t) =Cα k −(α+1) ∆t hα k ∂u(x−kh,t) kh + O(h2 ) hα X ∂x h = − C α ∆t (kh)α+1 k ( ) ∂u(x−kh,t) X O(h2−α ) hα X ∂x h + , 0 < α < 1, = − Cα ∆t (kh)α k 1+α k
(6.8)
k
in which the Taylor expansion of u(x, t) at (x − kh) is used. Taking ∆t, h → 0 in (6.8) yields Z +∞ d ∂u η −α u(x − η, t)dη ∂t = −Cα C α dx 0 Z x d (x − s)−α u(s, t)ds, 0 < α < 1, = −Cα C α dx −∞ where C α =
α lim h ∆t, h→0 ∆t
(6.9)
is a positive constant related to α, if it indeed exists. If we set Cα C α =
κ Γ(1−α) , κ > 0, then the right-hand side of (6.9) corresponds to the left Riemann-Liouville derivative, and (6.9) is accordingly reduced to
∂u + κ RL Dα −∞,x u = 0, 0 < α < 1. ∂t
(6.10)
For a left-sided nonlocal jump, the particles jump to the left of their current positions. The density of particles at position x after a waiting time ∆t, u(x, t + ∆t) can be written as u(x, t + ∆t) =
∞ X k=0
p(k)u(x + kh, t).
(6.11)
6.1. Indirect approximations to the fractional diffusion operator
241
Similar to the case of the right-sided nonlocal jump, we can also derive the following convection equation depicting a left-sided nonlocal jump: ∂u + κ RL Dα x,+∞ u = 0, 0 < α < 1. ∂t
(6.12)
Consider the convection process characterized by two-sided nonlocal jumps. The density u(x, t + ∆t) is given by u(x, t + ∆t) =
∞ X
(pl (k)u(x + kh, t) + pr (k)u(x − kh, t)) ,
(6.13)
k=0
in which pl and pr are the probability densities for the left-sided and the right-sided jumps, respectively. Assume that the convection process is symmetric (namely, pl (y) = pr (y)), and pl (y) = pr (y) = Cα |y|−(α+1) with Cα > 0 and α ∈ (0, 1). In this case, Cα C α = Γ(1−α)2κcos πα (2) α with C α = lim h∆t > 0. Then it follows from equations (6.10), (6.12), and (6.13) that ∆t, h→0
∂u κ + ∂t 2 cos πα 2
α RL D−∞, x u
+ RL Dα x,+∞ u = 0, 0 < α < 1.
(6.14)
That is, ∂u − κ RZ Dα x u = 0, 0 < α < 1. ∂t
(6.15)
If there is a source term, we can get the following FCE: ∂u − κ RZ Dα x u = f (x, t), ∂t
(x, t) ∈ R × R+ ,
(6.16)
where α ∈ (0, 1), κ > 0 is convection velocity, and f (x, t) the source term. In the coming sections, we present numerical approximations to both the fractional diffusion operator (the case with 1 < α < 2) and the fractional convection operator (the case with 0 < α < 1). Note that in the following discussions we always denote Ψα = 2 cos1 πα and N = b−a h , (2) with h being the step size.
6.1 Indirect approximations to the fractional diffusion operator The definition of the Riesz derivative in equation (2.183) implies that it is a linear combination of the left- and right-sided Riemann-Liouville derivatives. As a result, suitable linear combinations of numerical approximations to the Riemann-Liouville derivative can evaluate the Riesz derivative. The corresponding accuracy therefore (at least) inherits those in the case of the RiemannLiouville derivative. Discussions on truncated errors of these indirect methods are omitted here.
6.1.1 Approximation based on the L2 method In [173], the left- and right-sided Riemann-Liouville derivatives with lower terminal a = 0 are (α) discretized based on the L2 method. Let dk = (k + 1)2−α − k 2−α , k = 0, 1, . . . , j − 1 or
242
Chapter 6. Numerical Riesz differentiation
k = 0, 1, . . . , N − j − 1. Then one has α RL D0,x f (x) x=x j x1−α f 0 (0) α x−α j j f (0) + + C D0,x f (x) x=x = j Γ(1 − α) Γ(2 − α) Z xj (2) x1−α f 0 (0) x−α f (xj − t)dt 1 j j f (0) + + = Γ(1 − α) Γ(2 − α) Γ(2 − α) 0 t1−α x−α f (0) x1−α f 0 (0) j ≈ j + Γ(1 − α) Γ(2 − α) Z j−1 X f (xj−k−1 ) − 2f (xj−k ) + f (xj−k+1 ) xk+1 α−1 t dt + h2 Γ(2 − α) xk k=0 ( h−α (1 − α)(2 − α)f (x0 ) (2 − α) [f (x1 ) − f (x0 )] ≈ + Γ(3 − α) jα j α−1 ) j−1 X (α) dk [f (xj−k+1 ) − 2f (xj−k ) + f (xj−k−1 )] , +
(6.17)
k=0
where x0 = 0. Similarly, it holds that α RL Dx,b f (x) x=x j ( −α (1 − α)(2 − α)f (xN ) (2 − α) [f (xN ) − f (xN −1 )] ≈ h + Γ(3 − α) (N − j)α (N − j)α−1 ) NX −j−1 (α) dk [f (xj+k−1 ) − 2f (xj+k ) + f (xj+k+1 )] . +
(6.18)
k=0
(x0 ) and f 0 (b) ≈ f (xN )−fh (xN −1 ) are utilized. Taking the Here the relations f 0 (0) ≈ f (x1 )−f h above two approximations into (2.183) gives following numerical scheme for the case with 1 < α < 2 and x ∈ [0, b]: α [RZ Dx f (x)]x=x(j (1 − α)(2 − α)f (x0 ) (2 − α) [f (x1 ) − f (x0 )] Ψα h−α + = − Γ(3 − α) jα j α−1 j−1 X (α) + dk [f (xj−k+1 ) − 2f (xj−k ) + f (xj−k−1 )] (6.19) k=0 (1 − α)(2 − α)f (xN ) (2 − α) [f (xN ) − f (xN −1 )] + + (N − j)α (N − j)α−1 ) NX −j−1 (α) dk [f (xj+k−1 ) − 2f (xj+k ) + f (xj+k+1 )] + O(h). + k=0
Remark 6.1. For the case with arbitrary interval [a, b], the above approximation still works after utilizing suitable affine transformations.
6.1. Indirect approximations to the fractional diffusion operator
243
6.1.2 Approximation based on spline interpolation In view of equations (2.183), (5.19), and (5.20), we can readily get an approximation to the Riesz derivative on the real axis with 1 < α < 2, based on spline interpolation. This numerical scheme reads as [RZ Dα x f (x)]x=xj j−1 X Ψ α alj−l,k − 2alj,k + alj+l,k f (xk ) ≈− α Γ(4 − α)h k=0 l l (6.20) + aj+1,j − 2aj,j f (xj ) + alj+1,j+1 f (xj+1 ) + arj−1,j−1 f (xj−1 ) + arj−1,j − 2arj,j f (xj ) N X arj−1,k − 2arj,k + arj,k f (xk ) , 1 < α < 2, + k=j+1
with the coefficients alj,k and arj,k are given by (5.18) and (5.21), respectively. In the particular case with a = −∞ and b = ∞, (5.30) and (5.31) also give [RZ Dα x f (x)]x=xj = −
∞ X Ψα qm [f (xj−m ) + f (xj+m )] + O(h2 ), Γ(4 − α)hα m=−1
(6.21)
where x ∈ R, 1 < α < 2, and the coefficients qm are given by (5.29). The truncated error is of O(h2 ) provided that f (4) (x) = 0 for x ≤ a ˜ and x ≥ ˜b with −∞ < a ˜ < ˜b < ∞.
6.1.3 Approximation based on Grünwald-Letnikov type formulae Based on Grünwald-Letnikov type approximations to the Riemann-Liouville derivative, a series of numerical evaluations of the Riesz derivative with integer-order accuracy can be obtained. These approximations are applicable for cases with both 1 < α < 2 and 0 < α < 1. Therefore, Grünwald-Letnikov type approximations of the fractional convection operator are omitted here when indirect evaluations of the Riesz derivative with 0 < α < 1 are introduced. (I) The classical Grünwald-Letnikov formulae Since the left- and right-sided Riemann-Liouville derivatives can be evaluated by [RL Dα a,x f (x)]x=xj =
j 1 X (α) ωk f (xj−k ) + O(h) hα
(6.22)
N −j 1 X (α) ωk f (xj+k ) + O(h), hα
(6.23)
k=0
and [RL Dα x,b f (x)]x=xj =
k=0
the classical Grünwald-Letnikov formula for the Riesz derivative is therefore in the form " j # N −j X Ψα X (α) (α) α [RZ Dx f (x)]x=xj = − α ωk f (xj−k ) + ωk f (xj+k ) + O(h). (6.24) h k=0
(α)
Here the coefficients are given by ωk
k=0
= (−1)k
α k
.
244
Chapter 6. Numerical Riesz differentiation
(II) The shifted Grünwald-Letnikov formulae Making use of the shifted Grünwald-Letnikov formulae of the left- and right-sided RiemannLiouville derivatives, α RL Da,x f (x) x=x j
=
j+1 X
(α)
ωk f (xj−k+1 ) + O(h)
(6.25)
k=0
and
NX −j+1 (α) α = ωk f (xj+k−1 ) + O(h), D f (x) RL x,b x=x j
(6.26)
k=0
we obtain the following 1st order shifted formula: [RZ Dα x f (x)]x=xj " j+1 # NX −j+1 X (α) Ψ α (α) = − ωk f (xj−k+1 ) + ωk f (xj+k−1 ) + O(h) hα k=0
(6.27)
k=0
(α)
with the coefficients given by ωk
= (−1)k
α k
.
Remark 6.2. We can choose other integer shifts, while the truncated error is still of O(h). In addition, the best performance appears when the shift p is chosen such that |p − α2 | is minimized. (III) The weighted and shifted Grünwald-Letnikov formulae We can also easily derive the weighted and shifted Grünwald-Letnikov formulae for the Riesz derivative. α+2 When f (x) ∈ L1 (R) and RL Dα+2 a,x f (x), RL Dx,b f (x), and their Fourier transforms belong 1 to L (R), Theorem 5.5 yields the following 2nd order numerical approximation: [RZ Dα x f (x)]x=xj j+l j+l X1 (α) X2 (α) = − Ψα ν1 ω f (x ) + ν ωk f (xj−k+l2 ) j−k+l1 2 k hα k=0 k=0 ! N −j+l N −j+l X 1 (α) X 2 (α) ωk f (xj+k−l1 ) + ν2 ωk f (xj+k−l2 ) + O(h2 ). +ν1 k=0
(6.28)
k=0
Here the coefficients are given by ν1 =
2l1 − α α − 2l2 , ν2 = 2(l1 − l2 ) 2(l1 − l2 )
(6.29)
with l1 and l2 being two arbitrary integers satisfying l1 − l2 6= 0. α+3 Provided that f (x) ∈ L1 (R) and RL Dα+3 a,x f (x), RL Dx,b f (x), and their Fourier transforms belong to L1 (R), then the 3rd order weighted and shifted Grünwald-Letnikov formula for the
6.1. Indirect approximations to the fractional diffusion operator
Riesz derivative is given by [RZ Dα x f (x)]x=xj j+s j+s X1 (α) X2 (α) Ψα κ ω f (x ) + κ ωk f (xj−k+s2 ) = − 1 j−k+s1 2 k hα k=0 k=0 j+s N −j+s X3 (α) X 1 (α) + κ ω f (x ) + κ ωk f (xj+k−s1 ) 3 j−k+s3 1 k k=0 k=0 ! N −j+s N −j+s X 2 (α) X 3 (α) ωk f (xj+k−s2 ) + κ3 ωk f (xj+k−s3 ) + O(h3 ). +κ2 k=0
Here ωkα =
245
(6.30)
k=0
(−1)k αk
and the integer shifts s1 , s2 , and s3 satisfy (s1 − s2 )(s2 − s3 )(s1 − s3 ) 6= 0.
(6.31)
12s2 s3 − (6s2 + 6s3 + 1)α + 3α2 , κ = 1 12(s2 s3 − s1 s2 − s1 s3 + s21 ) 12s1 s3 − (6s1 + 6s3 + 1)α + 3α2 , κ2 = 12(s1 s3 − s1 s2 − s2 s3 + s22 ) 12s1 s2 − (6s1 + 6s2 + 1)α + 3α2 . κ3 = 12(s1 s2 − s1 s3 − s2 s3 + s23 )
(6.32)
The weights are given by
6.1.4 Approximation based on fractional backward difference formulae and their modifications Now we turn to fractional backward difference formulae and their modifications for the RiemannLiouville derivative to evaluate the Riesz derivative. Most of the results are presented as follows. (I) The classical Lubich formulae Recall that, for the function f (x) satisfying f (k) (a+) = 0 (k = 0, 1, . . . , p − 1) and f (k) (b−) = 0 (k = 0, 1, . . . , p − 1), the classical Lubich formulae for the left- and right-sided RiemannLiouville derivatives are given by x−a [X h ] 1 (α) α $p,l f (x − lh) + O(hp ), p = 2, . . . , 6, RL Da,x f (x) = α h
(6.33)
l=0
and b−x [X h ] 1 (α) α $p,l f (x + lh) + O(hp ), p = 2, . . . , 6. RL Dx,b f (x) = α h
(6.34)
l=0
(α)
Here the coefficients $p,l are generated by (3.38), or its equivalent form in (5.207), say, !α p ∞ X X 1 (α) k (1 − z) = $p,l z l . (6.35) k k=1
l=0
246
Chapter 6. Numerical Riesz differentiation
The following assertion holds. Theorem 6.1. Assume that f (x) satisfies f (k) (a+) = 0 (k = 0, 1, . . . , p − 1) and f (k) (b−) = 0 (k = 0, 1, . . . , p − 1), p = 2, . . . , 6. Then the classical Lubich formula for the Riesz derivative is given by
α RZ Dx f (x)
x−a b−x [X [X h ] h ] Ψα (α) (α) $p,l f (x − lh) + $p,l f (x + lh) + O(hp ), =− α h l=0
(6.36)
l=0
(α)
in which the coefficients $p,l can be obtained through the relation p X 1 (1 − z)k k
!α =
k=1
∞ X
(α)
$p,l z l .
(6.37)
l=0 (α)
Remark 6.3. Here we present explicit expressions for the coefficients $p,l with p = 1, 2, 3. For (α)
p = 1, the coefficients $1,l are expressed as (α)
$1,l = (−1)l
α Γ(1 + α) = (−1)l . l Γ(l + 1)Γ(1 + α − l)
(6.38)
(α)
For p = 2, $2,l can be calculated by (α)
$2,l =
α X l k 1 3 (α) (α) $1,k $1,l−k . 2 3
(6.39)
k=0
When p = 3, it holds that k1 [X α X k1 −k2 k2 l 2 ] 7 11 2 (α) $ = 3,l 6 11 7 k1 =0 k2 =0
(6.40)
(−1)k2 (k1 − k2 )! (α) (α) × $1,l−k1 $1,k1 −k2 . k2 !(k1 − 2k2 )! (α)
It is shown in [33] that these coefficients $3,l (l ≥ 1) can also be computed by the following recurrence formula: l−1 1 X (α) (α) $3,l = [α(l − j) − j]$3,j ωl−j , (6.41) ω0 l j=0 in which ωj , j = 0, 1, 2, . . . , satisfy 3 ∞ X X 1 k (1 − z) = ωj z j , |z| < 1. k j=0
k=1
(6.42)
6.1. Indirect approximations to the fractional diffusion operator
247
(α)
It is evident that ωj = 0 for j ≥ 4. Precisely, we can compute $3,l as follows: α 11 (α) $3,0 , = 6 18α (α) (α) $3,1 = − $ , 11 3,0 i 3 h (α) (α) (α) $3,2 = −3(α − 1)$3,1 + 3α$3,0 , 11 3 6 (α) (α) (α) −3(α − l + 1)$3,l−1 + (2α − l + 2)$3,l−2 $ = 3,l 11l 2 1 (α) − (3α − l + 3)$3,l−3 , l ≥ 3. 3
(6.43)
(II) The shifted fractional backward difference formulae In view of Theorems 5.16 and 5.19, the following shifted fractional backward difference formulae can be readily obtained. Theorem 6.2. Assume that f (x) ∈ C 5 (R) and all the derivatives of f (x) up to order 6 belong to L1 (R). Then it holds that α RZ Dx f (x)
=−
∞ Ψα X (α) k2,l [f (x − (l − 1)h) + f (x + (l − 1)h)] + O(h2 ), hα
(6.44)
l=0
(α)
where k2,l are defined by (5.151). Theorem 6.3. Let p ≥ 3. Assume that f (x) ∈ C p+3 (R) and all the derivatives of f (x) up to order p + 4 belong to L1 (R). It holds that α RZ Dx f (x) ∞ i (6.45) Ψα X h (α) (α) = − k f (x − (l − 1)h) + k f (x + (l − 1)h) + O(hp ). p,l p,l α h l=0
(α)
Here the coefficients kp,l are given by (5.175). Remark 6.4. If f (x) is defined on [a, b] and satisfies the homogeneous conditions f (n) (a) = f (n) (b) = 0 with n = 0, 1, . . . , p − 1, by zero extension, [ x−a h +1] X Ψ α (α) α D f (x) = − kp,l f (x − (l − 1)h) RZ x α h l=0 (6.46) b−x +1 ] [ h X (α) + kp,l f (x + (l − 1)h) + O(hp ), p ≥ 2. l=0 (α)
Here the coefficients kp,l are given by (5.151) when p = 2 and by (5.175) when p ≥ 3. The above numerical approximations can be verified through the following example.
248
Chapter 6. Numerical Riesz differentiation
Table 6.1. The absolute errors and convergence orders of Example 6.4 by numerical scheme (6.46). Republished with permission of Elsevier from High-order numerical algorithms for Riesz derivatives via constructing new generating functions, H. F. Ding and C. P. Li, J. Sci. Comput., 71 (2017); permission conveyed through Copyright Clearance Center, Inc. α
1.1
1.3
1.5
1.7
1.9
h
the absolute errors
the convergence orders
1 20 1 40 1 80 1 160 1 320 1 20 1 40 1 80 1 160 1 320 1 20 1 40 1 80 1 160 1 320 1 20 1 40 1 80 1 160 1 320 1 20 1 40 1 80 1 160 1 320
1.024998E-02 2.637917E-03 6.696656E-04 1.687501E-04 4.235832E-05
∗ 1.9582 1.9779 1.9886 1.9942
1.086415E-02 2.765901E-03 6.986664E-04 1.756327E-04 4.403339E-05
∗ 1.9738 1.9851 1.9920 1.9959
1.019484E-02 2.571520E-03 6.466344E-04 1.621885E-04 4.061728E-05
∗ 1.9871 1.9916 1.9953 1.9975
8.591645E-03 2.150762E-03 5.387638E-04 1.348712E-04 3.374326E-05
∗ 1.9981 1.9971 1.9981 1.9989
6.328485E-03 1.576638E-03 3.939214E-04 9.847841E-05 2.462110E-05
∗ 2.0050 2.0009 2.0000 1.9999
Example 6.4. [38] Consider the function f (x) = x2 (1 − x)2 , x ∈ [0, 1]. The Riesz derivative of f (x) at x = 21 is given by [RZ Dα x f (x)]x= 1 = − 2
πα (α2 − 6α + 8)2α−1 sec . Γ(5 − α) 2
(6.47)
Table 6.1 lists the absolute errors and numerical convergence orders for scheme (6.46) with p = 2. From these results, we see that the convergence orders are in line with the theoretical analysis. (III) Fractional-compact formulae In view of Theorem 5.32 and Remark 5.19, we can easily derive the following 3rd order fractionalcompact formulae for the Riesz derivative.
6.1. Indirect approximations to the fractional diffusion operator
249
Theorem 6.5. Suppose that f (x) ∈ C 6 ([a, b]) , f (n) (a) = f (n) (b) = 0 with n = 0, 1, 2, and all derivatives of f (x) up to order 7 belong to L1 ([a, b]). Then, for any x ∈ [a, b], L α 3 L RZ Dα A2 f (x) + R Aα (6.48) x f (x) = −Ψα 2 f (x) + O(h ). R α Here L, L Aα 2 f (x), and A2 f (x) are given by (5.252), (5.259), and (5.260), respectively.
Theorem 6.6. Assume that f (x) ∈ C 6 ([a, b]) , f (n) (a) = f (n) (b) = 0, n = 0, 1, 2, and all derivatives of f (x) up to order 7 belong to L1 ([a, b]). Then it holds that h i L eα R eα Le RZ Dα f (x) = −Ψ A f (x) + A f (x) + O(h3 ). (6.49) α x 2 2 e L Aeα f (x), and R Aeα f (x) are given by (5.270), (5.275), and (5.276), respectively. Here L, 2 2 Recalling Theorem 5.35, the following 4th order fractional-compact formula for the Riesz derivative can be readily obtained. Theorem 6.7. Let f (x) ∈ C 7 ([a, b]), f (n) (a) = f (n) (b) = 0, n = 0, 1, 2, 3, and all derivatives of f (x) up to order 8 belong to L1 ([a, b]). Then for x ∈ [a, b], it holds that α H RZ Dx f (x) h i (6.50) (α) (α) L eα R α R eα = − Ψα σ e3,0 L Aα + A − σ A + A f (x) + O(h4 ). 2 2 2 2 3,0 (α)
3
2
(α)
+12α−4 ,σ e3,s = Here σ3,s = 3α −11α 12α2 H is defined in (5.277).
3α3 +11α2 +12α+4 , 12α2
and the fractional-compact operator
Here we illustrate the previous approximations through the following numerical example. Example 6.8. [36] Consider the function f (x) = x2 (1−x)2 , whose Riesz derivative at x = 12 is given by (6.47). Choosing different stepsizes h, we utilize (6.48), (6.49), and (6.50) to compute the Riesz derivative of f (x) at x = 21 . Tables 6.2–6.4 display numerical results for different α.
6.1.5 Approximation based on fractional average central difference formulae Taking the average fractional central difference formulae (5.304) and (5.311) into (2.183), we obtain the following approximations given by [40]. α+2 Theorem 6.9. Suppose that f (x) and the Fourier transforms of RL Dα+2 −∞,x f (x) and RL Dx,+∞ f (x) 1 are in L (R). Then it holds that Dα f (x) RZ x "∞ X Ψ α k α = − (−1) (f (x − kh) + f (x − (k − α)h)) 2hα k (6.51) k=0 # ∞ X α + (−1)k (f (x + kh) + f (x + (k − α)h)) + O(h2 ) k k=0
250
Chapter 6. Numerical Riesz differentiation
Table 6.2. The absolute errors and convergence orders of Example 6.8 by numerical scheme (6.48). Reprinted with permission from Fract. Calc. Appl. Anal., 20 (2017), at https://www. degruyter.com/view/j/fca α
1.1
1.5
1.9
h
the absolute errors
the convergence orders
1 20 1 40 1 80 1 160 1 320 1 20 1 40 1 80 1 160 1 320 1 20 1 40 1 80 1 160 1 320
1.740717E-04 2.185595E-05 2.742123E-06 3.434158E-07 4.296784E-08
∗ 2.9936 2.9947 2.9973 2.9986
1.377134E-04 1.716087E-05 2.143372E-06 2.678606E-07 3.348027E-08
∗ 3.0045 3.0012 3.0003 3.0001
1.056422E-05 1.364672E-06 1.735867E-07 2.189369E-08 2.749249E-09
∗ 2.9526 2.9748 2.9871 2.9934
Table 6.3. The absolute errors and convergence orders of Example 6.8 by numerical scheme (6.49). Reprinted with permission from Fract. Calc. Appl. Anal., 20 (2017), at https://www. degruyter.com/view/j/fca α
1.1
1.5
1.9
h
the absolute errors
the convergence orders
1 20 1 40 1 80 1 160 1 320 1 20 1 40 1 80 1 160 1 320 1 20 1 40 1 80 1 160 1 320
5.290778E-02 6.548041E-03 8.150577E-04 1.016718E-04 1.269602E-05
∗ 3.0143 3.0061 3.0030 3.0015
1.263828E-02 1.583605E-03 1.966563E-04 2.448135E-05 3.053477E-06
∗ 2.9965 3.0095 3.0059 3.0032
2.787724E-03 3.697284E-04 4.637689E-05 5.791288E-06 7.231609E-07
∗ 2.9145 2.9950 3.0014 3.0015
6.1. Indirect approximations to the fractional diffusion operator
251
Table 6.4. The absolute errors and convergence orders of Example 6.8 by numerical scheme (6.50). Reprinted with permission from Fract. Calc. Appl. Anal., 20 (2017), at https://www. degruyter.com/view/j/fca α
1.1
1.5
1.9
h
the absolute errors
the convergence orders
1 20 1 40 1 80 1 160 1 320 1 20 1 40 1 80 1 160 1 320 1 20 1 40 1 80 1 160 1 320
8.281680E-07 5.167207E-08 3.218255E-09 2.007975E-10 1.240019E-11
∗ 4.0025 4.0050 4.0025 4.0173
5.084772E-07 3.725522E-08 2.454356E-09 1.567338E-10 9.262702E-12
∗ 3.7707 3.9240 3.9690 4.0807
9.867011E-08 7.533874E-09 5.041596E-10 3.322587E-11 6.561640E-12
∗ 3.7111 3.9014 3.9235 2.3402
uniformly for x ∈ R as h → 0+ . Furthermore, for f (x) defined on [a, b] with f (n) (a) = f (n) (b) = 0, n = 0, 1, one has α RZ Dx f (x) x−a [X h ] Ψ α k α (−1) (f (x − kh) + f (x − (k − α)h)) = − 2hα k k=0
+
b−x [X h ]
k=0
α (−1) (f (x + kh) + f (x + (k − α)h)) + O(h2 ). k k
Theorem 6.10. Let f (x) ∈ L1 (R), RL Dα+4 −∞,x f (x), forms be in L1 (R). Then there holds that
α+4 RL Dx,+∞ f (x),
and their Fourier trans-
α(3α + 1) 2 1+ δx RZ Dα x f (x) 24 "∞ Ψα X α =− α (−1)k (f (x − kh) + f (x − (k − α)h)) 2h k k=0 # ∞ X k α + (−1) (f (x + kh) + f (x + (k − α)h)) + O(h4 ) k k=0
(6.52)
(6.53)
252
Chapter 6. Numerical Riesz differentiation
uniformly for x ∈ R as h → 0+ . Here δx2 denotes the 2nd order central difference operator defined by δx2 f (xj ) = f (xj+1 ) − 2f (xj ) + f (xj−1 ). Furthermore, for f (x) defined on [a, b] with f (n) (a) = f (n) (b) = 0, n = 0, 1, 2, 3, one has α(3α + 1) 2 1+ δx RZ Dα x f (x) 24 x−a [X h ] Ψ k α = − α (−1) (f (x − kh) + f (x − (k − α)h)) 2hα k (6.54) k=0 b−x [ h ] X α (−1)k (f (x + kh) + f (x + (k − α)h)) + O(h4 ). + k k=0
The following numerical examples illustrate approximations (6.52) and (6.54). Example 6.11. [40] Consider the function f (x) = x2 (1 − x2 ), whose Riesz derivative at x = 1 2 is given by (6.47). Table 6.5 lists the absolute errors and convergence orders given by the numerical scheme (6.52) at x = 12 with different α. It confirms the 2nd order accuracy of (6.52).
Table 6.5. The absolute errors and convergence orders of Example 6.11 by numerical scheme (6.52). Reprinted by permission of Taylor & Francis Ltd (http://www.tandfonline.com) from “Highorder algorithms for Riesz derivative and their applications: Revisited” by H. F. Ding, C. P. Li, and Y. Q. Chen in Numerical Functional Analysis and Optimization, Vol. 38, No. 7, 2017. α
1.2
1.4
1.6
1.8
h
absolute error
convergence order
1 10 1 20 1 40 1 80 1 10 1 20 1 40 1 80 1 10 1 20 1 40 1 80 1 10 1 20 1 40 1 80
2.371107E-03 5.878877E-04 1.464631E-04 3.655831E-05
∗ 2.0119 2.0050 2.0023
3.614913E-03 8.891952E-04 2.207491E-04 5.500869E-05
∗ 2.0234 2.0101 2.0047
3.311900E-003 8.088550E-004 2.001866E-004 4.981273E-005
∗ 2.0337 2.0145 2.0068
1.748268E-03 4.265789E-04 1.055454E-04 2.625954E-05
∗ 2.0350 2.0149 2.0069
6.2. Direct approximations to the fractional diffusion operator
253
Example 6.12. [40] Consider the function f (x) = x6 (1 − x)6 , x ∈ [0, 1]. The corresponding Riesz derivative is 8
α RZ Dx f (x)
=
πα X Γ(` + 7) 1 x e sec x`+6−α + (1 − x)`+6−α . 2 2 Γ(` + 7 − α)
(6.55)
`=0
Table 6.6 displays the absolute errors and convergence orders at x = 21 given by the numerical scheme (6.54) with different α. It verifies that (6.54) is indeed of 4th order accuracy. Table 6.6. The absolute errors and convergence orders of Example 6.12 by numerical scheme (6.54). Reprinted by permission of Taylor & Francis Ltd (http://www.tandfonline.com) from “Highorder algorithms for Riesz derivative and their applications: Revisited” by H. F. Ding, C. P. Li, and Y. Q. Chen in Numerical Functional Analysis and Optimization, Vol. 38, No. 7, 2017. α
1.2
1.4
1.6
1.8
h
absolute error
convergence order
1 20 1 40 1 80 1 20 1 20 1 40 1 80 1 20 1 20 1 40 1 80 1 20 1 20 1 40 1 80 1 20
7.029869E-07 4.446255E-08 2.787163E-09 1.743236E-10
∗ 3.9828 3.9957 3.9990
4.372243E-08 3.985260E-09 2.690303E-10 1.712778E-11
∗ 3.4556 3.8888 3.9734
3.751536E-06 2.440706E-07 1.540687E-08 9.653215E-10
∗ 3.9421 3.9857 3.9964
1.592788E-05 1.034107E-06 6.524548E-08 4.087469E-09
∗ 3.9451 3.9864 3.9966
6.2 Direct approximations to the fractional diffusion operator Instead of deriving numerical approximations to the Riesz derivative based on those to the Riemann-Liouville derivative, we introduce some other methods for evaluating Riesz derivatives in direct ways.
6.2.1 Asymmetric centered difference operator Slightly different from the fractional average central difference operator, the symmetric fractional centered difference operator is defined by ∆α h f (x)
=
∞ X k=−∞
(α)
gk f (x − kh),
(6.56)
254
Chapter 6. Numerical Riesz differentiation (α)
where gk
=
(−1)k Γ(α+1) . Γ( α −k+1 )Γ( α2 +k+1) 2 1
The series (6.56) converges absolutely for any bounded
function f ∈ L (R). To see this, we recall the equalities N
X ck Γ(z + a) + z a−b O |z|−N −1 , | arg(z + a)| < π, |z| → ∞, = z a−b k Γ(z + b) z
(6.57)
k=0
and Γ(α + 1) sin (π(β − α)) Γ(α + 1)Γ(β − α) = . Γ(β + 1)Γ(α − β + 1) π Γ(β + 1)
(6.58)
Then we have (α)
gk by setting β =
α 2
= O |k|−α−1
(6.59)
− k. Here the coefficients ck are expressed in terms of generalized Bernoulli k
(b−a)k a−b+1 α Bk (a), with (z)n = Γ(z+n) polynomials by the formula ck = (−1) k! Γ(z) and Bk (x) given α ∞ P k by ezz−1 exz = Bkα (x) zk! , |z| ≤ 2π. Consequently, the series (6.56) converges absolutely k
for any bounded function f ∈ L1 (R). (α) For properties of the coefficients gk , we present the following assertions. (α)
Theorem 6.13. [17] Let gk
=
(−1)k Γ(α+1) Γ( α −k+1 )Γ( α2 +k+1) 2
be the coefficients of the centered finite
difference operator (6.56) with k = 0, ∓1, ∓2, . . . and α > −1. Then g0(α) ≥ 0, g (α) = g (α) ≤ 0 for all |k| ≥ 1. −k k (α)
Proof. It is evident that gk
(6.60)
(α)
= g−k . Let us consider the (k + 1)th coefficient (α)
gk+1 =
(−1)k+1 Γ(α + 1) . Γ α2 − k Γ α2 + k + 2
(6.61)
We can write the coefficients recursively as (α)
gk+1 =
1−
α 2
α+1 +k+1
(α)
gk
(6.62)
in view of the relation Γ(z + 1) = zΓ(z). Note that Γ(z) > 0 for z > 0. It holds that Γ(α) (α) ≥ 0, g = α 0 Γ 2 + 1 Γ α2 + 1 α+1 (α) (α) gk+1 = 1 − α gk ≤ 0, k = ∓1, ∓2, . . . . + k + 1 2 This ends the proof.
(6.63)
6.2. Direct approximations to the fractional diffusion operator (α)
Theorem 6.14. [37] If 1 < α < 2, then the coefficients gk (I) S(α)
α+4 α+2k
2(α+1)
(α)
< |gk | < S(α)
α+6 α+2(k+1)
α+1
255
satisfy the following properties.
, k ≥ 3,
Γ(α+1) . where S(α) = − Γ α −1 ( 2 )Γ( α2 +3) 2(α+1)
(II) P1 (m, n, α) (α+4) 2(2α+1)
0 independent of small |y|. We have ν(y) = 1 + O(y 2 )
(6.71)
for small y. Since f ∈ C 5 (R) and all derivatives up to order 5 belong to L1 (R), there exists a positive constant C0 such that b (6.72) f (ω) ≤ C0 (1 + |ω|)−5 . Therefore, we have b ω) =|ω|α |ν(ωh) − 1| fb(ω) ≤ |ω|α C|ωh|2 C0 (1 + |ω|)−5 φ(h,
(6.73)
≤C3 h2 |ω|α+2 (1 + |ω|)−5 = C3 h2 (1 + |ω|)α−3 ,
where C3 = CC0 is independent of ω. That is, the inverse Fourier transform of the function φb exists for 1 < α ≤ 2. Therefore the inverse Fourier transform gives
where
with C4 =
α −h−α ∆α h f (x) = RZ Dx f (x) + φ(h, x),
(6.74)
Z Z ∞ 1 ∞ −iωx b 1 b |φ(h, x)| = e φ(h, ω)dω ≤ φ(h, ω) dω 2π −∞ 2π −∞ Z ∞ 1 ≤ C3 h2 (1 + |ω|)α−3 dω = C4 h2 2π −∞
(6.75)
C3 π(2−α) .
This ends the proof.
6.2. Direct approximations to the fractional diffusion operator
257
Remark 6.5. (I) The fractional centered difference operator (6.56) can be regarded as a generalization of the 2nd order centered difference operator [41], i.e., lim ∆α h f (x) = f (x − h) − 2f (x) + f (x + h).
α→2
(6.76)
(II) For a function defined on [a, b] with homogeneous boundary conditions, one can also obtain a truncated version of (6.64) to evaluate the Riesz derivative.
6.2.2 Weighted and shifted centered difference operators We can also modify the symmetric fractional centered difference operator to obtain the weighted and shifted centered difference operators. Theorem 6.16. [40] Let f (x) lie in C 7 (R) and its derivatives up to order 7 belong to L1 (R). Set ∞ X
Lθ f (x) =
(α)
gk f (x − (k + θ)h),
θ = −1, 0, 1,
(6.77)
k=−∞
in which (α)
gk
=
Γ
α 2
(−1)k Γ(α + 1) . − k + 1 Γ α2 + k + 1
(6.78)
Then it holds that α RZ Dx f (x)
=
i α 1 hα α L f (x) − 1 + L f (x) + O(h4 ). L f (x) + −1 1 0 hα 24 12 24 (α)
Proof. Recall that the generating function for the coefficients gk P∞ (α) ikx . Applying the Fourier transform gives k=−∞ gk e
(6.79)
α is given by 2 sin x2 =
i α α 1 hα L−1 f (x) − 1 + L0 f (x) + L1 f (x) ; ω F hα 24 12 24 " ∞ ∞ 1 α X (α) i(k−1)ωh b α X (α) ikωh b = g e f (ω) − 1 + gk e f (ω) k 12 hα 24 k=−∞ k=−∞ # ∞ α X (α) i(k+1)ωh b + gk e f (ω) 24 k=−∞ α i 1 h α ωh b f (ω). = − α 1 + (1 − cos(ωh)) 2 sin h 12 2
(6.80)
h i b ω) =F 1 α L−1 f (x) − 1 + α L0 f (x) + α L1 f (x) ; ω φ(h, hα 24 12 24 + |ω|α fb(ω),
(6.81)
Set
258
Chapter 6. Numerical Riesz differentiation
then b ω) φ(h, ) ( i 2 sin ωh α h α α 2 fb(ω) (1 − cos(ωh)) 1− 1+ =|ω| ωh 12 n h i α α =|ω|α 1 − 1 + (ωh)2 − (ωh)4 + O(ωh)6 24 288 α α−1 1 2 4 6 × 1 − (ωh) + α + (ωh) + O(ωh) fb(ω) 24 1920 1152 11 α 4 6 α + (ωh) − O(ωh) fb(ω). = − |ω| α 1152 2880
(6.82)
Since f (x) ∈ C 7 (R) and its derivatives up to order 7 belong to L1 (R), there exists a positive e1 such that constant C e1 (1 + |ω|)−7 . |fb(ω)| ≤ C (6.83) Therefore, b ω)| ≤ C e2 h4 |ω|4+α |fb(ω)| |φ(h, e2 h4 (1 + |ω|)4+α |fb(ω)| ≤C e3 h4 (1 + |ω|)α−3 , ≤C
(6.84)
e3 = C e1 C e2 . where C Consequently, the inverse Fourier transform yields
α RZ Dx f (x)
h i = 1 α L f (x) − 1 + α L f (x) + α L f (x) − φ(h, x), −1 0 1 hα 24 12 24
(6.85)
where Z 1 b −iωx φ(h, ω)e dω |φ(h, x)| = 2π R Z 1 b ω)|dω ≤ |φ(h, 2π R e3 Z C α−3 ≤ (1 + |ω|) dω h4 2π R 4 e = Ch e = with C
e3 C (2−α)π .
(6.86)
In other words, the relation (6.79) is valid. The proof is thus completed.
Remark 6.6. For the function f ∈ C 7 ([a, b]) with all the derivative up to order 7 belonging to L1 ([a, b]), zero extension gives
6.2. Direct approximations to the fractional diffusion operator
[RZ Dα x f (x)]x=xj j−1 X α (α) gk f (xj − (k + 1)h) = 24hα k=−N +j+1 j−1 X α 1 (α) gk f (xj − kh) − 1 + α 12 h k=−N +j+1 j−1 X α (α) + gk f (xj − (k − 1)h) + O(h4 ), 24hα
259
(6.87)
k=−N +j+1
provided that f (n) (a) = f (n) (b) = 0, n = 0, 1, 2, 3. By the same reasoning as Theorem 6.16, we have the following assertions given by [41]. Theorem 6.17. Assume that f (x) ∈ C 9 (R) with derivatives up to order 9 belonging to L1 (R). Then it holds that Dα f (x) = 1 [A L f (x) + A L f (x) + A L f (x) RZ x 1 −2 2 −1 3 0 hα (6.88) 6 +A2 L1 f (x) + A1 L2 f (x)] + O(h ), in which
11 α A = − + α, 1 1152 2880 α 41 A2 = + α, 288 720 2 17α α A3 = − + +1 . 192 160
(6.89)
Theorem 6.18. Assume that f (x) lies in C 11 (R) with derivatives up to order 11 belonging to L1 (R). Then it holds that α RZ Dx f (x) 1 (6.90) = α [B1 L−3 f (x) + B2 L−2 f (x) + B3 L−1 f (x) + B4 L0 f (x) h +B3 L1 f (x) + B2 L2 f (x) + B1 L3 f (x)] + O(h8 ). Here
2 11α 191 α + + α, B = 1 82944 69120 362880 2 α 7α 211 B = − + + α, 2 13824 3840 30240 5α2 3α 7843 B3 = + + α, 27648 512 120960 5α3 29α2 5297α B4 = − + + +1 . 20736 3456 45360
(6.91)
260
Chapter 6. Numerical Riesz differentiation
Remark 6.7. Much higher-order difference operators, such as 10th order, 12th order, and so on, are also presented in [41], which are in the form α RZ Dx f (x)
= Hpα f (x) + O(hp ), p = 10, 12, . . . ,
(6.92)
with Hpα f (x)
1 X zθ,p = α h θ∈χ
∞ X
! (α) gk f (x
− (k + θ)h) .
(6.93)
k=−∞
Here θ ∈ χ = {0, ±1, ±2, ±3, . . .} and the coefficients zθ,p are determined by the Fourier transform method. The above approximations can be truncated to evaluate Riesz derivatives of functions defined on the bounded domain [a, b] whenever homogeneous boundary conditions are satisfied.
6.2.3 Compact centered difference operators Apart from the weighted and shifted centered difference operators, there exists another modification of the centered difference operator that is also of high-order accuracy, called the compact centered difference approach. Theorem 6.19. [37] Suppose that f (x) ∈ C 2n+3 (R), and all the derivatives of f (x) up to order 2n + 4 exist and belong to L1 (R). Then (δx0 − bn−1 δx2n−2 )RZ Dα x f (x) ! n−2 X ∆α 2l h f (x) + O(h2n ), n ∈ Z+ , = b δ − l x hα
(6.94)
l=0
where δx2l f (xj )
2l X
2l = (−1) f (xl+j−s ), l ≥ 0. s s=0 s
(6.95)
Specifically, δx0 is the identity operator, i.e., δx0 f (xj ) = f (xj ). The coefficients bl (l = 0, 1, . . . , n − 2) satisfy the equation n−2 l−1 n−1 X X X n−1−q X (−1)s+q (l − s)2q 2l ap s |ωh|2(p+q) b 2 l (2q)! s=0 q=0 p=0 l=0 ! n−1 X l 2l 2p +(−1) ap |ωh| l p=0 ! n−2 2n−2 X s 2n − 2 2(n − 1 − s) (−1) (−1)n−1 |ωh|2n−2 , =1 − bn−1 s (2n − 2)! s=0
(6.96)
6.2. Direct approximations to the fractional diffusion operator
261
and ap (p = 0, 1, . . .) satisfy the equation ∞ 2sin ωh α X 2p 2 ap |ωh| = ωh p=0 α α−1 1 2 = 1 − |ωh| + α|ωh|4 + 24 1920 1152 α−1 (α − 1)(α − 2) 1 6 − + + α|ωh| + · · · . 322560 46080 82944
(6.97)
Proof. If n = 1, then equation (6.94) is reduced to the 2nd order scheme
α RZ Dx f (x)
=−
∆α h f (x) + O(h2 ), hα
(6.98)
which was considered in [17]. The case n = 2 has already been mentioned in [41]. Here we consider the case with arbitrary positive integer n ≥ 3. Let now
Kn =
δx0
−
bn−1 δx2n−2
, Ln =
n−2 X
bl δx2l , n ≥ 3.
(6.99)
l=0
Taking the Fourier transform of Kn RZ Dα x f (x) gives 0 2n−2 F {Kn RZ Dα ) RZ Dα x f (x); ω} = F (δx − bn−1 δx x f (x); ω n−2 X 2n − 2 2(n − 1 − s)2n−2 = − |ω|α 1 − bn−1 (−1)s s (2n − 2)! s=0 !! n−1 2n−2 2n−2 n 2n 2n ×(−1) |ω| h + (−1) |ω| O(h ) fb(ω),
(6.100)
where we have used the relation 2n−2 δx f (x) 2n−2 X s 2n − 2 = (−1) f (x + (n − 1 − s)h) s s=0 n−2 X 2(n − 1 − s)2n−2 ∂ 2n−2 f (x) s 2n − 2 h2n−2 = (−1) 2n−2 (2n − 2)! ∂x s s=0 2n + ∂ f (x) O(h2n ). ∂x2n
(6.101)
262
Chapter 6. Numerical Riesz differentiation
Similarly, we also have ∆α h f (x) ;ω F Ln − hα ( ! ∞ ) n−2 X X (α) 1 2l =−F bl δx gk f (x − kh); ω hα l=0 k=−∞ ( n−2 ∞ l X X 1 X (α) b = − F g (−1)l−|s| l k hα l=0 k=−∞ s=−l ! ) 2l × f (x − (k + s)h) ; ω l − |s| ! ∞ n−2 l X (α) 2l 1 X X l−|s| isωh bl (−1) e gk eikωh fb(ω) =− α h l − |s| l=0 s=−l k=−∞ !! n−2 l−1 X X 2l 2l 1 bl 2(−1)s cos((l − s)ωh) + (−1)l =− α s l h s=0 l=0 ∞ X (α) × gk eikωh fb(ω) k=−∞ X ∞ n−2 l−1 X ((l − s)ωh)2q 1 X s 2l (−1)q b 2(−1) = − l α s q=0 h (2q)! s=0 l=0 !! X ∞ (α) l 2l +(−1) gk eikωh fb(ω) l k=−∞ n−1 n−2 l−1 2q X X X q ((l − s)ωh) α s 2l (−1) = − |ω| b 2(−1) l s q=0 (2q)! s=0 l=0 !! ! 2 sin ωh α 2l 2 +(−1)l + |ω|2n O(h2n ) fb(ω) l ωh n−1 n−2 l−1 2q X X X α s 2l q ((l − s)ωh) = − |ω| b 2(−1) (−1) l s q=0 (2q)! s=0 l=0 !! ! n−1 X 2l +(−1)l ap |ωh|2p + |ω|2n O(h2n ) fb(ω). l
(6.102)
p=0
The above two equations lead to α b h) = F {KnRZ Dα f (x); ω} − F Ln − ∆h f (x) ; ω δ(ω, x hα b1 h2n |ω|2n+α fb(ω). =C
(6.103)
6.2. Direct approximations to the fractional diffusion operator
263
Since f (x) ∈ C 2n+3 (R) and its derivatives up to order 2n + 4 belong to L1 (R), there exists b2 such that a positive constant C b2 (1 + |ω|)−(2n+4) . |fb(ω)| ≤ C
(6.104)
b h)| ≤C b1 |ω|α+2n h2n |fb(ω)| |δ(ω, b1 (1 + |ω|)α+2n h2n |fb(ω)| ≤C b3 (1 + |ω|)α−4 h2n , ≤C
(6.105)
Therefore,
b3 = C b1 C b2 . Furthermore, taking the inverse Fourier transform of both sides of (6.103) where C gives ∆α α h f (x) Kn RZ Dx f (x) − Ln − = |δ(ω, h)| hα Z 1 b −iωh = δ(ω, h)e dω 2π R Z 1 b (6.106) δ(ω, h) dω ≤ 2π R b3 Z C α−4 (1 + |ω|) dω h2n ≤ 2π R b 2n =Ch , i.e., δx0
2n−2
− bn−1 δx
α RZ Dx f (x)
=
n−2 X
bl δx2l
l=0
b= where n ∈ Z+ and C
b3 C (3−α)π .
!
∆α f (x) − hα h
+ O(h2n ),
(6.107)
This completes the proof.
Remark 6.8. In view of the analysis in [62, 125, 167], the conditions stated in Theorem 6.19 can be weakened as f (x) ∈ L2n+α (R), where L2n+α (R) =
Z f f ∈ L1 (R), and (1 + |ω|)2n+α |fb(ω)|dω < ∞ .
(6.108)
R
Remark 6.9. It should be noted that some suitable smooth conditions for the given function f (x) are necessary and cannot be dropped. Once these conditions are not met, the expected accuracy cannot be achieved. Example 6.21 will verify this fact. Remark 6.10. For function f (x) defined on the bounded interval [a, b] with f (k) (a) = f (k) (b) = 0, 0 ≤ k ≤ 2n − 1, we can define its zero extension outside of the boundary by ( f˜(x) =
f (x),
x ∈ [a, b],
0,
otherwise.
(6.109)
264
Chapter 6. Numerical Riesz differentiation
If the extended function f˜(x) defined on R satisfies the conditions of Theorem 6.19 or Remark 6.9, then the difference scheme (6.94) can be written as the following form for any point x ∈ [a, b]: (δx0 − bn−1 δx2n−2 )RZ Dα x f (x) x−a n−2 h −l X X 1 (α) gk f (x − kh) + O(h2n ), n ∈ Z+ . bl δx2l = − α h
(6.110)
k=− b−x h +l
l=0
Here we list the commonly used even-order fractional-compact numerical differential formulae: ∆α 0 h f (x) (δx0 − b1 δx2 )RZ Dα f (x) = b δ − + O(h4 ), 0 x x hα ∆α h f (x) + O(h6 ), − hα
(6.112)
∆α 0 2 4 h f (x) (δx0 − b3 δx6 )RZ Dα f (x) = (b δ + b δ + b δ ) − + O(h8 ), 0 x 1 x 2 x x hα
(6.113)
(δx0
(δx0
(6.111)
−
−
b2 δx4 )RZ Dα x f (x)
b4 δx8 )RZ Dα x f (x)
=
(b0 δx0
+
=
(b0 δx0
b1 δx2
+
+
b1 δx2 )
b2 δx4
+
b3 δx6 )
∆α h f (x) − + O(h10 ), hα
(6.114)
where b0 = 1, α b1 = − , 24 11 α b2 = + α, 2880 1152 191 11α α2 b3 = − + + α, 362880 69120 82944 2497 10181α 11α2 α3 b4 = + + + α. 29030400 348364800 3317760 7962624
(6.115)
Note also that when α → 2 equations (6.111)–(6.114) are reduced into classical compact formulae, 0 δx + 0 δx − δx0 + δx0 −
1 1 2 ∂ 2 f (x) δx = 2 δx2 f (x) + O(h4 ), 12 ∂x2 h 1 4 ∂ 2 f (x) 1 1 2 2 0 δx = δ − δ δ f (x) + O(h6 ), x 90 ∂x2 h2 12 x x 1 1 2 1 4 2 1 6 ∂ 2 f (x) 0 δx = δ − δ + δ δ f (x) + O(h8 ), x 560 ∂x2 h2 12 x 90 x x 1 8 ∂ 2 f (x) 1 1 2 1 4 1 6 2 0 δ = 2 δx − δx + δx − δ δ f (x) + O(h10 ). 3150 x ∂x2 h 12 90 560 x x (6.116)
6.2. Direct approximations to the fractional diffusion operator
265
Example 6.20. [37] Consider the function fn (x) = x2n (1 − x)2n , x ∈ [0, 1], n = 2, 3, 4, 5. Its Riesz derivative is α RZ Dx f (x)
= −Ψα
2n X
(−1)l
l=0
(2n)!(2n + l)![x2n+l−α + (1 − x)2n+l−α ] . l!(2n − l)!Γ(2n + l + 1 − α)
(6.117)
Utilize numerical schemes (6.111)–(6.114) to compute the Riesz derivative of f (x) at x = The absolute error is denoted by E(h) and the experimental convergence order (ECO) is calculated by E(h1 ) h1 ECO = log log . (6.118) E(h2 ) h2 1 2.
The numerical results are displayed in Tables 6.7–6.10. We can see that these experimental orders are in line with the theoretical analysis.
Table 6.7. The absolute error E(h) and the experimental convergence order (ECO) of function f2 (x) by numerical scheme (6.111). Numerical methods for partial differential equations by JOHN/WILEY & SONS, INC. Reproduced with permission of JOHN/WILEY & SONS, INC. in the format Book via Copyright Clearance Center. α
1.1
1.5
1.9
h
E(h)
ECO
1 20 1 40 1 80 1 160 1 320 1 20 1 40 1 80 1 160 1 320 1 20 1 40 1 80 1 160 1 320
1.985528E-06 1.247417E-07 7.806460E-09 4.880592E-10 3.050456E-11
∗ 3.9925 3.9981 3.9995 4.0000
5.712995E-06 3.588944E-07 2.245955E-08 1.404168E-09 8.777425E-11
∗ 3.9926 3.9982 3.9995 3.9998
1.486627E-05 9.335587E-07 5.841643E-08 3.652104E-09 2.284402E-10
∗ 3.9932 3.9983 3.9996 3.9988
Example 6.21. [37] Consider the function f (x) = x(1 − x) for x ∈ [0, 1]. This function fails to meet the conditions of Theorem 6.19 and Remark 6.8. We numerically compute RZ Dα x f (x) by using the 4th order formula (6.111) and display the numerical results in Table 6.11. We can see that the expected convergence order of (6.111) cannot be obtained, which verifies the assertion in Remark 6.9.
266
Chapter 6. Numerical Riesz differentiation
Table 6.8. The absolute error E(h) and the experimental convergence order (ECO) of function f3 (x) by numerical scheme (6.112). Numerical methods for partial differential equations by JOHN/WILEY & SONS, INC. Reproduced with permission of JOHN/WILEY & SONS, INC. in the format Book via Copyright Clearance Center. α
1.1
1.5
1.9
h
E(h)
ECO
1 20 1 24 1 28 1 32 1 36 1 20 1 24 1 28 1 32 1 36 1 20 1 24 1 28 1 32 1 36
3.120201E-08 1.057512E-08 4.223802E-09 1.904343E-09 9.422903E-10
∗ 5.9345 5.9537 5.9656 5.9735
1.123916E-07 3.810009E-08 1.521978E-08 6.862760E-09 3.396074E-09
∗ 5.9333 5.9528 5.9648 5.9728
3.675466E-07 1.245861E-07 4.976658E-08 2.244006E-08 1.110461E-08
∗ 5.9338 5.9530 5.9649 5.9727
Table 6.9. The absolute error E(h) and the experimental convergence order (ECO) of function f4 (x) by numerical scheme (6.113). Numerical methods for partial differential equations by JOHN/WILEY & SONS, INC. Reproduced with permission of JOHN/WILEY & SONS, INC. in the format Book via Copyright Clearance Center. α
1.1
1.5
1.9
h
E(h)
ECO
1 30 1 34 1 38 1 42 1 46 1 30 1 34 1 38 1 42 1 46 1 30 1 34 1 38 1 42 1 46
3.442344E-11 1.279869E-11 5.303531E-12 2.397829E-12 1.164521E-12
∗ 7.9048 7.9206 7.9315 7.9393
1.459781E-10 5.427806E-11 2.248828E-11 1.016534E-11 4.936491E-12
∗ 7.9044 7.9220 7.9335 7.9401
5.601931E-10 2.082444E-10 8.624075E-11 3.895758E-11 1.890060E-11
∗ 7.9062 7.9260 7.9401 7.9506
6.2. Direct approximations to the fractional diffusion operator
Table 6.10. The absolute error E(h) and the experimental convergence order (ECO) of function f5 (x) by numerical scheme (6.114). Numerical methods for partial differential equations by JOHN/WILEY & SONS, INC. Reproduced with permission of JOHN/WILEY & SONS, INC. in the format Book via Copyright Clearance Center. α
1.1
1.5
1.9
h
E(h)
ECO
1 30 1 34 1 38 1 42 1 46 1 30 1 34 1 38 1 42 1 46 1 30 1 34 1 38 1 42 1 46
2.669378E-12 6.799466E-13 2.057061E-13 7.247668E-14 2.983513E-14
∗ 9.5568 9.5521 9.3790 8.8684
1.316180E-11 3.294738E-12 9.747428E-13 3.301910E-13 1.257671E-13
∗ 9.6784 9.7306 9.7325 9.6443
5.640743E-11 1.401455E-11 4.097488E-12 1.354007E-12 4.868649E-13
∗ 9.7309 9.8250 9.9555 10.2198
Table 6.11. The numerical results of Example 6.21 by using the 4th order fractional-compact formula (6.111). Numerical methods for partial differential equations by JOHN/WILEY & SONS, INC. Reproduced with permission of JOHN/WILEY & SONS, INC. in the format Book via Copyright Clearance Center. α
1.1
1.5
1.9
h
E(h)
ECO
1 10 1 20 1 40 1 80 1 160 1 10 1 20 1 40 1 80 1 160 1 10 1 20 1 40 1 80 1 160
5.900848E-04 1.470732E-04 3.674047E-05 9.183380E-06 2.295736E-06
∗ 2.0044 2.0011 2.0003 2.0001
7.105214E-04 1.766378E-04 4.409782E-05 1.102061E-05 2.754912E-06
∗ 2.0081 2.0020 2.0005 2.0001
2.863584E-04 7.095794E-05 1.770032E-05 4.422636E-06 1.105502E-06
∗ 2.0128 2.0032 2.0008 2.0002
267
268
Chapter 6. Numerical Riesz differentiation
6.3 Indirect approximations to the fractional convection operator Similar to the case of the fractional diffusion operator, fractional convection operators can be indirectly evaluated by combining numerical approximations to the left- and right-sided RiemannLiouville derivatives. In the following, we present numerical approximations to the Riesz derivative with 0 < α < 1 through indirect approaches.
6.3.1 Approximation based on the L1 method + Let h = b−a N with N ∈ Z . We denote xj = a + jh, j = 0, 1, . . . , N . The L1 methods for the left- and right-sided Riemann-Liouville derivatives are given by j−1 X (α) α D f (x) = bj−k−1 [f (xk+1 ) − f (xk )] RL a,x x=xj k=0 (6.119) −α (x − a) f (a) j + + O(h2−α ), 0 < α < 1, Γ(1 − α)
and N −1 X (α) Dα f (x) bk−j [f (xk ) − f (xk+1 )] = RL x,b x=xj k=j
(6.120)
(b − xj )−α f (b) + + O(h2−α ), 0 < α < 1, Γ(1 − α)
(α)
where the coefficients bk
are given by (α)
bk
=
h−α (k + 1)1−α − k 1−α . Γ(2 − α)
Hence, Riesz derivative with 0 < α < 1 can be approximated by (xj − a)−α f (a) (b − xj )−α f (b) + [RZ Dα f (x)]x=xj = − Ψα x Γ(1 − α) Γ(1 − α) j−1 X (α) + bj−k−1 [f (xk+1 ) − f (xk )] k=0 N −1 X (α) + bk−j [f (xk ) − f (xk+1 )] + O(h2−α ).
(6.121)
(6.122)
k=j
6.3.2 Approximation based on L2-1σ formulae In [95], the L2-1σ formulae are reformulated in the forms α C Da,x f (x) x=x j+σ j
h−α X (α,σ) = dj−k [f (xk+1 ) − f (xk )] + O(h3−α ) Γ(2 − α) k=0
(6.123)
6.3. Indirect approximations to the fractional convection operator
269
and α C Dx,b f (x) x=x j+σ 0 N −1 h−α X ˜(α,σ0 ) = − dk−j [f (xk+1 ) − f (xk )] + O(h3−α ), Γ(2 − α)
(6.124)
k=j
(α,σ)
with 0 ≤ j ≤ N − 1, σ = 1 − α2 , and σ 0 = α2 . When j = 0, d0 (α,σ) (α,σ 0 ) d˜ =c . For j ≥ 1, the coefficients are given by 0
(α,σ)
= c0
. When j = N − 1,
0
(α,σ) dk
=
(α,σ) (α,σ) c + c˜1 , 0 (α,σ) ck (α,σ) ci
+
−
k = 0,
(α,σ) (α,σ) c˜k+1 − c˜k , (α,σ) c˜i ,
1 ≤ k ≤ j − 1,
(6.125)
k = j,
in which the second case of the right-hand side of equation (6.125) does not exist if j = 1, and
0
(α,σ ) d˜k =
(α,σ0 ) (α,σ 0 ) c0 + c˜2 , (α,σ 0 ) ck+1
0
(α,σ )
−
(α,σ 0 ) c˜k+1 0
(α,σ )
+
ck+1 − c˜k+1 ,
k = 0, (α,σ 0 ) c˜k+2 ,
1 ≤ k ≤ N − 2 − j,
(6.126)
k = N − 1 − j,
in which the second case of the right-hand side of equation (6.126) does not exist if j = N − 2. Here, (α,σ) = σ 1−α , c0 c(α,σ) = (k + σ)1−α − (k − 1 + σ)1−α , k ≥ 1, k 1 (α,σ) c˜k = (k + σ)2−α − (k − 1 + σ)2−α 2 − α 1 − (k + σ)1−α + (k − 1 + σ)1−α , k ≥ 1, 2
(6.127)
(α,σ0 ) = (1 − σ 0 )1−α , c0 (α,σ 0 ) = (k − σ 0 )1−α − (k − 1 − σ 0 )1−α , k ≥ 1, ck 1 (α,σ 0 ) c˜k = (k − σ 0 )2−α − (k − 1 − σ 0 )2−α 2−α 1 − (k − σ 0 )1−α + (k − 1 − σ 0 )1−α , k ≥ 1. 2
(6.128)
and
Combine equations (6.123) and (6.124), and let 2σ − 2 + α + 2σ 0 − α = 0 which gives σ + σ 0 = 1. Furthermore, assume that σ = σ 0 , i.e., σ = σ 0 = 21 ; then xj+σ = xj+ 12 = xj+σ0 .
270
Chapter 6. Numerical Riesz differentiation
One can get the following (3 − α)th order scheme for Riesz derivatives at x = xj+ 21 , 0 ≤ j ≤ N − 1 [95]: [RZ Dα x f (x)]x=xj+ 1 2 (x 1 − a)−α f (a) (b − xj+ 21 )−α f (b) 1 j+ 2 = − + 2 cos( πα Γ(1 − α) Γ(1 − α) 2 ) j h−α X (α, 12 ) dj−k [f (xk+1 ) − f (xk )] + Γ(2 − α) k=0 N −1 X 1 (α, ) d˜k−j2 [f (xk+1 ) − f (xk )] + O(h3−α ), −
(6.129)
k=j
1
1
(α, ) (α, ) in which dj−k2 and d˜k−j2 are defined by (6.125) and (6.126) with σ = σ 0 = 12 , respectively.
6.3.3 Approximation based on fractional backward difference formulae and their modifications Fractional backward difference formulae and their modifications for the Riesz derivative are applicable for cases with both 0 < α < 1 and 1 < α < 2. Therefore, those for the fractional convection operator are the same as the ones for the fractional diffusion case. Here we only present the following classical Lubich formula. Theorem 6.22. Assume that f (x) satisfies f (k) (a+) = 0 (k = 0, 1, . . . , p − 1) and f (k) (b−) = 0 (k = 0, 1, . . . , p − 1), p = 2, . . . , 6. Then the classical standard Lubich formula for the Riesz derivative is given by α RZ Dx f (x) x−a b−x [X [X h ] h ] (6.130) Ψα (α) (α) $p,l f (x − lh) + $p,l f (x + lh) + O(hp ). = − hα l=0 l=0 (α)
Here 0 < α < 1 and the coefficients $p,l are generated by (3.38), or its equivalent form in (5.207), say, !α p ∞ X X 1 (α) k (1 − z) = $p,l z l , |z| < 1. (6.131) k k=1
l=0
The following numerical example proposed by [35] illustrates the numerical approximation (6.130). Example 6.23. [35] Consider the function fn (x) = xn (1 − x)n , x ∈ [0, 1], n = 2, 3, 4, 5, 6. Its Riesz derivative is analytically expressed as α RZ Dx fn (x)
n X (−1)l n!(n + l)! xn+l−α + (1 − x)n+l−α = Ψα . l!(n − l)!Γ(n + l + 1 − α) l=0
(6.132)
6.3. Indirect approximations to the fractional convection operator
271
Using numerical scheme (6.130) to evaluate the Riesz derivative of fn (x) gives the numerical results in Tables 6.12–6.16, which imply that the experimental convergence orders are in line with the theoretical orders p (p = 2, 3, 4, 5, 6). Table 6.12. The absolute errors and convergence orders of Example 6.23 by numerical scheme (6.130) with p = n = 2. Reprinted with permission from Fract. Calc. Appl. Anal., 19 (2016), at https://www.degruyter.com/view/j/fca α
0.2
0.4
0.6
0.8
h
the absolute error
the convergence order
1 20 1 40 1 80 1 160 1 320 1 20 1 40 1 80 1 160 1 320 1 20 1 40 1 80 1 160 1 320 1 20 1 40 1 80 1 160 1 320
2.381267E-04 5.900964E-05 1.460639E-05 3.628491E-06 9.039358E-07
∗ 2.0127 2.0144 2.0092 2.0051
7.097814E-04 1.696639E-04 4.123703E-05 1.014980E-05 2.516782E-06
∗ 2.0647 2.0407 2.0225 2.0118
1.638369E-03 3.728453E-04 8.824023E-05 2.141683E-05 5.272483E-06
∗ 2.1356 2.0791 2.0427 2.0222
3.782418E-03 7.826735E-04 1.747182E-04 4.102708E-05 9.923174E-06
∗ 2.2728 2.1634 2.0904 2.0477
6.3.4 Fractional-compact difference method Recalling asymptotic behaviors of the Lubich formulae for the Riemann-Liouville derivative, which can be readily derived through Fourier transform, we obtain the following fractionalcompact difference formulae. (I) The 2nd order fractional-compact difference method (α)
Let χ(z) =
W1
(e−z ) zα
=
(1−e−z )α zα
χ(z) = 1 −
(α)
with W1 (z) given by (3.38). Then Taylor expansion gives
α 3α2 − α 2 z+ z + O(|z|3 ), 2 24
|z| < 1.
(6.133)
Define the difference operator A(α) as α A(α) f (x) = 1 − δx f (x) 2 with δx f (x) = f (x + h) − f (x).
(6.134)
272
Chapter 6. Numerical Riesz differentiation
Table 6.13. The absolute errors and convergence orders of Example 6.23 by numerical scheme (6.130) with p = n = 3. Reprinted with permission from Fract. Calc. Appl. Anal., 19 (2016), at https://www.degruyter.com/view/j/fca α
0.2
0.4
0.6
0.8
h
the absolute error
the convergence order
1 40 1 60 1 80 1 100 1 120 1 40 1 60 1 80 1 100 1 120 1 40 1 60 1 80 1 100 1 120 1 40 1 60 1 80 1 100 1 120
3.146678E-07 1.576085E-07 7.991483E-08 4.501080E-08 2.761993E-08
∗ 1.7052 2.3608 2.5726 2.6786
4.349687E-06 1.491902E-06 6.709309E-07 3.560502E-07 2.108270E-07
∗ 2.6391 2.7779 2.8394 2.8742
2.194879E-05 6.996810E-06 3.050074E-06 1.590859E-06 9.316704E-07
∗ 2.8196 2.8861 2.9169 2.9347
1.067572E-04 3.282279E-05 1.407258E-05 7.270149E-06 4.231299E-06
∗ 2.9088 2.9439 2.9598 2.9688
Using the Fourier analysis, one can readily prove that [95] A(α) RL Dα −∞,x f (x) =
∞ 1 X (α) $1,k f (x − kh) + O(h2 ) hα
(6.135)
∞ 1 X (α) $1,k f (x + kh) + O(h2 ). hα
(6.136)
k=0
and A(α) RL Dα x,+∞ f (x) =
k=0
(α)
(α)
Here the coefficients $1,k are generated by W1 (z), i.e., (1 − z)α =
∞ X
(α)
$1,k z k , |z| < 1.
k=0
Hence, we have the following 2nd order fractional-compact method for RZ Dα x f (x) with 0 < α < 1: A(α) RZ Dα x f (x) "∞ # ∞ (6.137) X Ψα X (α) (α) 2 = − $ f (x − kh) + $ f (x + kh) + O(h ). 1,k 1,k hα k=0
k=0
6.3. Indirect approximations to the fractional convection operator
273
Table 6.14. The absolute errors and convergence orders of Example 6.23 by numerical scheme (6.130) with p = n = 4. Reprinted with permission from Fract. Calc. Appl. Anal., 19 (2016), at https://www.degruyter.com/view/j/fca α
0.2
0.4
0.6
0.8
h
the absolute error
the convergence order
1 20 1 25 1 30 1 35 1 40 1 20 1 25 1 30 1 35 1 40 1 20 1 25 1 30 1 35 1 40 1 20 1 25 1 30 1 35 1 40
3.254967E-06 1.421712E-06 7.061638E-07 3.863551E-07 2.279637E-07
∗ 3.7121 3.8381 3.9123 3.9509
1.307893E-05 5.433994E-06 2.610192E-06 1.395202E-06 8.089264E-07
∗ 3.9362 4.0217 4.0635 4.0821
4.165885E-05 1.647646E-05 7.642179E-06 3.980841E-06 2.261840E-06
∗ 4.1569 4.2137 4.2309 4.2336
1.466022E-04 5.460563E-05 2.420431E-05 1.216174E-05 6.707129E-06
∗ 4.4258 4.4625 4.4647 4.4568
Furthermore, for f (x) defined on [a, b] with f (n) (a) = f (n) (b) = 0, n = 0, 1, its Riesz derivative at x = xj = a + jh can be evaluated by A(α) [RZ Dα x f (x)]x=xj " j # N −j (6.138) X Ψα X (α) (α) 2 = − $ f (x ) + $ f (x ) + O(h ). j−k j+k 1,k 1,k hα k=0
k=0
(II) The 3rd order fractional-compact difference method Now we present the 3rd order fractional-compact method based on the difference scheme (6.130) (α)
when p = 2. Let χ e(z) = holds:
W2
(e−z ) zα
(α)
with W2
χ e(z) = 1 −
defined by (3.38). Then the following expansion
α 2 z + O(|z|3 ). 3
(6.139)
Define the difference operator B (α) as α B (α) f (x) = 1 − δx2 f (x), 3 where δx2 f (x) = f (x + h) − 2f (x) + f (x − h).
(6.140)
274
Chapter 6. Numerical Riesz differentiation
Table 6.15. The absolute errors and convergence orders of Example 6.23 by numerical scheme (6.130) with p = n = 5. Reprinted with permission from Fract. Calc. Appl. Anal., 19 (2016), at https://www.degruyter.com/view/j/fca α
0.2
0.4
0.6
0.8
h
the absolute error
the convergence order
1 80 1 100 1 120 1 140 1 160 1 80 1 100 1 120 1 140 1 160 1 80 1 100 1 120 1 140 1 160 1 80 1 100 1 120 1 140 1 160
3.254967E-06 1.421712E-06 7.061638E-07 3.863551E-07 2.279637E-07
∗ 3.7121 3.8381 3.9123 3.9509
8.731739E-10 3.348011E-10 1.473288E-10 7.232096E-11 3.867341E-11
∗ 4.2959 4.5023 4.6160 4.6877
5.385482E-09 1.900398E-09 7.985621E-10 3.806168E-10 1.994064E-10
∗ 4.6680 4.7554 4.8071 4.8412
3.005433E-08 1.023908E-08 4.211445E-09 1.978515E-09 1.025811E-09
∗ 4.8256 4.8727 4.9008 4.9192
By virtue of the Fourier transform, we obtain the 3rd order fractional-compact scheme for the Riemann-Liouville derivative,
B (α) RL Dα −∞,x f (x) =
∞ 1 X (α) $2,k f (x − kh) + O(h3 ) hα
(6.141)
∞ 1 X (α) $2,k f (x + kh) + O(h3 ). hα
(6.142)
k=0
and B (α) RL Dα x,+∞ f (x) =
k=0
(α)
(α)
Here the coefficients $2,k are generated by W2 (z), i.e.,
3 1 − 2z + z 2 2 2
α =
∞ X k=0
(α)
$2,k z k , |z| < 1.
(6.143)
6.3. Indirect approximations to the fractional convection operator
275
Table 6.16. The absolute errors and convergence orders of Example 6.23 by numerical scheme (6.130) with p = n = 6. Reprinted with permission from Fract. Calc. Appl. Anal., 19 (2016), at https://www.degruyter.com/view/j/fca α
0.2
0.4
0.6
0.8
h
the absolute error
the convergence order
1 20 1 40 1 80 1 160 1 320 1 20 1 40 1 80 1 160 1 320 1 20 1 40 1 80 1 160 1 320 1 20 1 40 1 80 1 160 1 320
3.783855E-08 2.310553E-09 4.258651E-11 6.690552E-13 1.035841E-14
∗ 4.0335 5.7617 5.9921 6.0133
3.116503E-07 1.031745E-08 1.651582E-10 2.418476E-12 3.582209E-14
∗ 4.9168 5.9651 6.0936 6.0771
1.564617E-06 3.647148E-08 5.062291E-10 6.799919E-12 9.539537E-14
∗ 5.4229 6.1708 6.2181 6.1555
8.311643E-06 1.432632E-07 1.663072E-09 1.942938E-11 2.433601E-13
∗ 5.8584 6.4287 6.4195 6.3190
Then RZ Dα x f (x) with 0 < α < 1 can be approximated by (α) B RZ Dα x f (x) "∞ # ∞ X (α) X Ψ α (α) $2,k f (x − kh) + $2,k f (x + kh) + O(h3 ). = − hα k=0
(6.144)
k=0
Furthermore, for f (x) defined on [a, b] with f (n) (a) = f (n) (b) = 0, n = 0, 1, 2, its Riesz derivative at x = xj = a + jh can be evaluated by h i (α) α B D f (x) RZ x x=xj " j # (6.145) N −j X Ψα X (α) (α) 3 $2,k f (xj−k ) + $2,k f (xj+k ) + O(h ). = − hα k=0
k=0
6.3.5 Approximation based on fractional average central difference formulae The 2nd and 4th order methods (6.52) and (6.54) are also applicable for the case with 0 < α < 1. In other words, the following results hold.
276
Chapter 6. Numerical Riesz differentiation
α+2 Theorem 6.24. Suppose that f (x) and the Fourier transforms of RL Dα+2 −∞,x f (x) and RL Dx,+∞ f (x) 1 are in L (R). Then it holds for 0 < α < 1 that
α RZ Dx f (x) "∞ X Ψ α k α = − (−1) (f (x − kh) + f (x − (k − α)h)) 2hα k k=0 # ∞ X α (−1)k (f (x + kh) + f (x + (k − α)h)) + O(h2 ) + k
(6.146)
k=0
uniformly for x ∈ R as h → 0+ . Furthermore, for f (x) defined on [a, b] with f (n) (a) = f (n) (b) = 0, n = 0, 1, its Riesz derivative can be evaluated by α RZ Dx f (x) [ x−a h ] Ψα X k α (−1) (f (x − kh) + f (x − (k − α)h)) = − 2hα k k=0 (6.147) b−x [X h ] k α + (−1) (f (x + kh) + f (x + (k − α)h)) + O(h2 ). k k=0
α+4 Theorem 6.25. Assume that f (x) and the Fourier transforms of RL Dα+4 −∞,x f (x) and RL Dx,+∞ f (x) 1 are in L (R). Then it holds for 0 < α < 1 that α(3α + 1) 2 α 1 + δ x RZ Dx f (x) 24 "∞ Ψα X k α =− α (−1) (f (x − kh) + f (x − (k − α)h)) (6.148) 2h k k=0 # ∞ X k α (−1) (f (x + kh) + f (x + (k − α)h)) + O(h4 ) + k k=0
uniformly for x ∈ R as h → 0+ . Here δx2 denotes the 2nd order central difference operator defined by δx2 f (xj ) = f (xj+1 ) − 2f (xj ) + f (xj−1 ). Furthermore, for f (x) defined on [a, b] with f (n) (a) = f (n) (b) = 0, n = 0, 1, 2, 3, its Riesz derivative can be evaluated by α(3α + 1) 2 α δ 1 + x RZ Dx f (x) 24 x−a [X h ] Ψ α k α = − (−1) (f (x − kh) + f (x − (k − α)h)) k 2hα (6.149) k=0 b−x [X h ] α + (−1)k (f (x + kh) + f (x + (k − α)h)) + O(h4 ). k k=0
6.3. Indirect approximations to the fractional convection operator
277
The following numerical examples illustrate schemes (6.147) and (6.149) Example 6.26. [40] Consider the function f (x) = x2 (1 − x2 ), x ∈ [0, 1], whose Riesz derivative at x = 21 is given by (6.47). Table 6.17 lists the absolute errors and convergence orders given by the numerical scheme (6.147) at x = 21 with different α. It confirms the 2nd order accuracy of (6.147).
Table 6.17. The absolute errors and convergence orders of Example 6.26 by numerical scheme (6.147). Reprinted by permission of Taylor & Francis Ltd (http://www.tandfonline.com) from “High-order algorithms for Riesz derivative and their applications: Revisited” by H. F. Ding, C. P. Li, and Y. Q. Chen in Numerical Functional Analysis and Optimization, Vol. 38, No. 7, 2017. α
0.2
0.4
0.6
0.8
h
absolute error
convergence order
1 10 1 20 1 40 1 80 1 10 1 20 1 40 1 80 1 10 1 20 1 40 1 80 1 10 1 20 1 40 1 80
3.630966E-03 9.120270E-04 2.285315E-04 5.719787E-05
∗ 1.9932 1.9967 1.9984
5.124542E-03 1.289681E-03 3.234889E-04 8.100606E-05
∗ 1.9904 1.9952 1.9976
4.629914E-03 1.164707E-03 2.920982E-04 7.314118E-05
∗ 1.9910 1.9954 1.9977
2.652282E-03 6.653815E-04 1.666617E-04 4.170691E-05
∗ 1.9950 1.9973 1.9986
Example 6.27. [40] Consider the function f (x) = x6 (1 − x)6 , x ∈ [0, 1]. The corresponding Riesz derivative is 8
α RZ Dx f (x)
πα X Γ(` + 7) 1 = ex sec x`+6−α + (1 − x)`+6−α . 2 2 Γ(` + 7 − α)
(6.150)
`=0
Table 6.18 displays the absolute errors and convergence orders at x = 12 given by the numerical scheme (6.149) with different α. It verifies that (6.149) is indeed of the 4th order accuracy.
278
Chapter 6. Numerical Riesz differentiation
Table 6.18. The absolute errors and convergence orders of Example 6.27 by numerical scheme (6.149). Reprinted by permission of Taylor & Francis Ltd (http://www.tandfonline.com) from “High-order algorithms for Riesz derivative and their applications: Revisited” by H. F. Ding, C. P. Li, and Y. Q. Chen in Numerical Functional Analysis and Optimization, Vol. 38, No. 7, 2017. α
0.2
0.4
0.6
0.8
h
absolute error
convergence order
1 20 1 40 1 80 1 160 1 20 1 40 1 80 1 160 1 20 1 40 1 80 1 160 1 20 1 40 1 80 1 160
1.571776E-08 9.923695E-10 6.218021E-11 3.888658E-12
∗ 3.9854 3.9963 3.9991
6.255711E-08 3.955450E-09 2.479333E-10 1.550707E-11
∗ 3.9833 3.9958 3.9990
1.660570E-07 1.051535E-08 6.593628E-10 4.124394E-11
∗ 3.9811 3.9953 3.9988
3.501076E-07 2.219986E-08 1.392502E-09 8.711096E-11
∗ 3.9792 3.9948 3.9987
6.4 Direct approximations to the fractional convection operator 6.4.1 Asymmetric centered difference operator The fractional centered difference operator defined in (6.56) can also directly approximate the Riesz fractional derivative with 0 < α < 1, as the following assertion states. Theorem 6.28. [97] Let f ∈ C 4 (R) and all of its derivatives up to order 5 belong to L1 (R). Set ∆α h f (x)
=
∞ X
(α)
gk f (x − kh),
(6.151)
k=−∞
where (α)
gk
=
Γ
α 2
(−1)k Γ(α + 1) . − k + 1 Γ α2 + k + 1
(6.152)
Then it holds that α RZ Dx f (x)
when h → 0.
=−
1 α ∆ f (x) + O(h2 ), x ∈ R, 0 < α < 1, hα h
(6.153)
6.4. Direct approximations to the fractional convection operator
as
279
For f (x) defined on a finite interval [a, b], we can define the zero extension function of f (x) ( f (x), x ∈ [a, b], f˜(x) = (6.154) 0, x ∈ R\[a, b].
If f˜(x) satisfies the conditions in the above theorem, then RZ Dα x f (x) can be approximated as [RZ Dα x f (x)]x=xj h i h i = RZ Dα f˜(x) ˜ + O(h2 ) = −h−α ∆α h f (x) x x=xj x=xj (6.155) N X 1 (α) gj−k f (xk ) + O(h2 ), 0 < α < 1. = − hα k=0
6.4.2 Weighted and shifted centered difference operator In [40], the weighted and shifted centered difference operator was applied to numerical approximation of the Riesz derivative with 1 < α < 2, in the form of (6.79). The corresponding results for the case with 0 < α < 1 can also be obtained by virtue of the Fourier transform. Therefore, we have the following 4th, 6th, and 8th order fractional centered difference methods for the fractional convection operator. Theorem 6.29. Let f (x) lie in C 6 (R) and its derivatives up to order 7 belong to L1 (R). Set ∞ X
Lθ f (x) =
(α)
gk f (x − (k + θ)h),
θ = −1, 0, 1,
(6.156)
k=−∞
in which (α)
gk
=
Γ
α 2
(−1)k Γ(α + 1) . − k + 1 Γ α2 + k + 1
Then, for 0 < α < 1, we have α RZ Dx f (x) i h = 1 α L f (x) − 1 + α L f (x) + α L f (x) + O(h4 ). −1 0 1 α h 24 12 24
(6.157)
(6.158)
Theorem 6.30. [41] Assume that f (x) ∈ C 9 (R) and its derivatives up to order 9 belong to L1 (R). Then, for 0 < α < 1, Dα f (x) = 1 [A L f (x) + A L f (x) + A L f (x) RZ x 1 −2 2 −1 3 0 hα (6.159) 6 +A2 L1 f (x) + A1 L2 f (x)] + O(h ), in which
α 11 A = − + α, 1 1152 2880 α 41 A2 = + α, 288 720 2 α 17α A3 = − + +1 . 192 160
(6.160)
280
Chapter 6. Numerical Riesz differentiation
Theorem 6.31. [41] Assume that f (x) lies in C 11 (R) and its derivatives up to order 11 belong to L1 (R). Then, for 0 < α < 1,
Here
α RZ Dx f (x) 1 = α [B1 L−3 f (x) + B2 L−2 f (x) + B3 L−1 f (x) + B4 L0 f (x) h +B3 L1 f (x) + B2 L2 f (x) + B1 L3 f (x)] + O(h8 ).
(6.161)
2 α 11α 191 B = + + α, 1 82944 69120 362880 2 α 7α 211 B2 = − 13824 + 3840 + 30240 α, 5α2 3α 7843 B3 = + + α, 27648 512 120960 29α2 5297α 5α3 B4 = − + + +1 . 20736 3456 45360
(6.162)
For f (x) defined on a finite interval [a, b], we can also define the zero extension function of f (x) as in (6.154). Suppose that the zero extension f˜(x) in (6.154) satisfies the conditions in the above theorems; then RZ Dα x f (x) at x = xj can be derived as h i α˜ α = D [ D f (x)] f (x) RZ RZ x x x=x j x=xj j+1 j X 1 α α X (α) (α) g f (x ) − 1 + gk f (xj−k ) = j−k+1 k hα 24 12 k=−N +j+1 k=−N +j j−1 X α (α) + gk f (xj−k−1 ) + O(h4 ) (6.163) 24 k=−N +j−1 " N −1 N +1 1 α X (α) α X (α) = g f (x ) + gj−k f (xk−1 ) k+1 j−k hα 24 24 k=−1 k=1 # N X α (α) gj−k f (xk ) + O(h4 ), 0 < α < 1, − 1 + 12 k=0
[RZ Dα x f (x)]x=xj " N −2 N −1 X X 1 (α) (α) gj−k f (xk+2 ) + A2 gj−k f (xk+1 ) = hα A1 k=−2 k=−1 N N +1 X X (α) (α) + A g f (x ) + A gj−k f (xk−1 ) 3 k 2 j−k k=1 k=0 # N +2 X (α) gj−k f (xk−2 ) + O(h6 ), 0 < α < 1, +A1 k=2
(6.164)
6.4. Direct approximations to the fractional convection operator
and
[RZ Dα x f (x)]x=xj " N −3 N −2 X (α) X 1 (α) B g f (x ) + B gj−k f (xk+2 ) = 1 k+3 2 j−k α h k=−3 k=−2 N −1 N X (α) X (α) + B3 gj−k f (xk+1 ) + B4 gj−k f (xk ) k=−1 k=0 N +1 N +2 X (α) X (α) + B g f (x ) + B gj−k f (xk−2 ) 3 k−1 2 j−k k=1 k=2 # N +3 X (α) +B gj−k f (xk−3 ) + O(h8 ), 0 < α < 1. 1 k=3
The numerical simulations can be easily tested and so are omitted here.
281
(6.165)
Chapter 7
Numerical fractional Laplacian
In Section 2.6, the integral and the directional definitions of the fractional Laplacian were introduced. For physical interpretations of the fractional Laplacian, here we take the one-dimensional CTRW case, which has been investigated in [64], as an example to show how the integral definition of the fractional Laplacian emerges. Consider a column of Np particles on the x axis at the position x = h2 , and another column of Np particles at the mirror image position, x = − h2 . Here h > 0 is the length of every subinterval of the equispaced division on the x axis. The total number of particles is 2Np . Let q (0 ≤ q ≤ 1) denote the probability of each particle making a random step to the right. Then each particle makes a random step to the left with probability 1 − q (if none of the particles remain unmoved). Assume that during each period of time τ , every single jump of the particles takes place with a migration lengthof h. After n steps have been made, these 2Np particles may be located at x = xi = i − 21 h, i = 0, ±1, ±2, . . . Denote the number of particles being at the position Pn+1 x = xi by mi (n). It is evident that i=−n mi (n) = 2Np . Define a discrete number-density distribution mi (n) , i = −n, . . . , n + 1, (7.1) ui (n) = 2Np to quantify the population dynamics. Then the particle number conservation yields that n+1 X
ui (n) = 1.
(7.2)
i=−n
As a result, expectation and variance of position of these particles after n steps are given by x ¯(n) =
n+1 X
xi ui (n)
(7.3)
i=−n
and s(n) =
n+1 X
2
(xi − x ¯(n)) ui (n),
(7.4)
i=−n
respectively. Let now the probability p = 21 , i.e., every particle has the same chance of stepping to the left or to the right. Then after one more step, the number of particles being at the position x = xi satisfies 1 1 ui (n + 1) = ui−1 (n) + ui+1 (n), (7.5) 2 2 283
284
Chapter 7. Numerical fractional Laplacian
or equivalently 1 [ui−1 (n) − 2ui (n) + ui+1 (n)] . 2
(7.6)
ui−1 (n) − 2ui (n) + ui+1 (n) ui (n + 1) − ui (n) , =κ τ h2
(7.7)
ui (n + 1) − ui (n) = Therefore, it holds that
2
where κ = h2τ is the Einstein diffusion coefficient. Equation (7.7) has become the discrete formula for classical one-dimensional diffusion. In this case, the diffusion of those point particles obeys a Gaussian distribution . We always assume that these particles are able to jump with arbitrary length, rather than only h, within the period τ . Suppose that the probability of particles jumping to the left (or the right) with the length of kh, k = 0, ±1, ±2, . . . obeys the power-law distribution πk (s) = where ζs =
P∞
1 r=1 r s
1 1 , s ∈ (0, 1), k 6= 0, · ζ1+2s |k|1+2s
(7.8)
is the Riemann-zeta function. It is evident that ∞ X
πk (s) =
k=−∞
1 2ζ1+2s
In this case, the relation ui (n + 1) =
P.V.
∞ X
∞ X k=−∞
1 = 1. |k|1+2s
(7.9)
ui−k (n)πk (s)
(7.10)
k=−∞
is valid. As a result, it holds that ui (n + 1) − ui (n) = κs τ
∞ X 1 ui+k (n) − ui (n) P.V. 2s h |k|1+2s
! ,
(7.11)
k=−∞
2s
1 where κs = 2ζ1+2s · hτ is the generalized Einstein diffusion coefficient. The right-hand side of the above equation is usually expressed by a singular integral in the sense of principle value as h → 0, i.e., ∞ X ui+k (n) − ui (n) 1 P.V. 2s h |k|1+2s k=−∞ Z ∞ u(x + v, nτ ) − u(x, nτ ) ≈P.V. dv (7.12) |v|1+2s −∞ Z ∞ u(x, nτ ) − u(y, nτ ) dy =P.V. |x − y|1+2s −∞ = − (−∆)s u, h → 0+ .
The discrete formula (7.11) is in fact for the one-dimensional fractional Laplacian diffusion ut = −κs (−∆)s u.
(7.13)
Experiments and numerical simulations indicate that the diffusion of the particles is not Gaussian. In effect, the probability of the waiting time between each two steps meets the long-tail
285
distribution, i.e., each of the particles can stay at one position for a long time before making the next movement. Then the left-hand side of (7.11) is replaced by the fractional temporal derivative. The corresponding mathematical model turns into s Dα 0,t u(x, t) = −(−∆) u(x, t).
(7.14)
Here the notation Dα 0,t denotes a certain fractional temporal derivative, such as the Caputo or Riemann-Liouville derivative. Apart from the three previously mentioned definitions in Section 2.6, the fractional Laplacian can be defined through other approaches. Recently, Kwa´snicki claimed that there are at least ten equivalent definitions for the fractional Laplacian. Consider the fractional Laplace operator L = −(−∆)s in Rd with s ∈ (0, 1) and d ≥ 1. Let X be any of the Lebesgue spaces Lp , p ∈ [1, ∞), the space L0 of continuous functions vanishing at infinity, or the space Lbu of bounded uniformly continuous functions, and let f ∈ X . Then the following definitions of Lf ∈ X are equivalent [78]. (I) Fourier definition LF :
F {LF f (x); ω} = −|ω|2s fˆ(ω)
(7.15)
if X = Lp , p ∈ [1, 2]. (II) Distributional definition Lw : Z Z (Lw f (y)) ϕ(y)dy = Rd
f (y)Lϕ(y)dy
(7.16)
Rd
for all ϕ in the Schwartz space S, with Lϕ defined, for example, as in (I). (III) Bochner’s definition LB : LB f (x) =
1 |Γ(−s)|
∞
Z 0
Z
(f (x + z) − f (x)) kt (z)dz t−1−s dt
(7.17)
Rd
with kt (x) being the Gauss-Weierstrass kernel (or also called the heat kernel) d
kt (x) = (4πt)− 2 e−
|x|2 4t
.
(7.18)
∆(tI − ∆)−1 f (x)ts−1 dt
(7.19)
(IV) Balakrishnan’s definition LBb : sin(πs) LBb f (x) = π
Z
∞
0
with ∆ and (tI − ∆)−1 being the Laplace operator and its resolvent, respectively. (V) Integral definition LI : LI f (x) = with the limit in X .
22s Γ( d+2s 2 ) d 2
π |Γ(−s)|
Z lim+
r→0
Rd \Br
f (x + z) − f (x) dz |z|d+2s
(7.20)
286
Chapter 7. Numerical fractional Laplacian
(VI) Dykin’s definition LD : LD f (x) = lim+ r→0
22s Γ( d+2s 2 ) d 2
π |Γ(−s)|
Z Rd \B(x,r)
f (x + z) − f (x) dz, |z|d (|z|2 − r2 )s
(7.21)
with the limit in X . (VII) Quadratic form definition LQ : hLQ f, ϕi = E(f, ϕ)
(7.22)
s
for all ϕ in the Sobolev space H , where Z Z 22s Γ( d+2s (f (y) − f (x))(ϕ(y) − ϕ(x)) 2 ) E(f, ϕ) = dxdy d |x − y|d+2s d d 2 2π |Γ(−s)| R R
(7.23)
if X = L2 . (VIII) Semigroup definition LS : LS f (x) = lim+ t→0
Pt f − f , t
(7.24)
where Pt f = f ∗ pt with pt (x) being a symmetric kernel function satisfying F(pt )(ξ) = 2s e−t|ξ| . (IX) Definition as the inverse of the Riesz potential LR : Z Γ( d−2s LR f (x + z) 2 ) dz = −f (x) d 22s π 2 Γ(s) Rd |z|d−2s
(7.25)
d if 2s < d and X = Lp , p ∈ [1, 2s ).
(X) Definition through harmonic extension LH : 1 2 s 2− 1 2 ∆x u(x, y) + 4s cs y s ∂y u(x, y) = 0, u(x, 0) = f (x), ∂y u(x, 0) = LH f (x),
y > 0, (7.26)
−2s
|Γ(−s)| where cs = 2 Γ(s) and where u(·, y) is a function of class X which depends continuously on y ∈ [0, ∞), and ku(·, y)kX is bounded in y ∈ [0, ∞).
Remark 7.1. In (III), (V), (VI), (VIII), and (X), convergence in the uniform norm can be relaxed to pointwise convergence to a function in X when X = L0 or X = Lbu . Finally, for X = Lp with p ∈ [1, ∞), norm convergence in (V), (VI), (VIII), or (X) implies pointwise convergence for almost all x. Remark 7.2. Besides the above definitions of fractional Laplacian in terms of integrals, the spectral definition is also used due to its quite applicability and flexibility in the real world. Numerical fractional Laplacian is very limited and is just the beginning. In this chapter, we mainly introduce numerical algorithms available in this respect. The case with one dimension is presented as follows.
7.1. Approximations based on regularization and interpolation
287
7.1 Approximations based on regularization and interpolation In [140], the one-dimensional integral fractional Laplacian of a rapidly decaying function f (x) was rewritten as a regularized integral over a semi-infinite domain, i.e., s (−∆) f (x) Z ∞ (7.27) f (x + y) − 2f (x) + f (x − y) = − C(1, s) dy, x ∈ R, s ∈ (0, 1). y 1+2s 0 Here C(1, s) =
22s sΓ( 1+2s 2 ) 1
π 2 Γ(1−s)
. For the proof of the above expression we refer the reader to Lemma
2.19. Denote
∞
Z J (x) = 0
f (x + y) − 2f (x) + f (x − y) dy; y 1+2s
(7.28)
then (−∆)s f (x) = −C(1, s)J (x).
(7.29)
To compute the integral J (x) with arbitrary accuracy, we adopt a suitable cutoff length, ω(x) > 0, and break up the semi-infinite domain into two parts, say, J (x) = J1 (x) + f 00 (x)J2 (x) + J3 (x). Here Z
ω(x)
J1 (x) = 0
f (x + y) − 2f (x) + f (x − y) − f 00 (x)y 2 dy y 1+2s
is a regularized integral in a finite interval, Z ω(x) J2 (x) = y 1−2s dy = 0
1 2−2s (ω(x)) 2 − 2s
(7.30)
(7.31)
(7.32)
is a standard integral, and Z
∞
J3 (x) = ω(x)
f (x + y) − 2f (x) + f (x − y) dy y 1+2s
(7.33)
is a regular integral over a semi-infinite domain. The integrand of the first integral, J1 (x), 1 (4) f (x)y 3−2s . remains nonsingular as the integration variable y tends to zero, behaving as 12 The integral exists for any function belonging to the Schwartz space S. As a result, the integral defining J1 (x) can be computed by conventional methods such as the trapezoidal rule. The third integral can be approximated as Z ∞ dy , (7.34) J3 (x) ≈ −2f (x) 1+2s y ω(x) if we choose ω(x) |x|. This approximation yields that J3 (x) ≈ −f (x)
1 1 . s (ω(x))2s
(7.35)
It should be noted that the accuracy of this approximation improves as ω(x) becomes larger. Next, we introduce another numerical approach. In [68], an expression was considered for the discrete fractional Laplacian at x = xj = jh, j = 0, ±1, ±2, . . ., via regularizing the singular
288
Chapter 7. Numerical fractional Laplacian
integral. Note that the integral in the integral definition (2.213) with d = 1 can be divided into the singular part and the tail, i.e., Z f (x) − f (y) (−∆)s f (x) = C(1, s)P.V. dy d+2s R |x − y| (7.36) = IS f (x) + IT f (x), 0 < s < 1, where
Z IS f (x) = C(1, s)P.V. |y|h
f (x) − f (x − y) dy |y|1+2s
f (x) − f (x − y) dy |y|1+2s
(7.37)
(7.38)
with 0 < h < 1. Whenever f has bounded 2nd order derivatives, the singular part can be symmetrized as Z h 2f (x) − f (x − y) − f (x + y) IS f (x) = C(1, s) dy, (7.39) y 1+2s 0 which is no longer a principle value integral. Assume that f ∈ C 4 (R); the Taylor expansion with exact remainder f (x ± y) = f (x) ± f 0 (x)y +
y 2 00 y3 y4 f (x) ± f 000 (x) + f (4) (ξ(y)) 2 6 24
(7.40)
yields Z h (4) Z h y2 1 f (ξ(y))y 4 00 I f (x) = −C(1, s)f (x) dy − C(1, s) dy S 1+2s 12 0 y 1+2s 0 y Z h ¯ Z h y4 y2 f (4) (ξ) 00 = −C(1, s)f (x) dy − C(1, s) dy 1+2s 1+2s 12 0 y 0 y ¯ h4−2s h2−2s 00 f (4) (ξ) = −C(1, s) f (x) − C(1, s) 2 − 2s 12 4 − 2s for some ξ¯ ∈ (x − h, x + h). Utilizing the central difference ¯ f (x + h) − 2f (x) + f (x − h) f (4) (ξ) h2 + 2 h 12 to replace f 00 (x) in the above expression for IS f (x), we obtain f 00 (x) =
IS f (x) = −C(1, s)
f (x + h) − 2f (x) + f (x − h) M4 h4−2s − (2 − 2s)h2s 12 4 − 2s
¯ being bounded. This leads to with M4 = C(1, s)f (4) (ξ) [IS f (x)]x=xj 2f (xj ) − f (xj+1 ) − f (xj−1 ) ≈C(1, s) (2 − 2s)h2s −2s
C(1, s)h = [f (xj ) − f (xj+1 )] 2 − 2s −2s + C(1, s)h [f (xj ) − f (xj−1 )], h → 0. 2 − 2s
(7.41)
(7.42)
(7.43)
(7.44)
7.1. Approximations based on regularization and interpolation
289
In the tail region |y| ≥ h, we denote g(y) = f (xj ) − f (xj − y), |y| ≥ h, and approximate it by the interpolation X X Pg(y) = g(xi )Pi (y − xi ) = [f (xj ) − f (xj − xi )]Pi (y − xi ) i∈Z
(7.45)
(7.46)
i∈Z
for some basis functions, Pi being defined so that Pi (0) = 1 and Pi (xk ) = 0 for k 6= 0. Let the basis functions Pi be Lagrange basis polynomials extended to finite overlapping domains. In this case, it holds that [IT f (x)]x=xj Z Pg(y) ≈C(1, s) dy 1+2s |y| |y|≥h Z X (7.47) Pi (y − xi ) =C(1, s) [f (x ) − f (x )] dy j j−i 1+2s |y| |y|≥h i6=0 X = $ e i [f (xj ) − f (xj−i )] i6=0
with
Z $ e i = C(1, s)
|y|≥h
Pi (y − xi ) dy, i 6= 0. |y|1+2s
(7.48)
Furthermore, if Pi (y) is symmetric, we have $ ei = $ e −i , i 6= 0.
(7.49)
And a discrete approximation can be obtained in the form of [(−∆)s f (x)]x=xj ∞ X ≈ $i [2f (xj ) − f (xj−i ) − f (xj+i )] i=1 X $i [f (xj ) − f (xj−i )] =
(7.50)
i6=0
with
( $i =
$ e1 +
C(1,s)h−2s , 2−2s
|i| = 1, |i| > 1.
$ e i,
For example, if we use the piecewise linear interpolation X Ph1 g(x) = g(xi )Th (x − xi )
(7.51)
(7.52)
i∈Z
with Th (x) being the tent function ( Th (x) =
1− 0,
|x| h ,
|x| ≤ h, otherwise,
(7.53)
290
Chapter 7. Numerical fractional Laplacian
then the weights in (7.50) can be calculated as C(1,s) − F 0 (1) + F (2) − F (1) $1 = $−1 = 2−2s , h2s $i = F (i + 1) − 2F (i) + F (i − 1) , |i| > 1, h2s
(7.54)
where ( F (x) =
C(1,s)|x|1−2s (2s−1)2s ,
s 6= 12 ,
−C(1, s) log |x|,
s = 12 .
(7.55)
If we choose the piecewise quadratic interpolation Ph2 g(x) =
X
X
g(xi )Qh (x − xi ) +
i even
g(xi )Rh (x − xi ),
(7.56)
i odd
where Qh (x) is the quadratic Lagrange polynomial which interpolates (0, 1, 0) at (−h, 0, h), ( Qh (x) =
1−
x2 h2 ,
0,
|x| ≤ h,
(7.57)
otherwise,
and Rh (x) is a piecewise quadratic Lagrange polynomial which interpolates (0, 0, 1) on the left and (1, 0, 0) on the right, ( Rh (x) =
1−
3|x| 2h
+
x2 2h2 ,
0,
|x| ≤ 2h, otherwise,
(7.58)
then the weights in (7.50) can be calculated as 1 C(1, s) G0 (3) + 3G0 (1) 00 $±1 = 2s − G (1) − + G(3) − G(1) , h 2 − 2s 2 $i = h22s [G0 (i + 1) + G0 (i − 1) −G(i + 1) + G(i − 1)] , |i| > 1, if j is even, h 0 0 0 $i = h12s − G (i+2)+6G2(i)+G (i−2) +G(i + 2) − G(i − 2) , |i| > 1, if j is odd.
(7.59)
Here G(x) =
C(1,s)|x|2−2s (2−2s)(2s−1)2s ,
s 6=
C(1, s)(x − x log |x|),
s = 12 .
1 2
(7.60)
For the truncated errors of the approximation (7.50), we presented the following assertion given by [68]. Theorem 7.1. Suppose that f ∈ C 4 (R). Then the truncated error for the approximation (7.50) is O(h2−2s ) using weights obtained from exact quadrature via linear interpolation, and is O(h3−2s ) using weights obtained from exact quadrature via quadratic interpolation.
7.2. Approximation based on the weighted trapezoidal rule
291
7.2 Approximation based on the weighted trapezoidal rule In [44], a finite difference method was proposed for the fractional Laplacian on one-dimensional bounded domain Ω = (−l, l) with extended homogeneous Dirichlet boundary condition, i.e., f (x) = 0 for x ∈ Ωc . The fractional Laplacian of this type frequently appears in fractional Poisson problems with extended Dirichlet boundary condition of the form ( (−∆)s f (x) = g(x) for x ∈ Ω, (7.61) f (x) = 0 for x ∈ Ωc . Note that the integral definition of the fractional Laplacian also reads as Z ∞ f (x + y) − 2f (x) + f (x − y) dy, 0 < s < 1. (−∆)s f (x) = −C(1, s) y 1+2s 0
(7.62)
Denote L = |Ω| = 2l. Then the integral on the right-hand side of (7.62) can be divided into two parts, in the form "Z L f (x − y) − 2f (x) + f (x + y) s (−∆) f (x) = − C(1, s) dy y 1+2s 0 (7.63) # Z ∞ f (x − y) − 2f (x) + f (x + y) dy . + y 1+2s L For any x ∈ (−l, l) and y ≥ L, (x ± y) ∈ Ωc and thus f (x ± y) = 0. It follows that the second integral on the right-hand side of (7.63) can be exactly calculated as Z ∞ f (x − y) − 2f (x) + f (x + y) dy y 1+2s L (7.64) Z ∞ 1 dy = − 2s f (x). = − 2f (x) 1+2s sL L y It remains to numerically evaluate the first integral on the right-hand side of (7.63). 2l Define h = N with N ∈ Z+ , and denote yk = kh for 0 ≤ k ≤ N with y0 = 0. If we introduce the function φγ (x, y) =
f (x − y) − 2f (x) + f (x + y) yγ
(7.65)
with γ ∈ (2s, 2] being a splitting parameter, then the first integral on the right-side hand of (7.63) can be viewed as a weighted integral of φγ (x, y), with respect to y. In other words, Z L Z L f (x − y) − 2f (x) + f (x + y) dy = φγ (x, y)y γ−(1+2s) dy, (7.66) y 1+2s 0 0 where y γ−(1+2s) represents the weight function. For 2 ≤ k ≤ N , the weighted trapezoidal rule yields that Z yk φγ (x, y)y γ−(1+2s) dy y k−1 Z yk 1 (7.67) ≈ [φγ (x, yk−1 ) + φγ (x, yk )] y γ−(1+2s) dy 2 y k−1 1 γ−2s = ykγ−2s − yk−1 [φγ (x, yk−1 ) + φγ (x, yk )] 2(γ − 2s)
292
Chapter 7. Numerical fractional Laplacian
for any γ ∈ (2s, 2]. For k = 1 and γ = 2, one has Z
y1
φγ (x, y)y γ−(1+2s) dy =
y0
Z
y1
φ2 (x, y)y 1−2s dy ≈
y0
y12−2s φ2 (x, y1 ), 2 − 2s
(7.68)
where φ2 (x, y1 ) can be regarded as the central difference approximation to f 00 (x). When γ ∈ (2s, 2), we can approximate the integral with k = 1 as Z y1 φγ (x, y)y γ−(1+2s) dy y0 Z y1 1 y γ−(1+2s) dy lim φγ (x, y) + φγ (x, y1 ) ≈ (7.69) y→0 2 y0 y γ−2s φγ (x, y1 ), γ ∈ (2s, 2), = 1 2(γ − 2s) provided that f is suitably smooth (e.g., f ∈ C 2 (R)). Here the equality lim φγ (x, y) ≈ lim y 2−γ f 00 (x) = 0, γ ∈ (2s, 2)
y→0
y→0
(7.70)
is utilized. Combining the above discussions, one obtains the numerical approximation to the fractional Laplacian (−∆)s as (−∆)sh,γ f (x) (N X γ−2s C(1, s) γ−2s =− yk − yk−1 [φγ (x, yk−1 ) + φγ (x, yk )] 2(γ − 2s) k=2 o + χγ y γ−2s φγ (x, y1 ) + C(1, s) f (x), x ∈ (−l, l), 1 sL2s
(7.71)
where the coefficient χγ = 1 for γ ∈ (2s, 2), while χγ = 2 if γ = 2. Define grid points xj = −l + jh with 0 ≤ j ≤ N . For j = 1, 2, . . . , N − 1, the fully discrete scheme is (−∆)sh,γ f (x) x=x j −1 NX N ν − (N − 1)ν (k + 1)ν − (k − 1)ν + =C s,h,γ kγ Nγ k=2 ! ν +(2ν + χγ − 1) + f (xj ) sN 2s 2 ν + χγ − 1 − [f (xj+1 ) + f (xj−1 )] 2 N −1 ν ν 1 X (|i − j| + 1) − (|i − j| − 1) − f (x ) , i 2 i=1 |i − j|γ i6=j,j±1
(7.72)
7.2. Approximation based on the weighted trapezoidal rule
where ν = γ − 2s and the coefficient is given by Cs,h,γ = scheme can also be expressed in the matrix form
293 C(1,s) νh2s .
Furthermore, the above
(−∆)sh,γ f = Af .
(7.73)
Here the vector f = (f (x1 ), f (x2 ), . . . , f (xk−1 ))T , and A is the matrix representation of the fractional Laplacian, defined as N −1 X (k + 1)ν − (k − 1)ν N ν − (N − 1)ν + γ k Nγ k=2 ν , j = i, +(2ν + χγ − 1) + sN 2s (7.74) Ai,j = Cs,h,γ ν ν (|i − j| + 1) − (|i − j| − 1) − , i 6= j, j ± 1, 2|i − j|γ ν − 2 + χγ − 1 , j = i ± 1, 2 for i, j = 1, 2, . . . , N − 1. It is easy to see that the matrix A is a symmetric Toeplitz matrix. The computation of Af can be achieved efficiently by using the fast Fourier transform [18, 161, 169], and the computational cost is O(N ln N ). Remark 7.3. [44] As s → 1, the scheme (7.72) with γ = 1 + s or γ = 2 reduces to the central difference scheme for the classical Laplacian −∆, i.e., [−∆h f (x)]x=xj = −
f (xj−1 ) − 2f (xj ) + f (xj+1 ) for j = 1, 2, . . . , N − 1. h2
(7.75)
Before presenting error estimates for scheme (7.72), we recall the following definitions for Hölder spaces. For an open set Ω ∈ R and β ∈ (0, 1), denote C 0,β (Ω) as the Hölder space for functions on Ω, i.e., |f (x) − f (y)| 0,β 0 C (Ω) = f ∈ C (Ω) sup < ∞ . (7.76) x,y∈Ω |x − y|β x6=y Furthermore, denote C n,β (Ω) =
n
f ∈ C n (Ω)| f (k) ∈ C 0,β (Ω) for k ∈ N and k ≤ n
o
(7.77)
for an integer n ≥ 0. Then the error estimates for scheme (7.72) which can be found in [44] are as follows. Theorem 7.2. Suppose that f ∈ C 1,s (R) with s ∈ (0, 1) has finite support on an open set Ω ∈ R, and (−∆)sh,γ defined in (7.72) is a finite difference approximation of the fractional Laplacian (−∆)s . Then, for any γ ∈ (2s, 2], it holds that
(−∆)s f (x) − (−∆)sh,γ f (x) ≤ Ch1−s , s ∈ (0, 1), (7.78) ∞,Ω with C being a positive constant depending on s and γ.
294
Chapter 7. Numerical fractional Laplacian
Remark 7.4. Theorem 7.2 shows that, for f ∈ C 1,s (R), the accuracy of the scheme (7.72) is O(h1−s ) for small size h; i.e., its convergence for low regularity function is low, especially as s → 1. This result is consistent with the central difference scheme for the classical Laplacian, which is not convergent if f ∈ C 1,1 (R) or even f ∈ C 2 (R). Theorem 7.3. Suppose that f ∈ C 3,s (R) with s ∈ (0, 1) has finite support on an open set Ω ∈ R, and (−∆)sh,γ defined in (7.72) is a finite difference approximation of the fractional Laplacian (−∆)s . If the parameter is chosen as γ = 2 or γ = 1 + s, it holds that
(−∆)s f (x) − (−∆)sh,γ f (x) ≤ Ch2 , s ∈ (0, 1), (7.79) ∞,Ω with C being a positive constant depending on s and γ. Remark 7.5. Theorem 7.3 shows that, for f ∈ C 3,s (R), if the splitting parameter is chosen as γ = 2 or γ = 1 + s, the scheme (7.72) has the accuracy of O(h2 ) uniformly for any s ∈ (0, 1). This result coincides with the behavior of the central difference method for the classical Laplacian. Indeed, for f ∈ C 3,1 (R) (corresponding to f ∈ C 3,s (R) as s → 1), by Taylor’s theorem and the mean value theorem, there exist x− ∈ [x − h, x] and x+ ∈ [x, x + h], such that |φ2 (x, h) − u00 (x)| =
h 000 + |f (x ) − f 000 (x− )| ≤ Ch2 . 6
(7.80)
Remark 7.6. For f ∈ C 3,s (R), if the splitting parameter γ ∈ / {2, 1 + s}, the accuracy of scheme (7.72) becomes s dependent. Here, we divide the discussion into two parts. (I) For
1 2
< s < 1, it holds that
(−∆)s f (x) − (−∆)sh,γ f (x) ≤ Ch2−2s , γ ∈ (2s, 2), γ 6= 1 + s; ∞,Ω
(7.81)
that is, the accuracy is O(h2−2s ). (II) For 0 < s ≤ 12 , it is challenging to obtain a uniform error bound for all γ ∈ (2s, 2). Numerical study shows that the convergence rate is almost O(h2−2s ), if γ 6= 2 or γ = 6 1 + s. Remarks 7.4 and 7.6 show that choosing the splitting parameter γ = 2 or γ = 1+s in scheme (7.72) leads to a better accuracy.
7.3 Some remarks on the numerical fractional Laplacian In this chapter, numerical methods for the fractional Laplacian are introduced only for the onedimensional case. But in effect the fractional Laplacian −(−∆)s on R is essentially the Riesz derivative, i.e., −(−∆)s f (x) = RZ D2s x f (x), s ∈ (0, 1), f (x) ∈ S(R).
(7.82)
See also (2.246). In equation (7.82), when s = 21 , RZ D2s x f (x) seems not to be meaningful due to the factor − 2 cos1 πs . But in this case, (2) 1
−(−∆) 2 f (x) = RZ D1x f (x)
(7.83)
7.3. Some remarks on the numerical fractional Laplacian
295
can be interpreted as 1 RZ Dx f (x)
1 = π
or [11, 85] 1 RZ Dx f (x)
1 = π
Z 0
∞
∞
f 0 (x + y) − f 0 (x − y) dy y
(7.84)
f (x + y) − 2f (x) + f (x − y) dy. y2
(7.85)
Z 0
Due to the above facts, if s ∈ (0, 12 ) ∪ ( 12 , 1), the numerical differentiation for −(−∆)s f (x) on 1 R is just the numerical Riesz differentiation RZ D2s x f (x) on R. See Chapter 6. If s = 2 , then 1 −(−∆) 2 f (x) can be approximated by the numerical methods displayed in Sections 7.1 and 7.2. We must emphasize that −(−∆)s and RZ D2s x on the whole line are equivalent. But they are generally not equivalent on any proper subset of R. In the multidimension case, numerical differentiation for −(−∆)s f (x), x ∈ Rd may be implemented via Monte Carlo methods [79], since high-dimensional numerical integration (the fractional Laplacian is expressed by an integral) is often difficult or even impossible due to the limits of computability. On the other hand, the fractional Laplacian is defined on the whole space. Approximating the fractional Laplacian on the whole space is often expensive or even inadvisable. Hence, numerical fractional Laplacian on the whole space Rd can be truncated [43]. These truncated techniques can be applied to approximate some other fractional derivatives, such as the Caputo derivative, Riemann-Liouville derivative, and so on [21]. Numerical methods for fractional Laplacian and fractional differential equations with fractional Laplacian (here the fractional Laplacian is in the sense of integral definition) are just the beginning, and their studies are very limited. We do hope this chapter can offer fresh stimuli for the fractional calculus community to promote and develop cutting-edge research in this respect.
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Index Lp space, 7 “quotient” space, 75, 78, 80 absolute continuity, 5, 20 absolutely convergent, 52 anomalous convection, 239 anomalous diffusion, 3, 46, 54, 60, 239 Beta function, 8, 13 binomial coefficient, 32, 64 Caputo derivative, 19, 36–38, 40, 42–46, 71, 105, 106, 116, 118–120, 127, 128, 133–135, 138, 139, 141, 142, 148, 150, 152–154, 157, 165–167, 169–171, 173, 175–183, 185–187, 236, 295 left-sided and right-sided, 37 Chebyshev polynomials, 95, 100, 178 classical Lubich formulae, 270 composite trapezoidal formula, 88 conditionally convergent, 52 conservation law, 54 continuous time random walk (CTRW), 3, 4, 46, 239, 240, 283 decoupled, 4, 19 convolution quadrature, 92, 93 convolution weights, 92 Crank-Nicolson method, 116 cubic Hermitian interpolation, 91 differential matrix, 98, 180, 182, 236 dimensional constant, 55
eigenfunction, 63 eigenvalue, 63, 181, 182, 224, 225 Einstein diffusion coefficient, 284 generalized, 284 Erdélyi-Kober type fractional integral, 16 Fick’s law, 46, 54 Fick’s second law, 20 finite-part integrals, 237, 238 first-degree compound quadrature formula, 236, 237 Fourier transform, 4, 13–15, 28, 29, 44, 48, 49, 57, 61, 62, 64, 95, 194–196, 199–201, 203, 211–214, 218, 221, 227, 231, 234, 235, 239, 244, 249, 251, 257, 260, 261, 271, 274, 276, 279 of fractional difference, 64 fast, 92, 93, 173, 175, 176, 213, 293 inverse, 49, 197, 201, 219, 232, 256, 258, 263 of Caputo derivative, 44 of Riemann-Liouville derivative, 28 of Riemann-Liouville integral, 13 of Riesz derivative, 48 Fourier’s law, 54 Fourier-Laplace transform, 4 inverse, 4 fractional average central difference operator, 234, 253 left and right, 233
309
fractional backward difference formulae, 93–95, 170, 171, 203, 220, 226 fractional convection, 46–48, 55, 239–241, 243, 268, 279 fractional convection equation (FCE), 48, 240, 241 fractional convection operator, 241, 268, 270, 278 fractional derivative of distributions, 70 fractional difference, 64 fractional differential matrix, 98 fractional diffusion, 55, 239 fractional diffusion equation (FDE), 19, 48 fractional diffusion operator, 241, 253, 268 fractional integral, 3–7, 13, 19, 24, 27, 42, 44, 56, 87, 92–94, 97, 98, 100–103, 105, 109, 169, 183 by a function, 17 fractional Laplacian, 19, 54–56, 58, 60, 61, 63, 283, 285, 288, 291–295 Balakrishnan’s definition, 285 Bochner’s definition, 285 Dykin’s definition, 286 quadratic form definition, 286 semigroup definition, 286 the distributional definition, 285 the directional definition, 61 the Riesz (integral) definition, 55 the spectral definition, 63 fractional Poisson problem, 54, 291 fractional-order lower
310 consistency, 38 fractional-order upper consistency, 38 Fubini-Tonelli Theorem, 57 Gamma function, 4, 8, 32, 59, 102, 103 Gauss hypergeometric function, 76, 221 Gauss interpolation, 95, 178 Gauss-Weierstrass kernel, 285 Gaussian distribution, 284 Gaussian process, 47 generating function, 93–95, 171, 173, 175, 176, 203, 211, 215–218, 220, 226, 228, 255, 257 Grünwald-Letnikov derivative, 19, 64, 65, 70, 194 left-sided and right-sided, 64 Grünwald-Letnikov fractional integral, 16 Hölder condition, 6 global, 6 local, 69, 70 Hölder exponent, 6 Hölder inequality, 10 Hölder space, 6, 7, 293 Hadamard derivative, 19, 65–68, 70, 81 left-sided and right-sided, 65, 237 Hadamard finite-part integral, 236, 237 Hadamard fractional integral, 16 Hadamard property, 69, 70 Hardy-Littlewood Theorem, 10, 16 harmonic extension, 286 heat kernel, 103, 285 Heaviside function, 31 hypersingular integral, 51, 52, 63 initial condition, 4, 30, 42, 45, 61, 62, 71, 72, 177, 182, 184 homogeneous and inhomogeneous, 176 integration by parts, 22, 38, 42, 53, 54, 59, 110, 139, 140, 155 Jacobi polynomials, 95–97
Index Jacobi-Gauss-Lobatto (JGL) point, 97, 98, 101, 180, 181, 236 jump length, 4 Gaussian, 4 jump length distribution nonlocal, 240 jump length variance, 3, 4, 239 Lévy distribution, 239 Lévy motion, 61, 62 Lagrange basis polynomial, 289 Lagrange interpolation, 139 linear, 139, 140 quadratic, 141 quadrature, 139 Lagrange interpolation polynomial, 157 Lagrange interpolation remainder theorem, 152 Lagrange polynomial, 154 Laplace transform, 4, 13, 14, 28–30, 44, 45, 67, 68, 71 of Caputo derivative, 44 of Riemann-Liouville derivative, 29 of Riemann-Liouville integral, 13 Laplacian, 46, 54, 293, 294 Lebesgue summable function, 5 Legendre polynomials, 95, 99, 178 Leibniz rule for fractional derivative, 31 linear spline, 167, 187 Liouville derivative, 69 left-sided and right-sided, 68 Liouville fractional integral on the half-axis, 16 on the real-axis, 16 Lipschitz boundary, 63 Lizorkin’s space, 15, 52 logarithmic memory principle, 102 long-tail distribution, 285 Marchaud derivative, 68–70 Marchaud fractional derivative left-sided and right-sided, 68 mean value theorem, 294 Mellin transform, 67, 68 inverse, 67, 68 of left-sided Hadamard derivative, 68
of right-sided Hadamard derivative, 68 midpoint formula, 88 Minkowski’s inequality, 10 Mittag-Leffler function, 82, 154, 166 Monte Carlo method, 295 non-Gaussian process, 47 nonlocal jump left-sided, 240 right-sided, 240, 241 nonlocal jumps two-sided, 241 null space, 75, 78, 79 numerical Caputo differentiation, 105 based on polynomial interpolation, 119 based on spline interpolation, 167 diffusive approximation, 183, 184 fractional linear multistep method, 170 L1 method, 105, 106, 130 modified, 115, 116, 118 on nonuniform grid, 109, 112 on uniform grid, 105 L1-2 formula, 127, 130, 131, 134, 136, 145, 152, 159 L2 method, 116–118 L2-1σ formula, 134, 136, 138, 146 L2C method, 116–118 spectral approximations, 178, 181 L1 method, 185 numerical fractional integration, 87 diffusive approximation, 87, 101 fractional linear multistep method, 87, 92, 93 fractional backward difference formulae, 93 fractional Euler method, 93 fractional trapezoidal rule, 93, 94 generalized Newton-Gregory formulae, 93, 94 Lubich formula, 93
Index fractional Newton-Cotes formula, 90, 91, 95 fractional rectangular formula, 87, 88, 91 left-sided, 88 right-sided, 88 weighted, 88 fractional Simpson’s formula, 89, 91 fractional trapezoidal formula, 89, 91 numerical method based on cubic Hermitian interpolation, 91 spectral approximation, 87, 95 based on Chebyshev polynomials, 100 based on Jacobi polynomials, 97 based on Legendre polynomials, 99 numerical fractional Laplacian, 283, 286, 294, 295 approximation based on weighted trapezoidal rule, 291 approximations based on regularization and interpolation, 287 numerical Riemann-Liouville differentiation, 185 approximation based on Grünwald-Letnikov type formulae shifted, 244 approximation based on L2 method, 241 based on finite-part integrals, 185, 236 based on spline interpolation, 185, 187 classical Grünwald-Letnikov formulae, 194, 195, 203, 211 classical Lubich formulae, 203, 204, 211 fractional average central difference method, 185, 231 fractional-compact formulae, 226, 228, 230 L2 method, 186 L2C method, 187 shifted fractional backward
311 difference formulae, 211, 214, 217, 218, 228 shifted Grünwald-Letnikov formulae, 195, 198 spectral approximations, 185, 236 weighted and shifted Grünwald-Letnikov formulae, 198, 201, 202, 220 weighted and shifted Lubich formulae, 220, 221, 224 fractional backward difference formulae and their modifications, 185 L1 method, 185, 186 modified, 186 on nonuniform grids, 186 on uniform grids, 185 spectral approximations, 236 numerical Riesz differentiation, 239 approximation based on fractional average central difference formulae, 249, 275 approximation based on fractional backward difference formulae and their modifications, 245, 270 classical Lubich formulae, 245 fractional-compact formulae, 248 shifted fractional backward difference formulae, 247 approximation based on Grünwald-Letnikov type formulae, 243 classical, 243 shifted, 244 weighted and shifted, 244, 245 approximation based on L1 method, 268 approximation based on L2 method, 241 approximation based on L2-1σ formulae, 268 approximation based on spline interpolation, 243 asymmetric centered difference operator, 253
compact centered difference operator, 260 weighted and shifted centered difference operator, 257 ordinary differential equation (ODE), 103, 184 piecewise constant function, 87, 88, 105 piecewise cubic interpolation, 169 piecewise linear function, 89 piecewise linear interpolation, 169 piecewise quadratic polynomial, 89 Pochhammer symbol, 76 power-law distribution, 46, 47, 240, 284 principle value, 55, 284, 288 probability density function (PDF), 3, 4, 46, 239 pseudo-differential operator, 54 resolvent, 285 Richardson extrapolation, 170 Riemann-Lebesgue lemma, 196 Riemann-Liouville derivative, 19–22, 27–31, 33, 37, 38, 40, 42, 43, 45, 46, 49, 65, 68, 71, 73, 95, 105, 106, 170–172, 185–188, 194, 195, 201, 202, 213, 217, 231–233, 236, 237, 240, 241, 245, 253, 271, 285, 295 left-sided and right-sided, 20 Riemann-Liouville integral, 3–5, 13, 22, 26, 29, 30, 88, 103, 105, 178, 185 left-sided and right-sided, 4 Riemann-zeta function, 284 Riesz derivative, 19, 46–52, 54, 58, 60, 74, 239, 241, 243, 245, 246, 248, 249, 252, 253, 255, 265, 268, 270, 271, 277, 279, 294 Riesz fractional integral, 14, 15 Riesz fractional integration, 14 Riesz kernel, 14 Riesz potential, 14, 15, 52, 286 Schwartz space, 15, 48, 51, 52, 54, 55, 285, 287
312 shifted fractional backward difference formulae, 171 short memory principle, 102 Sobolev space, 100 Sobolev Theorem, 15 spectral radius, 181–183 splitting parameter, 291, 294 starting weights, 92–95, 225 Stirling’s formula, 233 subdiffusion, 19, 116 superconvergent, 138 Taylor expansion, 9, 56, 107,
Index 111, 119, 137, 160, 168, 190, 194, 197, 231, 240, 271, 288 Taylor series, 30, 39, 41, 93, 173, 175, 176 Taylor’s Theorem, 294 tent function, 289 the finite part of integrals, 69, 70 three-term recurrence relation, 96 Toeplitz matrix symmetric, 293 trapezoidal rule, 287
weighted, 291 unconditional stability, 164 practical sense, 164 uniform convergence, 53 waiting time, 3, 4, 239, 240, 284 long-tailed, 4 Poissonian, 239 zero extension, 172, 195, 201, 217, 222, 228, 236, 247, 258, 263, 279, 280
ue to its ubiquity across a variety of fields in science and engineering, fractional calculus has gained momentum in industry and academia. While a number of books and papers introduce either fractional calculus or numerical approximations, no current literature provides a comprehensive collection of both topics. This monograph introduces fundamental information on fractional calculus and provides a detailed treatment of existing numerical approximations. Theory and Numerical Approximations of Fractional Integrals and Derivatives: • presents an inclusive review of fractional calculus in terms of theory and numerical methods; • systematically examines almost all existing numerical approximations for fractional integrals and derivatives; • considers the relationship between the fractional Laplacian and the Riesz derivative, a key component absent from other related texts; and • highlights recent developments, including the authors’ own research and results. The book’s core audience spans several fractional communities, including those interested in fractional partial differential equations, the fractional Laplacian, and applied and computational mathematics. Advanced undergraduate and graduate students will find the material suitable as a primary or supplementary resource for their studies. Changpin Li is a professor in the mathematics department at Shanghai University. His research interests include numerical methods and computations for fractional partial differential equations and fractional dynamics. A 2012 recipient of the Riemann–Liouville Award for Best FDA Paper (theory), Li is editor-in-chief of the De Gruyter book series Fractional Calculus in Applied Sciences and Engineering and serves on the editorial boards of several journals. Min Cai is a Ph.D. student in the mathematics department at Shanghai University whose main research interests include numerical methods and computations for fractional partial differential equations.
OT163
Theory and Numerical Approximations of Fractional Integrals and Derivatives Changpin Li Min Cai
Changpin Li Min Cai
Society for Industrial and Applied Mathematics 3600 Market Street, 6th Floor Philadelphia, PA 19104-2688 USA +1-215-382-9800 • Fax +1-215-386-7999 [email protected] • siam.org
Theory and Numerical Approximations of Fractional Integrals and Derivatives
D
OT163 ISBN 978-1-611975-87-1 90000
9781611975871
OT163_LI_COVER_V2A_modified.indd 1
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