FRACTIONAL CALCULUS
Models and Numerical Methods
Series on Complexity, Nonlinearity and Chaos
ISSN 2010-0019
Series Editor: Albert C.J. Luo
(Southern Illinois University E...
Series on Complexity, Nonlinearity and Chaos – Vol. 3

FRACTIONAL CALCULUS
Models and Numerical Methods

Dumitru Baleanu
Ç...
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Sui...
November 23, 2011

11:8

World Scientific Book - 9in x 6in

To our families for their support and encouragement

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Contents

Preface
1.

Preliminaries
1.1
1.2

1.3

2.

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2.3

2.4
2.5
2.6
2.7
2.8

2.1.2 Gr¨ nwald-Letnikov methods . . . . . . . . .
u
2.1.3 Lubich...
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4.2

4.3

4.4

4.5

4.6

4.7

Stirling Functions s(α,...
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4.7.4
4.7...
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Contents

A.1
A.2
A.3
A.4
A.5

The Adams-Bashforth-Mo...
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Preface

Fractional Calculus deals with the study of so-called ...
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During ...
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pletely different solution. Note that, at tha...
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I...
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modern numerical methods used to solve fra...
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Fractional Calculus: Models and Numerical Methods
...
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areas of science and engineering, in genera...
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Fractional Calculus: Models and Numerical Methods

Until no...
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there is a substantial demand for efficient n...
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variou...
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tion already leads to an infinite memory i...
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Fractional Calculus: Models and Numerical Methods

...
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Chapter 1

Preliminaries

This chapter is preliminary in ...
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1.1...
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3

Preliminaries

It has the commutative property
h ∗ ϕ =...
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The...
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The n-dimensional Laplace transform of ...
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We ...
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This function has the property
m−1

ψ(z...
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If ...
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The Gauss hypergeometric series (1.2.25...
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He...
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α ∈ C (Re(α) > 0) are defined by
RL α
I...
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th...
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α
Likewise, the equality (RL Db− y)(x) ...
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be...
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The rules for fractional integration b...
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T...
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In particular, when 0 < Re(α) < 1 then...
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an...
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and
n−1
α
α
(RL Ib− C Db− y)(x) = y(x)...
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If 0 < Re(α) < 1 and x ∈ R, then
α
(L ...
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α
...
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α
α
The Liouville fractional derivativ...
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α
...
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Th...
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valid for the matrix exponential funct...
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0<...
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0<α<1
1
α=0.5
α=0.7
α=0.9

0.9
0.8
0.7

eα

−x

0.6
0.5
0...
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integrals (1.3.125) and (1.3.126), int...
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H

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and
H α β
I− t

(x) = (−β)−α xβ

H

α
...
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In...
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(1) If Re(α) > 0 and the series f (x) ...
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1.3.6

F...
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For Re(α) > 0 and Re(β) > 0, we have
Γ...
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1.3.7

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Gr¨nwald-Letnikov fractional de...
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Chapter 2

A Survey of Numerical Methods for
the Solution of Or...
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Transcript of "Fractional calculus"

  1. 1. FRACTIONAL CALCULUS Models and Numerical Methods
  2. 2. Series on Complexity, Nonlinearity and Chaos ISSN 2010-0019 Series Editor: Albert C.J. Luo (Southern Illinois University Edwardsville, USA) Aims and Scope The books in this series will focus on the recent developments, findings and progress on fundamental theories and principles, analytical and symbolic approaches, computational techniques in nonlinear physical science and nonlinear mathematics. Topics of interest in Complexity, Nonlinearity and Chaos include but not limited to: · · · · · · New findings and discoveries in nonlinear physics and mathematics, Complexity and mathematical structures in nonlinear physics, Nonlinear phenomena and observations in nature, Computational methods and simulations in complex systems, New theories, and principles and mathematical methods, Stability, bifurcation, chaos and fractals in nonlinear physical science. Vol. 1 Ray and Wave Chaos in Ocean Acoustics: Chaos in Waveguides D. Makarov, S. Prants, A. Virovlyansky & G. Zaslavsky Vol. 2 Applications of Lie Group Analysis in Geophysical Fluid Dynamics Nail H. Ibragimov & Ranis N. Ibragimov Vol. 3 Fractional Calculus: Models and Numerical Methods D. Baleanu, K. Diethelm, E. Scalas & J. J. Trujillo Linda - Fractional Calculus.pmd 1 11/30/2011, 9:25 AM
  3. 3. Series on Complexity, Nonlinearity and Chaos – Vol. 3 FRACTIONAL CALCULUS Models and Numerical Methods Dumitru Baleanu Çankaya University, Turkey & Institute of Space Sciences, Romania Kai Diethelm Technische Universität Braunschweig, Germany & GNS mbH, Germany Enrico Scalas Università del Piemonte Orientale, Italy & Basque Center for Applied Mathematics, Spain Juan J. Trujillo University of La Laguna, Spain World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TA I P E I • CHENNAI
  4. 4. Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Series on Complexity, Nonlinearity and Chaos — Vol. 3 FRACTIONAL CALCULUS: MODELS AND NUMERICAL METHODS Copyright © 2012 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN-13 978-981-4355-20-9 ISBN-10 981-4355-20-8 Printed in Singapore. Linda - Fractional Calculus.pmd 2 11/30/2011, 9:25 AM
  5. 5. November 23, 2011 11:8 World Scientific Book - 9in x 6in To our families for their support and encouragement v book
  6. 6. November 23, 2011 11:8 World Scientific Book - 9in x 6in This page intentionally left blank book
  7. 7. November 23, 2011 11:8 World Scientific Book - 9in x 6in book Contents Preface 1. Preliminaries 1.1 1.2 1.3 2. xiii 1 Fourier and Laplace Transforms . . . . . . . . . . . . . . . Special Functions and Their Properties . . . . . . . . . . . 1.2.1 The Gamma function and related special functions 1.2.2 Hypergeometric functions . . . . . . . . . . . . . . 1.2.3 Mittag-Leffler functions . . . . . . . . . . . . . . . Fractional Operators . . . . . . . . . . . . . . . . . . . . . 1.3.1 Riemann-Liouville fractional integrals and fractional derivatives . . . . . . . . . . . . . . . . 1.3.2 Caputo fractional derivatives . . . . . . . . . . . . 1.3.3 Liouville fractional integrals and fractional derivatives. Marchaud derivatives . . . . . . . . . 1.3.4 Generalized exponential functions . . . . . . . . . 1.3.5 Hadamard type fractional integrals and fractional derivatives . . . . . . . . . . . . . . . . . . . . . . 1.3.6 Fractional integrals and fractional derivatives of a function with respect to another function . . . . . 1.3.7 Gr¨ nwald-Letnikov fractional derivatives . . . . . u A Survey of Numerical Methods for the Solution of Ordinary and Partial Fractional Differential Equations 2.1 Approximation of Fractional Operators . . . . . . . . . . . 2.1.1 Methods based on quadrature theory . . . . . . . vii 2 5 5 8 9 10 10 16 20 24 30 36 39 41 42 44
  8. 8. November 23, 2011 11:8 viii 2.3 2.4 2.5 2.6 2.7 2.8 2.1.2 Gr¨ nwald-Letnikov methods . . . . . . . . . u 2.1.3 Lubich’s fractional linear multistep methods Direct Methods for Fractional ODEs . . . . . . . . . 2.2.1 The basic idea . . . . . . . . . . . . . . . . . 2.2.2 Quadrature-based direct methods . . . . . . Indirect Methods for Fractional ODEs . . . . . . . . 2.3.1 The basic idea . . . . . . . . . . . . . . . . . 2.3.2 An Adams-type predictor-corrector method . 2.3.3 The Cao-Burrage-Abdullah approach . . . . Linear Multistep Methods . . . . . . . . . . . . . . . Other Methods . . . . . . . . . . . . . . . . . . . . . Methods for Terminal Value Problems . . . . . . . . Methods for Multi-Term FDE and Multi-Order FDS Extension to Fractional PDEs . . . . . . . . . . . . . 2.8.1 General formulation of the problem . . . . . 2.8.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efficient Numerical Methods 3.1 3.2 4. book Fractional Calculus: Models and Numerical Methods 2.2 3. World Scientific Book - 9in x 6in Methods for Ordinary Differential Equations . . . . . . 3.1.1 Dealing with non-locality . . . . . . . . . . . . . 3.1.2 Parallelization of algorithms . . . . . . . . . . . 3.1.3 When and when not to use fractional linear multistep formulas . . . . . . . . . . . . . . . . . 3.1.4 The use of series expansions . . . . . . . . . . . 3.1.5 Adams methods for multi-order equations . . . . 3.1.6 Two classes of singular equations as application examples . . . . . . . . . . . . . . . . . . . . . . Methods for Partial Differential Equations . . . . . . . . 3.2.1 The method of lines . . . . . . . . . . . . . . . . 3.2.2 BDFs for time-fractional equations . . . . . . . 3.2.3 Other methods . . . . . . . . . . . . . . . . . . . 3.2.4 Methods for equations with space-fractional operators . . . . . . . . . . . . . . . . . . . . . . Generalized Stirling Numbers and Applications 4.1 Introduction 48 49 54 55 56 58 58 60 64 66 69 74 76 83 83 87 93 . . . 93 93 99 . 104 . 106 . 108 . . . . . 117 123 124 127 136 . 138 141 . . . . . . . . . . . . . . . . . . . . . . . . . 141
  9. 9. November 23, 2011 11:8 World Scientific Book - 9in x 6in Contents 4.2 4.3 4.4 4.5 4.6 4.7 Stirling Functions s(α, k), α ∈ C . . . . . . . . . . . . . . 4.2.1 Equivalent definitions . . . . . . . . . . . . . . . . 4.2.2 Multiple sum representations. The Riemann Zeta function . . . . . . . . . . . . . . . . . . . . . . . General Stirling Functions s(α, β) with Complex Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Definition and main result . . . . . . . . . . . . . 4.3.2 Differentiability of the s(α, β); The zeta function encore . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Recurrence relations for s(α, β) . . . . . . . . . . Stirling Functions of the Second Kind S(α, k) . . . . . . . 4.4.1 Stirling functions S(α, k), α ≥ 0, and their representations by Liouville and Marchaud fractional derivatives . . . . . . . . . . . . . . . . 4.4.2 Stirling functions S(α, k), α < 0, and their representations by Liouville fractional integrals . . 4.4.3 Stirling functions S(α, k), α ∈ C, and their representations . . . . . . . . . . . . . . . . . . . . 4.4.4 Stirling functions S(α, k), α ∈ C, and recurrence relations . . . . . . . . . . . . . . . . . . . . . . . 4.4.5 Further properties and first applications of Stirling functions S(α, k), α ∈ C . . . . . . . . . . . . . . 4.4.6 Applications of Stirling functions S(α, k) (α ∈ C) to Hadamard-type fractional operators . . . . . . Generalized Stirling Functions S(n, β), β ∈ C . . . . . . . 4.5.1 Definition and some basic properties . . . . . . . . 4.5.2 Main properties . . . . . . . . . . . . . . . . . . . Generalized Stirling Functions S(α, β), α, β ∈ C . . . . . . 4.6.1 Basic properties . . . . . . . . . . . . . . . . . . . 4.6.2 Representations by Liouville fractional operators . 4.6.3 First application . . . . . . . . . . . . . . . . . . . 4.6.4 Special examples . . . . . . . . . . . . . . . . . . . Connections Between s(α, β) and S(α, k) . . . . . . . . . 4.7.1 Coincidence relations . . . . . . . . . . . . . . . . 4.7.2 Results from sampling analysis . . . . . . . . . . . 4.7.3 Generalized orthogonality properties . . . . . . . . book ix 145 145 151 153 153 164 166 168 168 172 173 176 179 185 191 191 199 205 206 213 215 219 223 223 225 227
  10. 10. November 23, 2011 11:8 x World Scientific Book - 9in x 6in Fractional Calculus: Models and Numerical Methods 4.7.4 4.7.5 5. book The s(α, k) connecting two types of fractional derivatives . . . . . . . . . . . . . . . . . . . . . . 229 The representation of a general fractional difference operator via s(α, k) . . . . . . . . . . . 233 Fractional Variational Principles 239 5.1 241 241 5.2 6. CTRW and Fractional Diffusion Models 6.1 6.2 6.3 7. Fractional Euler-Lagrange Equations . . . . . . . . . . . . 5.1.1 Introduction and survey of results . . . . . . . . . 5.1.2 Fractional Euler-Lagrange equations for discrete and continuous systems . . . . . . . . . . . . . . . 5.1.3 Fractional Lagrangian formulation of field systems 5.1.4 Fractional Euler-Lagrange equations with delay . 5.1.5 Fractional discrete Euler-Lagrange equations . . . 5.1.6 Fractional Lagrange-Finsler geometry . . . . . . . 5.1.7 Applications . . . . . . . . . . . . . . . . . . . . . Fractional Hamiltonian Dynamics . . . . . . . . . . . . . . 5.2.1 Introduction and overview of results . . . . . . . . 5.2.2 Fractional Hamiltonian analysis for discrete and continuous systems . . . . . . . . . . . . . . . . . 5.2.3 Fractional Hamiltonian formulation for constrained systems . . . . . . . . . . . . . . . . . 5.2.4 Applications . . . . . . . . . . . . . . . . . . . . . 267 269 273 289 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 289 The Definition of Continuous-Time Random Walks . . . . 290 Fractional Diffusion and Limit Theorems . . . . . . . . . . 311 Applications of CTRW to Finance and Economics 7.1 7.2 7.3 7.4 7.5 243 245 246 253 255 258 266 266 Introduction . . . . . . . . . . . Models of Price Fluctuations in Simulation . . . . . . . . . . . . Option Pricing . . . . . . . . . Other Applications . . . . . . . Appendix A Source Codes . . . . . . . . . . . Financial Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 . . . . . . . . . . . . . . . . . . . . 317 321 322 324 332 335
  11. 11. November 23, 2011 11:8 World Scientific Book - 9in x 6in book xi Contents A.1 A.2 A.3 A.4 A.5 The Adams-Bashforth-Moulton Method . . . . . . . . Lubich’s Fractional Backward Differentiation Formulas Time-fractional Diffusion Equations . . . . . . . . . . Computation of the Mittag-Leffler Function . . . . . . Monte Carlo simulation of CTRW . . . . . . . . . . . . . . . . . . . . . 335 346 355 358 359 Bibliography 363 Index 397
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  13. 13. November 23, 2011 11:8 World Scientific Book - 9in x 6in Preface Fractional Calculus deals with the study of so-called fractional order integral and derivative operators over real or complex domains and their applications. It has its roots in 1695, in a letter from de l’Hospital to Leibniz. Questions such as “What is understood by Fractional Derivative?” √ or “What does the derivative of order 1/3 or 2 of a function mean?” motivated many brilliant scientists to focus their attention on this topic during the 18th and 19th centuries. For instance, we can mention Euler (1738, [211]), Laplace (1812, [329]), Fourier (1822, [226]), Abel (1823, [3]), Liouville (1832–1855, [347, 348]), Gr¨nwald (1867, [252]), Letnikov (1868– u 1872, [337–339]), Riemann (1876, [478]), Laurent (1884, [333]), or Heaviside (1893–1912, [268, 269]). It is well known that Abel implicitly applied fractional calculus in 1823 in connection with the tautochrone problem, which was modeled through a certain integral equation with a weak singularity of the type that appears in the so-called Riemann-Liouville fractional integral [4]. Therefore he can be considered the first scholar who investigated an interesting physical problem using techniques from what we today call fractional calculus. Later, Liouville tried to apply his definitions of fractional derivatives to different problems [347]. On the other hand, in 1882 Heaviside introduced a so-called operational calculus which reconciliated the fractional calculus with the explicit solution of some diffusion problems. Particularly, his techniques were applied to the theory of the transmission of electrical currents in cables [268]. For more historic facts about the development of the fractional calculus during these two centuries, the monographs by Oldham and Spanier [433], by Ross [490], by Miller and Ross [400] and Samko et al. [501] can be consulted. xiii book
  14. 14. November 23, 2011 xiv 11:8 World Scientific Book - 9in x 6in Fractional Calculus: Models and Numerical Methods During the 20th century, up to 1985, we can list some of the pioneer researchers in this topic, such as Weyl (1917, [577]), Hardy (1917–1928, [259, 260]), Littlewood (1925–1928, [261, 262]), Levy (1937, [340]), Zygmud (1934–1945, [601, 602]), M. Riesz (1936–1949, [479, 480]), Doetsch (1937, [189]), Erd´lyi (1939–1965, [207, 208]), Kober (1940, e [320]), Widder (1941, [581]), Rabotnov (1948–1980, [453, 474]), Feller (1943–1971, [213, 214]), Maraval (1956–1971, [376, 377]), Sneddon (1957– 1979, [526–528]), Gorenflo (since 1965, [245]), Caputo (since 1966, [131]), Dzherbashyan (1970, [197]), Samko (since 1967, [500]), Srivastava (since 1968, [530, 531]), Oldham (1969, [432]), Osler (1970, [436]), Caputo and Mainardi (since 1971, [136]), Love (since 1971, [351]), Oldham and Spanier (1974, [433]), Mathai and Saxena (since 1978, [383]), Ross (since 1974, [489]), McBride (since 1979, [385]), Nigmatullin (since 1979, [425]), Oustaloup (since 1981, [437, 438]), Bagley and Torvik (since 1983, [558, 558]), among others. As we will argue below, around 1985, new and fertile applications of fractional differential equations emerged and this field became part of applied sciences and engineering. A fractional derivative is just an operator which generalizes the ordinary derivative, such that if the fractional derivative is represented by the operator symbol Dα then, when α = 1, it coincides with the ordinary differential operator D. As a matter of fact, there are many different ways to set up a fractional derivative, and, nowadays, it is usual to see many different definitions. Here we must remark that, when we speak of fractional calculus, or fractional operators, we are not speaking of fractional powers of operators, except when we are working in very special functional spaces, such as the Lizorkin spaces. Fractional differential equations, that is, those involving real or complex order derivatives, have assumed an important role in modeling the anomalous dynamics of many processes related to complex systems in the most diverse areas of science and engineering. However the interest in the specific topic of fractional calculus surged only at the end of the last century. The theoretical interest in fractional differential equations as a mathematical challenge can be traced back to 1918, when O’Shaughnessy [435] gave an explicit solution to the differential equation y (α = y/x, after he himself had suggested the problem. In 1919, Post [457] proposed a com- book
  15. 15. November 23, 2011 11:8 World Scientific Book - 9in x 6in Preface book xv pletely different solution. Note that, at that time, this problem was not rigorously defined, since there was no mention of what fractional derivative was being used in the proposed differential equation. This explains why both authors found such different solutions, and why neither of them was wrong. As one would expect, since a fractional derivative is a generalization of the ordinary derivative, it is going to lose many of its basic properties; for example, it loses its clear geometric or physical interpretation, the index law is only valid when working in specific functional spaces, the derivative of the product of two functions is difficult to compute, and the chain rule cannot be straightforwardly applied. It is natural to ask, then, what properties of fractional derivatives make them so suitable for modeling certain complex systems. We think the answer lies in the property exhibited by such systems of “non-local dynamics”, that is, the processes’ dynamics have a certain degree of memory and fractional operators are non-local, while the ordinary derivative is a local operator. In 1974, after a joint research activity, Oldham and Spanier published the first monograph devoted to fractional operators and their applications in problems of mass and heat transfer [433]. In 1974, the First Conference on Fractional Calculus and its Applications took place at the University of New Haven, organized by B. Ross who edited the corresponding proceedings [489]. We can think of this year as the beginning of a new age for fractional calculus. The stochastic interpretation for the fundamental solution of the ordinary diffusion equation in terms of Brownian motion has been known since the early years of 20th century. Physicists often mention Einstein as the pioneer in this field [200]. Indeed, Einstein’s paper on Brownian motion had a large success and motivated further experimental work on the atomic and molecular hypothesis. However, five years before Einstein, L. Bachelier published his thesis on price fluctuations at the Paris stock exchange [46]. In this thesis, the connection was already made clear between Brownian motion and the diffusion equation. These results considered the position of a diffusing object as the sum of independent and identically distributed random variables leading to a Gaussian distribution in the asymptotic limit by virtue of the central limit theorem which was refined in the first half of the 20th century as well [215].
  16. 16. November 23, 2011 xvi 11:8 World Scientific Book - 9in x 6in book Fractional Calculus: Models and Numerical Methods In 1949, Gnedenko and Kolmogorov [237] introduced a generalization of the classical central limit theorem for sums of random variables with infinite second moment converging to α-stable random variables. Almost simultaneously, L´vy and Feller also wrote seminal contributions leading to some e controversy on priority [215]. In 1965, Montroll and Weiss [409] introduced a process in physics, later called continuous time random walk (CTRW) by Scher [408, 512, 513]. This process turned out to be very useful for the theoretical description of anomalous diffusion phenomena associated to certain materials [85]. CTRWs (also known as compound renewal processes in the mathematical community) are a generalization of the above mentioned method for normal diffusion processes. Therefore, they became the tool of choice for many applied scientists in order to characterize and describe anomalous diffusive processes from the mid-20th century until today. The use of Laplace and Fourier integral transforms helps us in proving that, for a sub-diffusive process, the density function u(x, t) of finding the diffusing particle in x at time t is the fundamental solution of the following time-fractional diffusion equation: β ∆2 u = kDt u. x (1) Such a connection lets us consider the CTRW models of the subdiffusive process as fractional differential models. Among the papers dealing with this fact there are Balakrishnan (1985, [51]), Wyss (1986, [586]), Schneider and Wyss (1989, [587]), Fujita (1990, [228–230]), Shlesinger, Zalavsky, and Klafter (1993, [521]), Metzler, Gl¨ckle, and Nonnenmacher (1994, [394]), o Zaslavsky (1994, [593]), Engheta (1996, [203]), Klafter, Shlesinger, and Zumofen (1996, [316]), Metzler and Nonnenmacher (1997, [397]), Metzler, Barkai, and Klafter (1999, [393]), Hilfer (2000, [280]), Anton (1995, [282]), and many more. For a comprehensive review, we recommend the excellent papers by Metzler and Klafter (2000–2004, [395, 396]). In these papers, the reader finds an accurate description of fractional diffusion models as well as a clear explanation of the role played by other linear or non-linear diffusive fractional models, such as the fractional Fokker-Planck equation. The relationship between CTRWs and fractional diffusion will be dealt with in Chapters 6 and 7, as well as in several sections of Chapter 5. We must also point out that this is perhaps the first monograph presenting
  17. 17. November 23, 2011 11:8 World Scientific Book - 9in x 6in Preface book xvii modern numerical methods used to solve fractional differential equations (see Chapters 2 and 3). As a result of many investigations in different areas of applied sciences and engineering and as a consequence of the relationship between CTRWs and diffusion-type pseudo-differential equations, new fractional differential models were used in a great number of different applied fields. We can mention material science, physics, astrophysics, optics, signal processing and control theory, chemistry, transport phenomena, geology, bioengineering and medicine, finance, wave and diffusion phenomena, dissemination of atmospheric pollutants, flux of contaminants transported by subterranean waters through different strata, chaos, and so on. Also, the reader can find many more references, e.g., in the monographs, [204, 280, 384, 442, 453, 550, 471, 108, 309, 365, 496, 517] and [66, 130, 145, 172, 370, 407, 508, 553, 543, 315, 92, 371] The idea that physical phenomena, such as anomalous diffusive or wave processes, can be described with fractional differential models raises, at least, the following three fundamental questions: • Are mathematical models with fractional space and/or time derivatives consistent with the fundamental laws and well known symmetries of the nature? • How can the fractional order of differentiation be observed experimentally or how does a fractional derivative emerge from microscopic models? • Once a fractional calculus model is available, how can a fractional order equation be solved (exactly or approximately)? Of course, here, we must mention the very important contributions in nonlinear non-fractional differential models which were more studied by mathematicians than used by applied researchers, at least to describe the dynamics of processes within anomalous media, but this is beyond the scope of this book. However, we must remark the fractional differential models are a complementary tool to classical methods. The reader can consult the paper [105], where it is shown that strongly non-differentiable functions can be solutions of elementary fractional equations. During the last 25 years there has been a spectacular increase in the use of fractional differential models to simulate the dynamics of many different anomalous processes, especially those involving ultra-slow diffusion. The
  18. 18. November 23, 2011 xviii 11:8 World Scientific Book - 9in x 6in book Fractional Calculus: Models and Numerical Methods following table is only based on the Scopus database, but it reflects this state of affairs clearly: Table 1. Evolution in the number of publications on fractional differential equations and their applications. Words in title or abstract Fractional Brownian Motion Anomalous Diffusion Anomalous Relaxation Superdiffusion or Subdiffusion Anomalous Dynamics Anomalous Processes Fractional Models Fractional Relaxation Fractional Kinetics Fractional Dynamics Fractional Differential Equation Fractional Fokker-Planck Equation Fractional Diffusion Equation 1960–1980 1981–1990 1991–2000 2001–2010 2 185 21 0 11 38 261 23 22 24 532 626 70 121 128 1295 1205 61 521 443 1 1 74 943 On the other hand, and from a mathematical point of view, during the last five years we have been able to find many interesting publications connected with applications of classical fixed point theorems on abstract spaces to study the existence and uniqueness of solutions of many kinds of initial value problems and boundary value problems for fractional operators. See, e.g., [29, 327, 14, 424, 271, 13, 10, 547, 75, 177, 181]. For these reasons, we expect that fractional differential models will play an important role in the near future in the description of the dynamics of many complex systems. From our point of view, despite the attention given to it until the moment by many authors, only a few steps have been taken toward what may be called a clear and coherent theory of fractional differential equations that supports the widespread use of this tool in the applied sciences in a manner analogous to the classical case. Therefore we can find here a great number of both theoretical and applied open problems. For example, we think that three important kinds of such challenges, among others, are the following: • In spite of the fact that there were several attempts to formulate a deterministic approach of fractional differential models in many different
  19. 19. November 23, 2011 11:8 World Scientific Book - 9in x 6in Preface book xix areas of science and engineering, in general there have not been many rigorous justifications of such models. A deterministic approach to fractional differential models including a clear justification, as in the classical case, is an important open problem. Such an objective comes motivated mainly by the need to take into account the macroscopic behavior of anomalous processes not connected to stochastic theories. An example could be when studying the dynamics of ultrasonic waves through very irregular media, and there are many other potential examples. The first steps in this direction may have been done recently; see for example, [30, 486, 590, 589, 360, 485]. In addition, we recall the paper [428] where memory, expressed in terms of fractional operators, emerges from the initial model that does not “hint” to the presence of memory. Also, as an example, one can point to the literature on dielectric relaxation based on fractional kinetics [426] where differential equations containing non-integer operators describing the kinetic phenomena emerge from the self-similar structure of the medium considered. This approach recently received also its experimental confirmation [427]. • The introduction of a suitable fractional Laplacian for Dirichlet and Neumann problems associated to isotropic and anisotropic media. We must remark that, in the literature, at least three different approaches were used to solve such a problem in the isotropic case, namely the application of the well-known fractional power of operators, the hyper-singular inverse of one of the Riesz fractional integral operators of potential, and the characterization by means of the corresponding Fourier transform. The first two cases do not allow to work in wide functional spaces, whereas in the third one, the possibility exists not to determine with enough rigor the fractional Laplacian in the spatial field. We refer to [379, 499, 126, 125, 137, 127] for further details. • The development of suitable and well-founded numerical methods to solve fractional ordinary and partial differential equations, so that applied researchers can refine their results, as in the classical case. We will devote two chapters of this book to this problem, where the reader can find a number of relevant references. In any case, this is still an important open field whose development will allow quicker advances in applied fields.
  20. 20. November 23, 2011 xx 11:8 World Scientific Book - 9in x 6in Fractional Calculus: Models and Numerical Methods Until now, our attention was focused on fractional differential equations and their applications. However, fractional operators had been mainly used for other objectives in the past. We will illustrate this fact with the following two examples connected with potential theory: the n-dimensional fractional operators introduced by Riesz (1932–1945), as a generalization of the Riemann-Liouville integral operators, were used to write the solution of certain ordinary partial differential equations explicitly [479]; Erd´lyi e and Sneddon (1960–1966) used the so-called Erd´lyi-Kober fractional ine tegral operators to solve explicitly certain dual integral equations (see [208, 527]). Following such ideas, many other authors used fractional operators to generalize certain classical theories or to simplify classical problems. So, many special functions were expressed in terms of elementary functions by using fractional operators by Kiryakova [314]; the singularities of certain known ordinary differential equations were avoided in the framework of fractional calculus (see [487, 484]), or Riewe, Agrawal, Klimek, Baleanu et al. have initiated a fractional generalization of variational theory (see [481, 317, 15, 54, 62, 468, 34, 42, 298, 68, 542]), etc. Remarkable are the results obtained in control theory through the fractional generalization of the well-known PID controllers (see, for example, [442, 549, 550, 456, 441, 496, 407, 130]). Therefore, we can use the label “fractional calculus models” when we refer to a generalization of a classical theory in the framework of fractional calculus. In this sense, in Chapter 4 and in a part of Chapter 5, we develop fractional generalizations of important classical theories. Below, we are going to explain such issues with more detail. This book consists of a total of seven chapters, one appendix and an extensive bibliography. Chapter 1 contains preliminary material that can be skipped by informed readers. This chapter is here to help the reader in reducing the use of external sources. As we have seen, fractional-order models are a generalization of classical integer-order models. However, it turns out that these models are also in need of more general techniques in order to provide analytical solutions in closed form and/or qualitative studies of the solutions. As in the classical case, such techniques are not enough in many practical relevant cases. Thus book
  21. 21. November 23, 2011 11:8 World Scientific Book - 9in x 6in Preface book xxi there is a substantial demand for efficient numerical techniques to handle fractional derivatives and integrals and equations involving such operators. Many algorithms were proposed for this purpose in the last few decades, but they tend to be scattered across a large number of different publications and, moreover, an appropriate and rigorous convergence analysis is often not available. Thus, a user who needs a numerical scheme for a particular problem often has difficulties in finding a suitable method. As a partial remedy to this state of affairs, in Chapters 2 and 3 we collect the most important numerical methods for practically relevant tasks. We have focused our attention on those algorithms whose behavior is well understood and that have proven to be reliable and efficient. The fourth chapter is devoted to generalize the classical theory of Stirling numbers of first s(n, k) and second kind S(n, k) in the framework of the fractional calculus, basically using fractional differential and integral operators. Such special numbers play a very important role in connection with many applications, in particular in computing finite difference schemes and in numerical approximation methods. Such generalizations have been an open problem whose solution was approached by Butzer and collaborators, [111–115, 265], during the last years of the 20th century. We have worked out the mentioned generalization with respect to both parameters, n and k, so that they can be real or complex numbers, but keeping almost any known property corresponding to the classical numbers. Moreover, in this chapter, we introduce a number of important applications; for instance we connect the generalized Stirling functions with the corresponding infinity differences and with the fractional Hadamard derivative or with the fractional Liouville operators. On the other hand, we must remark that our treatment of this issue is not the ultimate one, even if we believe that our theory can open new and interesting perspectives to apply such results to approach to the calculus of infinite difference or fractional difference equations. The latter could be very important in the context of modeling the dynamics of anomalous processes. Classical calculus of variations as a branch of mathematics is recognized for its fundamental contributions in various areas of physics and engineering. The history of variational calculus started already with problems wellknown to Greek philosophers as well as scientists and contains illustrative contributions to the evolution of the science and engineering. During the last decade, when fractional calculus started to be applied intensively to
  22. 22. November 23, 2011 xxii 11:8 World Scientific Book - 9in x 6in Fractional Calculus: Models and Numerical Methods various problems related to real world applications, it was pointed out that it should be applied also to variational problems. As a result of this fusion the theory of fractional variational principles was created. This new theory consists of two parts, the first one is related to the mathematical generalization of the classical theory of calculus of variations and the second one involves the applications. The fractional Euler-Lagrange equations recently studied are a new set of differential equations involving both the left and the right fractional derivatives. As a result of interaction among fractional calculus, delay theory and time scales calculus, we observed that the new theory started to be generalized according to new results obtained in these fields. Also the applications of this new theory in so called fractional differential geometry started to be reported as well as with some promising generalizations of the classical formalisms in physics and in control theory. An important feature of fractional variational principles is that they contain classical ones as a particular case when fractional operators converge to ordinary differential operators. Besides, fractional optimal control is largely developed and fractional numerical methods started to be applied to solve the fractional Euler-Lagrange equations. At this stage, we are confident that fractional variational principles will lead to new discoveries in several fields. In Chapter 5, we introduce the reader of this book to the extension of variational calculus within the framework of fractional calculus, presenting a theory of fractional variational principles. Moreover, in the first part of the mentioned chapter, we consider the study of solutions for the corresponding fractional Euler-Lagrange equations, a new set of fractional differential equations involving both the left and right fractional derivatives. In the second part of this chapter, we study the discrete and continuous case of the called fractional Hamiltonian dynamics, which generalizes the classical dynamics of Hamiltonian systems. As already discussed, there is a deep connection between the fractional diffusion equation and the stochastic models for anomalous diffusion called CTRWs (continuous-time random walks). These processes, as discussed by physicists, are an instance of semi-Markov processes. This mild generaliza- book
  23. 23. November 23, 2011 11:8 World Scientific Book - 9in x 6in Preface book xxiii tion already leads to an infinite memory in time. Considered non-physical by several authors, spatial non-locality is connected to the power-law behavior of the distribution of jumps. All these phenomena are described by means of a suitable stochastic process, the fractional compound Poisson process with symmetric α-stable jumps which makes quite simple the proof of the generalized central limit theorem. This is the subject we study in Chapter 6. In Chapter 7, we present an overview of the application of CTRWs to finance. In particular we give a brief presentation on the application of CTRWs, and implicitly fractional models, to option pricing and we point the reader to other applications such as insurance risk evaluation and economic growth models. When tick-by-tick prices are considered, not only price jumps, but also inter-trade durations seem to vary at random. Therefore, as a first approximation, it is possible to describe durations as independent and identically distributed random variables. In this framework, position is replaced by log-price and jumps in position by tick-by-tick log-returns. The interesting case comes when inter-trade durations do not follow a exponential distribution. The material covered in the seven chapters is complemented by an appendix where we explicitly provide the implementation of the algorithms described in the previous chapters, in several common programming languages. Finally we include an extensive bibliography which, however, is far from being exhaustive. During the time we have dedicated to write this monograph in this present form, the authors have gratefully received invaluable suggestions and comments from researches at many different academic institutions and research centers around the world. Special mention ought to be made of the help and assistance so generously and meticulously provided by colleagues Thabet Abdeljawad, Mohamed Herzallah, Fahd Jarad, Sami I. Muslih, Eqab M. Rabei, Margarita Rivero, Luis Rodr´ ıguez-Germ´, and a Luis V´zquez. a Here, we would like to remember the great inspiration and support we received from Professors Om P. Agrawal at SIU Carbondale, Paul Butzer at RWTH Aachen University, Rudolf Gorenflo at Freie Universit¨t Berlin, a
  24. 24. November 23, 2011 11:8 xxiv World Scientific Book - 9in x 6in book Fractional Calculus: Models and Numerical Methods Anatoly A. Kilbas at State University of Belarus, Raoul R. Nigmatullin at University of Kazan, Hari M. Srivastava at the University of Victoria, J. A. Tenreiro Machado at ISEP Porto, George Zaslavsky at Courant Institute, Neville J. Ford at the University of Chester and Alan D. Freed at Saginaw Valley State University. Finally, the authors would like to thankfully acknowledge the financial grants and support for this book project, which were awarded by the MICINN of Spain (Projects No. MTM2007/60246 and MTM2010/16499), the Belarusian Foundation for Funding Scientific Research (Project No. F10MC-24), the Research Promotion Plan 2010 of Universitat Jaume I and the Basque Center for Applied Mathematics in Spain, Cankaya Univer¸ sity of Turkey, and Technische Universit¨t Braunschweig, Germany, among a others. August 2011 Dumitru Baleanu Kai Diethelm Enrico Scalas Juan J. Trujillo
  25. 25. November 23, 2011 11:8 World Scientific Book - 9in x 6in book Chapter 1 Preliminaries This chapter is preliminary in character and contents. We give here the definitions and some properties of several fractional integrals and fractional derivatives of different types. Also we give the definition and properties of some special functions which will be used in this book. More detailed information about the content of this chapter may be found, for example, in the works of Erd´lyi et al. [209], Copson [149], e Riesz [480], Doetsch [189], Sneddon [526], Zemanian [595], McBride [385], Samko et al. [501], Kiryakova [314], Podlubny [453], Butzer et al. [116–118], Kilbas et al. [309], Caponetto et al. [130], Diethelm [172], Mainardi [370], Monje et al. [407], Duarte Ortigueira [192] and Tarasov [543]. In general, the results we present in this chapter will be considered for “suitable functions”. Precise details can be found, e.g., in the above mentioned references. First of all, let Ω = [a, b] (−∞ ≤ a < b ≤ ∞) be a finite or infinite interval of the real axis R. We denote by Lp (a, b) (1 ≤ p ≤ ∞) the set of those Lebesgue complex-valued measurable functions f on Ω for which f p < ∞, where 1/p b f p p = a |f (t)| dt (1 ≤ p < ∞) (1.0.1) and f ∞ = esssupa≤x≤b |f (x)|. Here esssup|f (x)| is the essential maximum of the function |f (x)|. 1 (1.0.2)
  26. 26. November 23, 2011 2 11:8 World Scientific Book - 9in x 6in book Fractional Calculus: Models and Numerical Methods 1.1 Fourier and Laplace Transforms In this section we present definitions and some properties of one- and multidimensional Fourier and Laplace transforms. We begin with the one-dimensional case. The Fourier transform of a function ϕ(x), of a real variable, is defined by ∞ (F ϕ)(κ) = F [ϕ(x)](κ) = ϕ(κ) = ˆ eiκx ϕ(x)dx, (1.1.1) −∞ with x, κ ∈ R. The inverse Fourier transform is given by the formula (F −1 g)(x) = F −1 [g(κ)](x) = 1 1 g (−x) = ˆ 2π 2π ∞ e−iκx g(κ)dκ. (1.1.2) −∞ Each of the transforms (1.1.1) and (1.1.2) is inverse to the other one, F −1 F ϕ = ϕ, F F −1 g = g, (1.1.3) also the following simple relation is valid (F F ϕ)(x) = ϕ(−x). (1.1.4) The rate of decrease of (F ϕ)(x) at infinity is connected with the smoothness of the function ϕ(x). Other well known properties of the Fourier transform are F [Dn ϕ(x)](κ) = (−iκ)n (F ϕ)(κ) (n ∈ N) (1.1.5) and Dn (F ϕ)(κ) = (iκ)n F [ϕ(x)](κ) (n ∈ N) (1.1.6) where Dn denotes the classical differential operator of order n. The Fourier convolution operator of two functions h and ϕ is defined by the integral ∞ h ∗ ϕ = (h ∗ ϕ)(x) = −∞ h(x − t)ϕ(t)dt (x ∈ R). (1.1.7)
  27. 27. November 23, 2011 11:8 World Scientific Book - 9in x 6in book 3 Preliminaries It has the commutative property h ∗ ϕ = ϕ ∗ h. (1.1.8) and is connected to the Fourier transform operator by (F (h ∗ ϕ)) (κ) = (F h)(κ) · (F ϕ)(κ). (1.1.9) The n-dimensional Fourier transform of a function ϕ(x) of x ∈ Rn is defined by (F ϕ)(κ) = F [ϕ(x)](κ) = ϕ(κ) = ˆ eiκ·x ϕ(x)dx, (1.1.10) Rn with k ∈ Rn , while the corresponding inverse Fourier transform is given by the formula (F −1 g)(x) = F −1 [g(κ)](x) = = 1 (2π)n 1 ˆ n g (−x) (2π) e−ix·κ g(κ)dκ. (1.1.11) Rn The integrals in (1.1.10) and (1.1.11) have the same properties as those of the one-dimensional ones in (1.1.1) and (1.1.2). They converge absolutely, e.g., for functions ϕ, g ∈ L1 (Rn ) and in the norm of the space L2 (Rn ) for ϕ, g ∈ L2 (Rn ). If ∆ is the n-dimensional Laplace operator ∆= ∂2 ∂2 + ···+ 2 . ∂x2 ∂xn 1 (1.1.12) then (F [∆ϕ])(κ) = −|κ|2 (F ϕ)(κ) (κ ∈ Rn ). (1.1.13) Analogous to (1.1.7), the Fourier convolution operator of two functions h and ϕ is defined by h ∗ ϕ = (h ∗ ϕ)(x) = Rn h(x − t)ϕ(t)dt (x ∈ Rn ) , (1.1.14) whose Fourier transform is given by the formula (1.1.9), but with κ ∈ Rn .
  28. 28. November 23, 2011 4 11:8 World Scientific Book - 9in x 6in book Fractional Calculus: Models and Numerical Methods The Laplace transform of a function ϕ(t) of a variable t ∈ R+ = (0, ∞) is defined by ∞ (Lϕ)(s) = L[ϕ(t)](s) = ϕ(s) = ˜ e−st ϕ(t)dt 0 (s ∈ C), (1.1.15) if the integral converges. Here C is the complex plane. If the integral (1.1.15) is convergent at the point s0 ∈ C, then it converges absolutely for s ∈ C such that Re(s) > Re(s0 ). The infimum σϕ of values s for which the Laplace integral (1.1.15) converges is called the abscissa of convergence. The inverse Laplace transform is given for x ∈ R+ by the formula γ+i∞ 1 2πi (L−1 g)(x) = L−1 [g(s)](x) = esx g(s)ds. (1.1.16) γ−i∞ with γ = Re(s) > σϕ .The direct and inverse Laplace transforms are inverse to each other for “sufficiently good” functions ϕ and g, that is L−1 Lϕ = ϕ and LL−1 g = g. (1.1.17) Some simple properties of the Laplace transform analogous to those given for the Fourier transform are the following n−1 L[Dn ϕ(t)](s) = sn (Lϕ)(s) − sn−j−1 (Dj ϕ)(0) j=0 (n ∈ N). (1.1.18) and Dn (Lϕ)(s) = (−1)n L[tn ϕ(t)](s) (n ∈ N). (1.1.19) The Laplace convolution operator of two functions h(t) and ϕ(t), given on R+ , is defined for x ∈ R+ by the integral x h ∗ ϕ = (h ∗ ϕ)(x) = 0 h(x − t)ϕ(t)dt, (1.1.20) which has the commutative property h ∗ ϕ = ϕ ∗ h. (1.1.21) (L(h ∗ ϕ)(s) = (Lh)(s) · (Lϕ)(s). (1.1.22) and
  29. 29. November 23, 2011 11:8 World Scientific Book - 9in x 6in book 5 Preliminaries The n-dimensional Laplace transform of a function ϕ(t) of t ∈ Rn is + a simple generalization of the one-dimensional case, as with the Fourier transform. 1.2 Special Functions and Their Properties In this section we present the definitions and some properties of special known functions as the Euler Gamma function, Mittag-Leffler functions, etc. To get a more extensive study about this topic consult, e.g., the above mentioned books. 1.2.1 The Gamma function and related special functions The Euler Gamma function Γ(z) is defined by the so-called Euler integral of the second kind ∞ Γ(z) = tz−1 e−t dt (Re(z) > 0), (1.2.1) 0 where tz−1 = e(z−1) log(t) . This integral is convergent for all complex z ∈ C with Re(z) > 0. For this function we have the reduction formula Γ(z + 1) = zΓ(z) (Re(z) > 0); (1.2.2) using this relation, the Euler Gamma function is extended to the half-plane Re(z) ≤ 0 (Re(z) > −n; n ∈ N; z ∈ Z− = {0, −1, −2, . . .}) by 0 Γ(z) = Γ(z + n) . (z)n (1.2.3) Here (z)n is the Pochhammer symbol, defined for complex z ∈ C and nonnegative integer, with n ∈ N, by (z)0 = 1 and (z)n = z(z + 1) · · · (z + n − 1). (1.2.4) Equations (1.2.2) and (1.2.4) yield Γ(n + 1) = (1)n = n! with (as usual) 0! = 1. (n ∈ N0 ) (1.2.5)
  30. 30. November 23, 2011 6 11:8 World Scientific Book - 9in x 6in book Fractional Calculus: Models and Numerical Methods We also indicate some other properties of the Gamma function such as Γ(z)Γ(1 − z) = Γ 1 2 = π sin(πz) (z ∈ Z0 ; 0 < Re(z) < 1), (1.2.6) √ π, (1.2.7) the Legendre duplication formula 22z−1 1 Γ(2z) = √ Γ(z)Γ z + π 2 (z ∈ C), (1.2.8) and Stirling’s asymptotic formula, for | arg(z)| < π; |z| → ∞ Γ(z) = (2π)1/2 z z−1/2 e−z 1 + O 1 z . (1.2.9) In particular, Eq. (1.2.9) implies the well known results n! = (2πn)1/2 n e n 1+O 1 n (n ∈ N, n → ∞), |Γ(x + iy)| = (2π)1/2 |x|x−1/2 e−x−π[1−sign(x)y]/2 1 + O 1 x (1.2.10) , (1.2.11) when x → ∞, and |Γ(x + iy)| = (2π)1/2 |y|x−1/2 e−x−π|y|/2 1 + O 1 y , (1.2.12) when y → ∞. The quotient expansion of two Gamma functions at infinity is given by Γ(z + a) = z a−b 1 + O Γ(z + b) 1 z (| arg(z + a)| < π; |z| → ∞). (1.2.13) The digamma function is defined as the logarithmic derivative of the Gamma-function, ψ(z) = d Γ′ (z) log Γ(z) = dz Γ(z) (z ∈ C). (1.2.14)
  31. 31. November 23, 2011 11:8 World Scientific Book - 9in x 6in book 7 Preliminaries This function has the property m−1 ψ(z + m) = ψ(z) + k=0 1 z+k (z ∈ C; m ∈ N), (1.2.15) 1 z (1.2.16) which, for m = 1, yields ψ(z + 1) = ψ(z) + (z ∈ C). Another function, related with the digamma function, is the m-th polygamma function ψ m (z), which is given by ψ m (z) = d dz m ψ(z) (z ∈ CZ0 ) (1.2.17) The Beta function is defined by the Euler integral of the first kind 1 B(z, w) = 0 tz−1 (1 − t)w−1 dt (Re(z) > 0; Re(w) > 0), (1.2.18) This function is connected to the Gamma function by the relation B(z, w) = Γ(z)Γ(w) Γ(z + w) (z, w ∈ Z− = {0, −1, −2, ...}). 0 (1.2.19) The binomial coefficients are defined for α ∈ C and n ∈ N by α 0 = 1, α n = α(α − 1) · · · (α − n + 1) (−1)n (−α)n = . (1.2.20) n! n! In particular, when α = m, n ∈ N0 = {0, 1, · · · }, with m ≥ n, we have m n = m! n!(m − n)! (1.2.21) and m n =0 (m, n ∈ N0 ; 0 ≤ m < n) (1.2.22)
  32. 32. November 23, 2011 11:8 World Scientific Book - 9in x 6in 8 book Fractional Calculus: Models and Numerical Methods If α ∈ Z− = {−1, −2, −3, · · · }, the formula (1.2.20) is represented via the Gamma function by α n = Γ(α + 1) n!Γ(α − n + 1) (α ∈ C; α ∈ Z− ; n ∈ N0 ). (1.2.23) Such a relation can be extended from n ∈ N0 to arbitrary complex β ∈ C by α β = Γ(α + 1) Γ(α − β + 1)Γ(β + 1) (α, β ∈ C; α ∈ Z− ). (1.2.24) For more information on Gamma and Beta functions we refer to the standard works [5, 40, 423]. 1.2.2 Hypergeometric functions The Gauss hypergeometric function 2 F1 (a, b; c; z) is defined in the unit disk as the sum of the hypergeometric series ∞ 2 F1 (a, b; c; z) = k=0 (a)k (b)k z k , (c)k k! (1.2.25) where |z| < 1; a, b ∈ C; c ∈ CZ− , and (a)k is the Pochhammer sym0 bol (1.2.4). Alternatively, the function can be given by the Euler integral representation 2 F1 (a, b; c; z) = Γ(c) Γ(b)Γ(c − b) 1 0 tb−1 (1 − t)c−b−1 (1 − zt)−a dt, (1.2.26) when 0 < Re(b) < Re(c) and | arg(1 − z)| < π. The confluent hypergeometric function is defined by ∞ 1 F1 (a; c; z) = k=0 (a)k z k , (c)k k! (1.2.27) where z, a ∈ C, c ∈ CZ− and c = 0; but, in contrast to the hypergeometric function in (1.2.25), this series is convergent for any z ∈ C. It has the integral representation 1 F1 (a; c; z) = Γ(c) Γ(a)Γ(c − a) for 0 < Re(a) < Re(c). 1 0 ta−1 (1 − t)c−a−1 ezt dt (1.2.28)
  33. 33. November 23, 2011 11:8 World Scientific Book - 9in x 6in book 9 Preliminaries The Gauss hypergeometric series (1.2.25) and (1.2.27) are extended to the generalized hypergeometric series defined by ∞ p Fq (a1 , . . . , ap ; b1 , . . . , bq ; z) = k=0 (a1 )k · · · (ap )k z k , (b1 )k · · · (bq )k k! (1.2.29) where al , bj ∈ C, bj = 0, −1, −2, . . . (l = 1, . . . , p; j = 1, . . . , q). This series is absolutely convergent for all values of z ∈ C if p ≤ q. 1.2.3 Mittag-Leffler functions In this section we present the definitions and some properties of the MittagLeffler functions. The Mittag-Leffler function of one parameter Eα (z) is defined by ∞ Eα (z) = k=0 zk Γ(αk + 1) (z ∈ C; Re(α) > 0) . (1.2.30) In particular, for α = 1, 2 we have √ E1 (z) = ez and E2 (z) = cosh( z). (1.2.31) The Mittag-Leffler function of two parameters Eα,β (z), generalizing the one in (1.2.30), is defined by ∞ Eα,β (z) = k=0 zk Γ(αk + β) (z, β ∈ C; Re(α) > 0) . (1.2.32) In particular when β = 1, Eα,1 (z) = Eα (z) (z ∈ C; Re(α) > 0) (1.2.33) √ ez − 1 sinh( z) √ E1,2 (z) = , and E2,2 (z) = . z z (1.2.34) and The function Eα,β (z) has the integral representation Eα,β (z) = 1 2π C tα−β et dt, tα − z (1.2.35)
  34. 34. November 23, 2011 11:8 10 World Scientific Book - 9in x 6in book Fractional Calculus: Models and Numerical Methods Here the path of integration C is a loop which starts and ends at −∞ and encircles the circular disk |t| ≤ |z|1/α in the positive sense | arg(t)| ≤ π on C. The following is a Mittag-Leffler function which generalizes the MittagLeffler functions (1.2.30) and (1.2.32) defined, for z, α, β, ρ ∈ C and Re(α) > 0, by ∞ ρ Eα,β (z) = k=0 (ρ)k zk , Γ(αk + β) k! (1.2.36) where (ρ)k is the Pochhammer symbol (1.2.4). In particular, when ρ = 1, it coincides with the Mittag-Leffler function (1.2.32), that is, 1 Eα,β (z) = Eα,β (z) (z ∈ C). (1.2.37) ρ When α = 1, E1,β (z) coincides with the confluent hypergeometric function (1.2.27), apart from a constant factor [Γ(β)]−1 , i.e., ρ E1,β (z) = 1.3 1 1 F1 (ρ; β; z). Γ(β) (1.2.38) Fractional Operators In this section we give the definitions and some properties of fractional integrals and fractional derivatives of different kinds, such as RiemannLiouville, Caputo, Liouville, Hadamard, Marchaud and Gr¨nwaldu Letnikov. 1.3.1 Riemann-Liouville fractional integrals and fractional derivatives In this subsection we give the definitions of the Riemann-Liouville fractional integrals and fractional derivatives on a finite real interval and some of their properties. Let Ω = [a, b] (−∞ < a < b < ∞) be a finite interval on the real axis α α R. The Riemann-Liouville fractional integrals RL Ia+ f and RL Ib− f of order
  35. 35. November 23, 2011 11:8 World Scientific Book - 9in x 6in book 11 Preliminaries α ∈ C (Re(α) > 0) are defined by RL α Ia+ f (x) = RL α Ib− f (x) = 1 Γ(α) x a f (t)dt (x > a; Re(α) > 0) (x − t)1−α (1.3.1) f (t)dt (x < b; Re(α) > 0), (t − x)1−α (1.3.2) and 1 Γ(α) b x respectively. These integrals are called the left-sided and the right-sided fractional integrals. α α The Riemann-Liouville fractional derivatives RL Da+ y and RL Db− y of order α ∈ C (Re(α) ≥ 0) are defined by RL α Da+ y (x) = d dx n RL n−α Ia+ y (x) (x > a) (1.3.3) and RL α Db− y (x) = − d dx n RL n−α Ib− y (x) (x < b), (1.3.4) respectively, with n = −[−Re(α)], where [•] means the integral part of the argument, that is  [Re(α)] + 1 n= α for α ∈ N0 , for α ∈ N0 . (1.3.5) In particular, when α = n ∈ N0 , then 0 0 (RL Da+ y)(x) = (RL Db− y)(x) = y(x), n (RL Da+ y)(x) = y (n) (x), n (RL Db− y)(x) = (−1)n y (n) (x), (1.3.6) (1.3.7) where y (n) (x) is the classical derivative of y(x) of order n. The particular cases when a = 0 in the left-sided fractional integral and derivative of Riemann-Liouville are often used in the literature, because in
  36. 36. November 23, 2011 11:8 World Scientific Book - 9in x 6in 12 book Fractional Calculus: Models and Numerical Methods this case such fractional operators have a straightforward Laplace transform. For the sake of simplicity, this special case will be written in this book with any of the following nomenclature RL α I0+ f = RL I α f = J α f (1.3.8) and RL α D0+ f = RL Dα f (1.3.9) If α, β ∈ C, Re(α) ≥ 0 and Re(β) > 0, the following properties can be directly verified: − a)β (x) = Γ(β + 1) (x − a)β+α , Γ(β + α + 1) (1.3.10) α Da+ (t − a)β (x) = Γ(β + 1) (x − a)β−α , Γ(β − α + 1) (1.3.11) − t)β (x) = Γ(β + 1) (b − x)β+α , Γ(β + α + 1) (1.3.12) α Db− (b − t)β (x) = Γ(β + 1) (b − x)β−α . Γ(β − α + 1) (1.3.13) (b − x)−α , Γ(1 − α) (1.3.14) α Db− (b − t)α−j (x) = 0. (1.3.15) RL α Ia+ (t RL RL α Ib− (b RL For 0 < Re(α) < 1 this reduces to RL α Da+ 1 (x) = (x − a)−α , Γ(1 − α) RL α Db− 1 (x) = and for j = 1, 2, . . ., n = −[−Re(α)], we obtain RL α Da+ (t − a)α−j (x) = 0, RL α From (1.3.15) we derive that the equality (Da+ y)(x) = 0 is valid if, and only if, n y(x) = j=1 cj (x − a)α−j , where n = [Re(α)] + 1 and cj ∈ R (j = 1, . . . , n) are arbitrary constants. α In particular, when 0 < Re(α) ≤ 1, the relation (RL Da+ y)(x) = 0 holds if, and only if, y(x) = c(x − a)α−1 with any c ∈ R.
  37. 37. November 23, 2011 11:8 World Scientific Book - 9in x 6in book 13 Preliminaries α Likewise, the equality (RL Db− y)(x) = 0 is valid if, and only if, n y(x) = j=1 dj (b − x)α−j , where dj ∈ R (j = 1, . . . , n) are arbitrary constants. In particular, when α 0 < Re(α) ≤ 1, the relation (RL Db− y)(x) = 0 holds if, and only if, y(x) = d(b − x)α−1 with any d ∈ R. The next results give us an alternative representation of the fractional α α derivatives RL Da+ and RL Db− , Re(α) ≥ 0, n = [Re(α)] + 1, for suitable functions y(x) n−1 α (RL Da+ y)(x) = k=0 y (k) (a) 1 (x − a)k−α + Γ(1 + k − α) Γ(n − α) x a y (n) (t)dt (x − t)α−n+1 (1.3.16) and n−1 (−1)n (−1)k y (k) (b) (b − x)k−α + Γ(1 + k − α) Γ(n − α) b y (n) (t)dt . α−n+1 x (t − x) k=0 (1.3.17) α The semigroup property of the fractional integration operators RL Ia+ α and RL Ib− establishes that, if Re(α) > 0 and Re(β) > 0, then the equations α (RL Db− y)(x) = β α+β β α+β α α (RL Ia+ RL Ia+ f )(x) = (RL Ia+ f )(x) and (RL Ib− RL Ib− f )(x) = (RL Ib− f )(x) (1.3.18) are satisfied at almost every point x ∈ [a, b] for f (x) ∈ Lp (a, b) (1 ≤ p ≤ ∞). If α + β > 1, then the relations in (1.3.18) hold at any point of [a, b]. Similarly, we have the following index rule m β α+β α (RL Da+ RL Da+ f )(x) = (RL Da+ f )(x) − β−j (RL Da+ f )(a+) j=1 (x − a)−j−α , Γ(1 − j − α) (1.3.19) if α, β > 0 such that n − 1 < α ≤ n, m − 1 < β ≤ m (n, m ∈ N) and α + β < n. For f (x) ∈ Lp (a, b) (1 ≤ p ≤ ∞), the composition relations β α−β β α−β α α (RL Da+ RL Ia+ f )(x) = Ia+ f (x) and (RL Db− RL Ib− f )(x) = RL Ib− f (x) (1.3.20)
  38. 38. November 23, 2011 14 11:8 World Scientific Book - 9in x 6in book Fractional Calculus: Models and Numerical Methods between fractional differentiation and fractional integration operators hold almost everywhere on [a, b] if Re(α) > Re(β) > 0. In particular, when β = k ∈ N and Re(α) > k, then α−k α−k α α (RL DkRL Ia+ f )(x) = RL Ia+ f (x) and (RL DkRL Ib− f )(x) = (−1)kRL Ib− f (x). (1.3.21) So, the fractional differentiation is an operation inverse to the fractional integration from the left, i.e., if Re(α) > 0, then the equalities α α α α (RL Da+ RL Ia+ f )(x) = f (x) and (RL Db− RL Ib− f )(x) = f (x) (1.3.22) hold almost everywhere on [a, b]. On the other hand, if Re(α) > 0, n = [Re(α)] + 1 and fn−α (x) = RL n−α ( Ia+ f )(x), the relation n α α (RL Ia+ RL Da+ f )(x) = f (x) − j=1 (n−j) fn−α (a) (x − a)α−j Γ(α − j + 1) (1.3.23) n−α holds almost everywhere on [a, b]. Also, if gn−α (x) = (RL Ib− g)(x), then the formula n α α (RL Ib− RL Db− g)(x) = g(x) − (n−j) j=1 (−1)n−j gn−α (a) (b − x)α−j Γ(α − j + 1) (1.3.24) holds almost everywhere on [a, b]. Let Re(α) ≥ 0, m ∈ N and D = d/dx, then if the fractional derivatives α+m α (Da+ y)(x) and (RL Da+ y)(x) exist, we have (RL Dm RL α+m α Da+ y)(x) = (RL Da+ y)(x), (1.3.25) α+m α and, if the fractional derivatives (RL Db− y)(x) and (RL Db− y)(x) exist, then (RL Dm RL α+m α Db− y)(x) = (−1)m (RL Db− y)(x). (1.3.26) In connection with the Laplace transform, if Re(α) > 0 and n = [Re(α)]+1, we have n−1 α (LRL D0+ y)(s) = sα (Ly)(s) − for (Re(s) > q0 ). n−α sn−k−1 Dk (RL I0+ y)(0+) k=0 (1.3.27)
  39. 39. November 23, 2011 11:8 World Scientific Book - 9in x 6in book 15 Preliminaries The rules for fractional integration by parts read as follows. (a) If ϕ(x) ∈ Lp (a, b) and ψ(x) ∈ Lq (a, b), then b ϕ(x) a RL α Ia+ ψ b (x)dx = ψ(x) a RL α Ib− ϕ (x)dx. (1.3.28) α (b) If f (x) = RL Ib− h1 (x) with some h1 (x) ∈ Lp (a, b) and g(x) = RL α Ia+ h2 (x) with some h2 (x) ∈ Lq (a, b), then b b f (x) a RL α Da+ g (x)dx = g(x) a RL α Db− f (x)dx. (1.3.29) Here we assume α > 0, p ≥ 1, q ≥ 1, and (1/p) + (1/q) ≤ 1 + α (p = 1 and q = 1 in the case when (1/p) + (1/q) = 1 + α). The generalized fractional Leibniz formula for the Riemann-Liouille derivative, applied to suitable functions on [a, b], reads ∞ RL α RL α−j ( Da+ f )(x)(Dj g)(x), j α Da+ (f g) (x) = j=0 (1.3.30) where α > 0. Below, we present three particular cases to illustrate this property. (a) Let 0 < α < 1, f (x) = x and g(x) a suitable function. Then 1−α α α [RL D0+ (f g)](x) = x(RL D0+ g)(x) + (RL I0+ g)(x) (1.3.31) (b) Let 0 < α < 1, f (x) = xα−1 and g(x) a suitable function. Then ∞ RL α D0+ (f g) (x) = j=1 α j Γ(α) j−1 (j) x g (x) Γ(j) (1.3.32) (c) Let p ∈ N, α > 0, and f (x) a suitable function. Then p α (RL D0+ tp f )(x) = j=0 α j α−j (Dj xp )(RL D0+ f )(x) (1.3.33)
  40. 40. November 24, 2011 13:43 World Scientific Book - 9in x 6in 16 book Fractional Calculus: Models and Numerical Methods The computation of a fractional Riemann-Liouville derivative of the composition of two suitable functions can be very complicated. The corresponding formula RL α Da+ (f (g)) (x) = ∞ + j=1 (x − a)−α f (g(x)) Γ(1 − α) α j!(x − a)j−α j Γ(j + 1 − α) j j [Di f (g)](x) r=1 (1.3.34) j 1 a ! r=1 r (Dr g)(x) r! ar , j where r=1 rar = j and r=1 ar = i, exhibits the complicated structure very clearly. The above relation is a consequence of the application of (1.3.30) and the well known Fa´ di Bruno formula (see, e.g., formula (24.1.2) in [5]) for a a natural n, viz. n (Dn f (g)) (x) = (Dm f ) (g(x)) (1.3.35) m=0 × (n; a1 , a2 , . . . , an ) {(Dg)(x)}a1 (D2 g)(x) a2 · · · {(Dn g)(x)}an summed over a1 + 2a2 + . . . + nan and a1 + a2 + . . . + an = m, see also Eqs. (5.2.26) and (5.2.29). 1.3.2 Caputo fractional derivatives The Caputo fractional derivatives are closely related to the RiemannLiouville derivatives. Let [a, b] be a finite interval of the real line R. For α ∈ C (Re(α) ≥ 0) the Caputo fractional derivatives are defined by α (C Da+ y)(x) = 1 Γ(n − α) x a y (n) (t)dt n−α = (RL Ia+ Dn y)(x) (x − t)α−n+1 (1.3.36) and (−1)n Γ(n − α) b y (n) (t)dt n−α = (−1)n (RL Ib− Dn y)(x), α−n+1 x (t − x) (1.3.37) where D = d/dx and n = −[−Re(α)], i.e., n = [Re(α)] + 1 for α ∈ N0 , and n = α for α ∈ N0 . These derivatives are called left-sided and right-sided Caputo fractional derivatives of order α. α (C Db− y)(x) =
  41. 41. November 23, 2011 11:8 World Scientific Book - 9in x 6in book 17 Preliminaries In particular, when 0 < Re(α) < 1 then α (C Da+ y)(x) = x 1 Γ(1 − α) a y ′ (t)dt 1−α = (RL Ia+ Dy)(x) (x − t)α (1.3.38) y ′ (t)dt 1−α = −(RL Ib− Dy)(x). (t − x)α (1.3.39) and α (C Db− y)(x) = − 1 Γ(1 − α) b x The connections between the Caputo and the Riemann-Liouville derivatives are given by the relations n−1 α (C Da+ y)(x) = RL α Da+ y(t) − k=0 y (k) (a) (t − a)k k! (x) (1.3.40) y (k) (b) (b − t)k k! (x), (1.3.41) and n−1 α (C Db− y)(x) = RL α Db− y(t) − k=0 respectively. In particular, when 0 < Re(α) < 1, the relations (1.3.40) and (1.3.41) take the following forms α (C Da+ y)(x) = RL α Da+ [y(t) − y(a)] (x), (1.3.42) α (C Db− y)(x) = RL α Db− [y(t) − y(b)] (x). (1.3.43) If α = n ∈ N0 and the usual derivative y (n) (x) of order n exists, then n n ( Da+ y)(x) coincides with y (n) (x), while (C Db− y)(x) coincides with y (n) (x) up to the constant factor (−1)n , i.e., C n n (C Da+ y)(x) = y (n) (x) and (C Db− y)(x) = (−1)n y (n) (x) (n ∈ N). (1.3.44) α α The Caputo derivatives (C Da+ y)(x) and (C Db− y)(x) have properties similar to those given in Eqs. (1.3.11) and (1.3.13) for the Riemann-Liouville fractional derivatives. If Re(α) > 0, n = −[−Re(α)] is given by (1.3.5) and Re(β) > n − 1, then C α Da+ (t − a)β (x) = Γ(β + 1) (x − a)β−α Γ(β − α + 1) (1.3.45)
  42. 42. November 23, 2011 11:8 World Scientific Book - 9in x 6in 18 book Fractional Calculus: Models and Numerical Methods and C α Db− (b − t)β (x) = Γ(β + 1) (b − x)β−α . Γ(β − α + 1) (1.3.46) However, for k = 0, 1, . . . , n − 1, we have α α (C Da+ (t − a)k )(x) = 0 and (C Db− (t − a)k )(x) = 0. (1.3.47) In particular, α α (C Da+ 1)(x) = 0 and (C Db− 1)(x) = 0. (1.3.48) On the other hand, if Re(α) > 0 and λ > 0, then C α Da+ eλt (x) = λα eλx (1.3.49) for any a ∈ R. Let Re(α) > 0 and let y(x) a suitable function, for example y(x) ∈ C[a, b]. Then If Re(α) ∈ N or α ∈ N, the Caputo fractional differentiaα α tion operators C Da+ and C Db− provide operations inverse to the Riemannα α Liouville fractional integration operators Ia+ and Ib− from the left, that is α α α α (C Da+ RL Ia+ y)(x) = y(x) and (C Db− RL Ib− y)(x) = y(x). (1.3.50) However, when Re(α) ∈ N and Im(α) = 0, we have α α (C Da+ RL Ia+ y)(x) = y(x) − α+1−n (RL Ia+ y)(a+) (x − a)n−α Γ(n − α) (1.3.51) α α (C Db− RL Ia+ y)(x) = y(x) − α+1−n (RL Ib− y)(b−) (b − x)n−α . Γ(n − α) (1.3.52) and On the other hand, if Re(α) > 0 and n = −[−Re(α)] is given by (1.3.5), then under sufficiently good conditions for y(x) n−1 α α (RL Ia+ C Da+ y)(x) = y(x) − k=0 y (k) (a) (x − a)k k! (1.3.53)
  43. 43. November 23, 2011 11:8 World Scientific Book - 9in x 6in book 19 Preliminaries and n−1 α α (RL Ib− C Db− y)(x) = y(x) − k=0 (−1)k y (k) (b) (b − x)k . k! (1.3.54) In particular, if 0 < Re(α) ≤ 1, then α α α α (RL Ia+ C Da+ y)(x) = y(x) − y(a) and (RL Ib− C Db− y)(x) = y(x) − y(b). (1.3.55) Under suitable conditions, the Laplace transform of the Caputo fracα tional derivative C D0+ y is given by n−1 α (LC D0+ y)(s) = sα (Ly)(s) − sα−k−1 (Dk y)(0). (1.3.56) k=0 In particular, if 0 < α ≤ 1, then α (LC D0+ y)(s) = sα (Ly)(s) − sα−1 y(0). (1.3.57) We have defined the Caputo derivatives on a finite interval [a, b]. Formulas (1.3.36) and (1.3.37) can be used for the definition of the Caputo fractional derivatives on the whole axis R. Thus the corresponding Caputo fractional derivative of order α ∈ C (with Re(α) > 0 and α ∈ N) can be defined as follows α (C D+ y)(x) = 1 Γ(n − α) x −∞ y (n) (t)dt (x − t)α+1−n (1.3.58) y (n) (t)dt , (t − x)α+1−n (1.3.59) and α (C D− y)(x) = (−1)n Γ(n − α) ∞ x with x ∈ R. α α The Caputo derivatives (C D+ y)(x) and (C D− y)(x) have properties similar to those that we will describe below for the operators known as Liouville derivatives. In particular, we mention the identities C α D+ eλt (x) = λα eλx and C α D− e−λt (x) = λα e−λx . (1.3.60)
  44. 44. November 23, 2011 11:8 20 1.3.3 World Scientific Book - 9in x 6in book Fractional Calculus: Models and Numerical Methods Liouville fractional integrals and fractional derivatives. Marchaud derivatives First of all, we present the definitions and some properties of the Liouville fractional integrals and fractional derivatives on the whole axis R. More detailed information may be found in the bibliography. The Liouville fractional integrals on R have the form α (L I+ f )(x) = 1 Γ(α) α (L I− f )(x) = 1 Γ(α) x (1.3.61) f (t)dt , (t − x)1−α −∞ f (t)dt (x − t)1−α (1.3.62) and ∞ x where x ∈ R and Re(α) > 0, while the fractional Liouville derivatives are defined as n d n−α (L I+ y)(x) dx n x 1 d y(t)dt = Γ(n − α) dx (x − t)α−n+1 −∞ α (L D+ y)(x) = (1.3.63) and α (L D− y)(x) = = − d dx n 1 Γ(n − α) n−α (L I− y)(x) − d dx n ∞ x y(t)dt , (t − x)α−n+1 (1.3.64) where n = −[−Re(α)], Re(α) ≥ 0 and x ∈ R. α α The expressions for L I+ f and L I− f in (1.3.61) and (1.3.62), and for L α L α D+ y and D− y in (1.3.63) and (1.3.64), are called Liouville left- and rightsided fractional integrals and fractional derivatives on the whole axis R, respectively. In particular, when α = n ∈ N0 , then 0 0 (L D+ y)(x) = (L D− y)(x) = y(x) (1.3.65) and n n (L D+ y)(x) = y (n) (x) and (L D− y)(x) = (−1)n y (n) (x) (n ∈ N), (1.3.66) where y (n) (x) is the usual derivative of y(x) of order n.
  45. 45. November 23, 2011 11:8 World Scientific Book - 9in x 6in Preliminaries book 21 If 0 < Re(α) < 1 and x ∈ R, then α (L D+ y)(x) = 1 d Γ(1 − α) dx x −∞ y(t)dt (x − t)α−[Re(α)] (1.3.67) y(t)dt . (t − x)α−[Re(α)] (1.3.68) and α (L D− y)(x) = − 1 d Γ(1 − α) dx ∞ x α α The Liouville fractional operators L I+ and L D+ of the exponential function eλx yield the same exponential function, both apart from a constant multiplication factor, i.e., if λ > 0 and Re(α) ≥ 0, α (L I+ eλt )(x) = λ−α eλx ; L α −λt I− e (x) = λ−α e−λx ; (1.3.69) (1.3.70) α (L D+ eλt )(x) = λα eλx ; (1.3.71) α D− e−λt (x) = λα e−λx . (1.3.72) and L On the other hand, if α > 0, β > 0, then, for “sufficiently good” functions, we have β α+β β α+β α α (L I+ L I+ f )(x) = (L I+ f )(x) and (L I− L I− f )(x) = (L I− f )(x); (1.3.73) α α α α (L D+ L I+ f )(x) = f (x), and (L D− L I− f )(x) = f (x). (1.3.74) If in addition α > β > 0, then the formulas β α−β β α−β α α (L D+ L I+ f )(x) = (L I0+ f )(x) and (L D− L I− f )(x) = (L I− f )(x) (1.3.75) hold. Furthermore, when β = k ∈ N and Re(α) > k, then α−k α−k α α (Dk L I+ f )(x) = L I+ f (x), and (Dk L I− f )(x) = (−1)k L I− f (x). (1.3.76)
  46. 46. November 23, 2011 11:8 World Scientific Book - 9in x 6in 22 book Fractional Calculus: Models and Numerical Methods α The Fourier transform (1.1.1) of the Liouville fractional integrals L I+ f L α and I− f is given for 0 < Re(α) < 1, by the following relations α (F L I+ f )(κ) = (F f )(κ) (−iκ)α (1.3.77) α (F L I− f )(κ) = (F f )(κ) . (iκ)α (1.3.78) and Here (∓iκ)α means (∓iκ)α = |κ|α e∓απi sgn(κ)/2 . (1.3.79) Moreover, if Re(α) > 0, then, for “sufficiently good” functions f (x), the equations (1.3.77) and (1.3.78) are valid as well as the following correα α sponding relations for the Liouville fractional derivatives L D+ f and L D− f α (F L D+ f )(κ) = (−iκ)α (F f )(κ) (1.3.80) α (F L D− f )(κ) = (iκ)α (F f )(κ), (1.3.81) and where (∓iκ)α is defined by (1.3.79). The rules for fractional integration by parts, for α > 0, and for “sufficiently good” functions, are given by ∞ ϕ(x) −∞ L α I+ ψ ∞ f (x) L −∞ ∞ ϕ(x) 0 f (x) 0 RL ψ(x) L α I− ϕ g(x) L (x)dx. (1.3.82) α D− f (x)dx. (1.3.83) ψ(x) L α I− ϕ (x)dx. (1.3.84) g(x) L α D− f (x)dx. (1.3.85) −∞ α D+ g (x)dx = RL α I0+ ψ ∞ ∞ (x)dx = ∞ −∞ ∞ (x)dx = α D0+ g (x)dx = 0 ∞ 0
  47. 47. November 23, 2011 11:8 World Scientific Book - 9in x 6in book 23 Preliminaries α α The Liouville fractional derivatives L D+ f and L D− f exist for suitable functions f , but they are not defined, for example, for constant functions. Nevertheless, they can be reduced in general to more convenient forms which admit fractional differentiation of a constant function. In this way α α we come to the Marchaud fractional derivatives M D+ f and M D− f of order α ∈ C, defined by M 1 κ(α, k) α D+ f (x) = ∞ 0 (∆k f )(x) t dt (k > Re(α) > 0) t1+α (1.3.86) (∆k f )(x) −t dt (k > Re(α) > 0), t1+α (1.3.87) and M α D− f (x) = 1 κ(α, k) ∞ 0 respectively. Here κ(α, k) is the constant ∞ κ(α, k) = 0 (1 − e−u )k du u1+α (k ∈ N, k > Re(α) > 0), (1.3.88) and (∆k f )(x) is the finite difference of order k of a function f (x) with h increment h k ∆k f h (−1)j (x) = j=0 k j f (x − jh). (1.3.89) Note that k ∆k 1 h (−1)j (x) = j=0 k j = (1 − 1)k = 0 (k ∈ N). (1.3.90) In particular, when h = −1, (1.3.89) coincides with the difference (1.3.89) with h = +1 except for the factor (−1)k , i.e. ∆k f (x) = (−1)k (∆k f )(x). −1 It is known that for a real number α > 0 the right-hand sides of (1.3.86) and (1.3.87) do not depend on the choice of k (k > α). Similarly, for complex α the right-hand sides of (1.3.86) and (1.3.87) do not depend on the choice of k (k > Re(α)).
  48. 48. November 23, 2011 11:8 World Scientific Book - 9in x 6in 24 book Fractional Calculus: Models and Numerical Methods α α The Marchaud fractional derivatives M D+ f and M D− f are defined for the constant function f = c ∈ C, and in accordance with (1.3.90) M α D+ c (x) = M α D− c (x) = 0 (α ∈ C, Re(α) > 0). (1.3.91) For “suitable functions” f , the Marchaud fractional derivatives coincide with the Liouville fractional derivatives for same α M α α D+ f = L D+ f ; M α α D− f = L D− f. In particular, they have the same properties as the Liouville derivatives over exponential functions ebt and e−bt , in the following sense α D+ ebt (x) = bα ebx , (1.3.92) α D− e−bt (x) = bα e−bx . (1.3.93) M M for α ∈ C (Re(α) > 0) and b ∈ C (Re(b) > 0). 1.3.4 Generalized exponential functions In this section we consider two special functions, which play a role as generalized exponentials. The first one is the Mittag-Leffler function ∞ Eα (λz α ) = k=0 λk z αk Γ(αk + 1) (z, λ ∈ C; Re(α) > 0) , (1.3.94) and the second one is defined in terms of the Mittag-Leffler type function by ∞ eλz = z α−1 Eα,α (λz α ) = α k=0 λk z α(k−1) Γ(α(k + 1)) (z, λ ∈ C, Re(α) > 0) . (1.3.95) The above functions satisfy the identity E1 (λz) = eλz = eλz 1 (z, λ ∈ C), (1.3.96) and therefore they are generalizations of the classical exponential function.
  49. 49. November 23, 2011 11:8 World Scientific Book - 9in x 6in book 25 Preliminaries Some others properties of the first of these functions, for z ∈ C{0}, α, λ ∈ C (Re(α) > 0), and n ∈ N, are the following lim Eα (λ(z − a)α ) = 1; (1.3.97) [Eα (λz α )] = z −n Eα,1−n (λz α ) ; (1.3.98) n+1 [Eα (λz α )] = n!z αn Eα,αn+1 (λz α ) ; (1.3.99) z→a+ n ∂ ∂z n ∂ ∂λ sα−1 sα − λ L [Eα (λz α )] (s) = (Re(s) > 0; |λs−α | < 1); (1.3.100) and ∂ ∂λ L tαn n Eα (λz α ) (s) = n!sα−1 . (sα − λ)n+1 (1.3.101) On the other hand, for the function eλz , we have the following properties α lim z→a+ ∂ ∂z ∂ ∂λ (z − a)1−α eλ(z−a) = α 1 ; Γ(α) (1.3.102) n eλz = z α−n−1 Eα,α−n (λz α ) ; α (1.3.103) n n+1 λz eα = n!z αn+α−1 Eα,(n+1)α (λz α ) ; L eλz (s) = α 1 sα − λ Re(s) > 0; |λs−α | < 1 ; (1.3.104) (1.3.105) and L ∂ ∂λ n eλz (s) = α n! . (sα − λ)n+1 (1.3.106)
  50. 50. November 23, 2011 26 11:8 World Scientific Book - 9in x 6in book Fractional Calculus: Models and Numerical Methods The generalized α-exponential functions do not have the index property, i.e., in general eλz eµz = e(λ+µ)z α α α Eα (λz)Eα (µz) = Eα ((λ + µ)z); (α = 1). (1.3.107) For example, if α = 2 and z = 1, then in accordance with the second relation in (1.2.34), we have eλ = 2 √ √ λ sinh( λ), (1.3.108) but √ √ √ √ [ λ sinh( λ)][ µ sinh( µ)] = (λ + µ) sinh( (λ + µ). (1.3.109) Let Mn (R) (n ∈ N) be the set of all matrices A = [ajk ] of order n × n with ajk ∈ R (j = 1, . . . , n). By analogy with (1.3.95), for α ∈ C{0} (Re(α) > 0), and A ∈ Mn (R), here we introduce a matrix α-exponential function defined by ∞ eAz = z α−1 α z αk , Γ((k + 1)α) (1.3.110) (z ∈ C; Re(α) > 0) . (1.3.111) Ak k=0 and also the function ∞ Eα (Az α ) = k=0 Ak z kα Γ(αk + 1) When α = 1 we have E1 (Az) = eAz = eAz 1 (z ∈ C). (1.3.112) Of course, in general, the semigroup property does not hold for the generalized matrix exponential functions eAz eBz = e(A+B)z α α α (z, α ∈ C; A, B ∈ Mn (R)) . (1.3.113) Similarly, the inversion formula (eAz )−1 = e−Az , (1.3.114)
  51. 51. November 23, 2011 11:8 World Scientific Book - 9in x 6in book 27 Preliminaries valid for the matrix exponential function eAz , is not true, in general, for matrix α-exponential function eAz , α (eAz )−1 = e−Az . α α (1.3.115) of the matrix A with elements ajk ∈ R If we define the norm A (j, k = 1, . . . , n) by A = max |ajk | j,k∈N (1.3.116) then, from (1.3.110), we derive the estimate for the norm of eAz . For any α fixed z ∈ C, the following relation holds ∞ eAz ≤ α A k=0 k |z|Re(α)k . |Γ((k + 1)α)| (1.3.117) When z = x > 0 and α > 0, the above formula takes the simpler form ∞ eAx ≤ α A k=0 k xαk . Γ((k + 1)α) (1.3.118) Corresponding properties can be proved for the other generalized matrix exponential functions Eα (Az α ). Particularly important are the following properties of the functions λ(z−a) Eα (λ(z − a)α ) and eα : C α Da+ Eα (λ(z − a)α ) (x) = λEα (λ(x − a)α ) (1.3.119) and RL α Da+ e(λ(z−a)) (x) = λe(λ(x−a)) α α (1.3.120) with α > 0, a ∈ R and λ ∈ C. Thus, the two generalized exponential functions are eigenfunctions of Caputo and Riemann-Liouville differential operators, respectively. The behavior of Eα (−x) and e−x for various values of α in the intervals α (0, 1) and (1, 2) can be seen from the following figures.
  52. 52. November 23, 2011 11:8 World Scientific Book - 9in x 6in 28 book Fractional Calculus: Models and Numerical Methods 0<α<1 1 α=0.5 α=0.7 α=0.9 0.9 0.8 0.7 Eα(−x) 0.6 0.5 0.4 0.3 0.2 0.1 0 0 2 4 6 8 10 x Fig. 1.1 Representation of Eα (−x) for some values of α ∈ (0, 1). 1<α<2 1 α=1.5 α=1.7 α=1.9 0.8 0.6 0.4 Eα(−x) 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0 10 20 30 40 50 x Fig. 1.2 Representation of Eα (−x) for some values of α ∈ (1, 2).
  53. 53. 11:8 World Scientific Book - 9in x 6in book 29 Preliminaries 0<α<1 1 α=0.5 α=0.7 α=0.9 0.9 0.8 0.7 eα −x 0.6 0.5 0.4 0.3 0.2 0.1 0 0 2 4 6 8 10 x Fig. 1.3 Representation of e−x for some values of α ∈ (0, 1). α 1<α<2 1 α=1.5 α=1.7 α=1.9 0.8 0.6 0.4 0.2 e−x α November 23, 2011 0 −0.2 −0.4 −0.6 −0.8 −1 0 10 20 30 40 50 x Fig. 1.4 Representation of e−x for some values of α ∈ (1, 2). α
  54. 54. November 23, 2011 11:8 World Scientific Book - 9in x 6in 30 book Fractional Calculus: Models and Numerical Methods 1.3.5 Hadamard type fractional integrals and fractional derivatives In this section we present the definitions and some properties of the Hadamard type fractional integrals and fractional derivatives. Let (a, b) (0 ≤ a < b ≤ ∞) be a finite or infinite interval of the halfaxis R+ , and let Re(α) > 0 and µ ∈ C. We consider the left-sided and right-sided integrals of fractional order α ∈ C (Re(α) > 0) defined by H α Ia+ f x 1 Γ(α) (x) = log x t log t x a α−1 f (t)dt (a < x < b) t (1.3.121) α−1 f (t)dt (a < x < b), t (1.3.122) and H α Ib− f (x) = b 1 Γ(α) x respectively. When a = 0 and b = ∞, these relations are given by H α I0+ f (x) = x 1 Γ(α) log x t α−1 f (t)dt (x > 0) t (1.3.123) t x α−1 log f (t)dt (x > 0). t (1.3.124) 0 and H α I− f (x) = ∞ 1 Γ(α) x More general fractional integrals than those in (1.3.123) and (1.3.124) are defined by H α I0+,µ f (x) = 1 Γ(α) x 0 t x µ log x t α−1 f (t)dt (x > 0) t (1.3.125) t x α−1 log f (t)dt (x > 0) t (1.3.126) and H α I−,µ f (x) = 1 Γ(α) ∞ x x t µ with µ ∈ C. The integral in (1.3.123) was introduced by Hadamard [254]. Therefore, the integrals (1.3.121), (1.3.122) and (1.3.123), (1.3.124) are often referred to as the Hadamard fractional integrals of order α. The more general
  55. 55. November 23, 2011 11:8 World Scientific Book - 9in x 6in book 31 Preliminaries integrals (1.3.125) and (1.3.126), introduced by Butzer et al. [117], are called the Hadamard type fractional integrals of order α. The left- and right-sided Hadamard fractional derivatives of order α ∈ C (Re(α) ≥ 0) on (a, b) are defined by H α Da+ y (x) = δ n = x H n−α Ia+ y n d dx (x) 1 Γ(n − α) x log a x t n−α+1 y(t)dt (1.3.127) t and H α Db− y (x) = (−δ)n = −x H n−α Ib− y n d dx (x) b 1 Γ(n − α) log x t x n−α+1 y(t)dt (1.3.128) t for a < x < b, respectively, where n = −[−Re(α)] and δ = xD (D = d/dx). When a = 0 and b = ∞, we have H H α D0+ y (x) = δ n α D− y (x) = (−δ)n H n−α I0+ y (x) (x > 0) ; H n−α I− y (x) (x > 0) . (1.3.129) (1.3.130) The Hadamard type fractional derivatives of order α with µ ∈ C, more general than those in (1.3.129) and (1.3.130), are defined for Re(α) ≥ 0 by H H α D0+,µ y (x) = x−µ δ n xµ α D−,µ y (x) = xµ (−δ)n x−µ H n−α I0+,µ y (x) (x > 0) ; H n−α I−,µ y (x) (x > 0) . (1.3.131) (1.3.132) 0 0 0 0 The Hadamard type operators H I0+,µ , H D0+,µ and H I−,µ , H D−,µ can be defined as the identity operator H 0 I0+,µ f 0 ≡ H D0+,µ f = f, H 0 I−,µ f 0 ≡ H D−,µ f = f, (1.3.133) and in particular, H 0 I0+ f 0 ≡ H D0+ f = f, H 0 I− f 0 ≡ H D− f = f. (1.3.134)
  56. 56. November 23, 2011 11:8 World Scientific Book - 9in x 6in 32 H book Fractional Calculus: Models and Numerical Methods For Re(α) > 0, n = −[−Re(α)], and 0 < a < b < ∞, we have that α Da+ y (x) = 0 is valid if, and only if, n y(x) = cj log j=1 x a α−j , (1.3.135) where cj ∈ R (j = 1, . . . , n) are arbitrary constants. In particular, when α 0 < Re(α) ≤ 1, the relation (Da+ y)(x) = 0 holds if, and only if, y(x) = x α−1 c log a for any c ∈ R. α On the other hand, the equality H Db− y (x) = 0 is valid if, and only if, n y(x) = dj log j=1 b x α−j , (1.3.136) where dj ∈ R (j = 1, . . . , n) are arbitrary constants. In particular, when α 0 < Re(α) ≤ 1, the relation H Db− y (x) = 0 holds if, and only if, y(x) = α−1 b d log x for any d ∈ R. It can also be directly verified that the Hadamard and Hadamard type fractional integrals and derivatives (1.3.123)–(1.3.126) and (1.3.129)– (1.3.132) of the power function xβ yield the same function, apart from a constant multiplication factor, that is, if Re(α) > 0, β, µ ∈ C, and Re(β + µ) > 0, then H α I0+,µ tβ (x) = (µ + β)−α xβ (1.3.137) α D0+,µ tβ (x) = (µ + β)α xβ . (1.3.138) and H On the other hand, if Re(β − µ) < 0, then (x) = (µ − β)−α xβ (1.3.139) α D−,µ tβ (x) = (µ − β)α xβ . (1.3.140) H α I−,µ tβ and H In particular, we have H α β I0+ t (x) = β −α xβ and H α D0+ tβ (x) = β α xβ (Re(β) > 0) (1.3.141)
  57. 57. November 23, 2011 11:8 World Scientific Book - 9in x 6in book 33 Preliminaries and H α β I− t (x) = (−β)−α xβ H α D− tβ (x) = (−β)α xβ (Re(β) < 0). (1.3.142) The Hadamard and Hadamard type fractional integrals (1.3.121)– (1.3.126) satisfy the semigroup property H α H β Ia+ Ia+ f and α+β = H Ia+ f and H α H β Ib− Ib− f α+β = H Ib− f, (1.3.143) for Re(α) > 0, Re(β) > 0 and 0 < a < b < ∞. If µ ∈ C, a = 0 and b = ∞, then β H α I0+,µ H I0+,µ f α+β = H I0+,µ f (1.3.144) α+β = H I−,µ f. (1.3.145) and H α H β I−,µ I−,µ f In particular, when µ = 0, H α H β I0+ I0+ f α+β = H I0+ f ; H α β I− J− f α+β = H I− f. (1.3.146) Now we give the properties of compositions between the operators of fractional differentiation (1.3.127)–(1.3.132) and fractional integration (1.3.121)–(1.3.126). If α ∈ C and β ∈ C are such that Re(α) > Re(β) > 0, then H β α−β α Da+ H Ia+ f = H Ia+ f and H α−β β α Db− H Ib− f = H I b− f (1.3.147) when 0 < a < b < ∞. In particular, if β = m ∈ N, then H α−m m α Da+ H Ia+ f = H Ia+ f and H α−m m α Db− H Ib− f = H Ib− f. (1.3.148) If µ ∈ C, a = 0 and b = ∞, then H β α−β α D0+,µ H I0+,µ f = H I0+,µ f (1.3.149) β α−β α D−,µ H I−,µ f = H I−,µ f. (1.3.150) and H
  58. 58. November 23, 2011 11:8 World Scientific Book - 9in x 6in 34 book Fractional Calculus: Models and Numerical Methods In particular, if β = m ∈ N, then H α−m m α D0+,µ H I0+,µ f = H I0+,µ f (1.3.151) α−m m α D−,µ H I−,µ f = H I−,µ f. (1.3.152) and H while, when µ = 0 and m ∈ N, β α−β α D0+ H I0+ f = H I0+ f ; H α−m m α D0+ H I0+ f = H I0+ f ; H H β α−β α D− H I− f = H I− f (1.3.153) α−m m α D− H I− f = H I− f. (1.3.154) and H The Hadamard and Hadamard type fractional derivatives (1.3.127), (1.3.128) and (1.3.131), (1.3.132) are operators inverse to the corresponding fractional integrals (1.3.121), (1.3.122) and (1.3.125), (1.3.126), that is, if Re(α) > 0 and 0 < a < b < ∞ then H α α Da+ H Ia+ f = f and H α α Db− H Ib− f = f, (1.3.155) whereas if Re(α) > 0, µ ∈ C, a = 0 and b = ∞, then H α α D0+,µ H I0+,µ f = f and H α α D−,µ H I−,µ f = f. (1.3.156) In particular, if µ = 0, then H α α D0+ H I0+ f = f and H α α D− H I− f = f. (1.3.157) The following property yields the formula for the composition of the α fractional differentiation operator H Da+ with the fractional integration opα erator H Ia+ . n α α (H Ia+ H Da+ y)(x) = y(x) − k=1 n−α (δ n−k (H Ia+ y))(a) x log Γ(α − k + 1) a α−k , (1.3.158) for Re(α) > 0, n = −[−Re(α)] and 0 < a < b < ∞. It is known that function series admit a term-by-term Riemann-Liouville fractional integration and differentiation under certain conditions. Similar assertions are true for Hadamard-type fractional integration and differenα α tiation operators H I0+,µ and H D0+,µ . Proposition 1.1. Let α ∈ C, µ > 0, l > 0, and let f (x) = fk (x) ∈ C([0, l]). ∞ k=0 fk (x),
  59. 59. November 23, 2011 11:8 World Scientific Book - 9in x 6in book 35 Preliminaries (1) If Re(α) > 0 and the series f (x) = ∞ fk (x) is uniformly converk=0 α gent on [0, l], then its termwise Hadamard-type integration H I0+,µ is admissible ∞ H α I0+,µ ∞ fk α (H I0+,µ fk )(x) (x) = k=0 (0 < x < l), (1.3.159) k=0 ∞ α and the series k=0 (H I0+,µ fk )(x) is also uniformly convergent on [0, l]; ∞ ∞ α (2) If Re(α) ≥ 0 and the series k=0 fk (x) and k=0 (H D0+,µ fk )(x) are uniformly convergent on [ǫ, l] (ǫ > 0), then the former series admits α termwise Hadamard-type fractional differentiation H D0+,µ by the formula ∞ H α D0+,µ ∞ fk H (x) = k=0 α D0+,µ fk (x) (0 < x < l). (1.3.160) k=0 Proposition 1.2. Let α ∈ C, µ > 0, and let f (x) be a convergent power series ∞ ak xk f (x) = k=0 (ak ∈ C, k ∈ N0 ). (1.3.161) α (1) If Re(α) > 0, then the Hadamard-type integral H I0+,µ f is also represented by the convergent power series ∞ H α I0+,µ f (µ + k)−α ak xk . (x) = (1.3.162) k=0 α (2) If Re(α) ≥ 0, then the Hadamard-type derivative H D0+,µ f is also represented by the convergent power series ∞ H α D0+,µ f (x) = (µ + k)α ak xk . (1.3.163) k=0 The radii of convergence of the series in (1.3.161), (1.3.162) and (1.3.163) coincide.
  60. 60. November 23, 2011 11:8 World Scientific Book - 9in x 6in 36 Fractional Calculus: Models and Numerical Methods 1.3.6 Fractional integrals and fractional derivatives of a function with respect to another function In this section we present the definitions and some properties of the fractional integrals and fractional derivatives of a function f with respect to another function g. The Hadamard fractional integral and derivative are particular cases of these new operators. Let (a, b) (−∞ ≤ a < b ≤ ∞) be a finite or infinite interval of the real line R and Re(α) > 0. Also let g(x) be an increasing and positive monotone function on (a, b], having a continuous derivative g ′ (x) on (a, b). The leftand right-sided fractional integrals of a function f with respect to another function g on [a, b] are defined by α (Ia+;g f )(x) = x g ′ (t)f (t)dt (x > a; Re(α) > 0) (1.3.164) [g(x) − g(t)]1−α b 1 Γ(α) g ′ (t)f (t)dt (x < b; Re(α) > 0), (1.3.165) [g(t) − g(x)]1−α a and α (Ib−;g f )(x) = 1 Γ(α) x respectively. When a = 0 and b = ∞, we shall use the following notations α (I0+;g f )(x) = α (I−;g f )(x) = 1 Γ(α) 1 Γ(α) x g ′ (t)f (t)dt (x > 0; Re(α) > 0), (1.3.166) [g(x) − g(t)]1−α ∞ g ′ (t)f (t)dt (x > 0; Re(α) > 0); (1.3.167) [g(t) − g(x)]1−α 0 x while, for a = −∞ and b = ∞, we have α (I+;g f )(x) = 1 Γ(α) α (I−;g f )(x) = 1 Γ(α) x −∞ ∞ x g ′ (t)f (t)dt (x ∈ R; Re(α) > 0), (1.3.168) [g(x) − g(t)]1−α g ′ (t)f (t)dt (x ∈ R; Re(α) > 0). (1.3.169) [g(t) − g(x)]1−α Integrals (1.3.164) and (1.3.165) are called the g-Riemann-Liouville fractional integrals on a finite interval [a, b], (1.3.166) and (1.3.167) the gLiouville fractional integrals on a half-axis R+ , while (1.3.168) and (1.3.169) are called the g-Liouville fractional integrals on the whole axis R. book
  61. 61. November 23, 2011 11:8 World Scientific Book - 9in x 6in book 37 Preliminaries For Re(α) > 0 and Re(β) > 0, we have Γ(β) [g(x) − g(a)]α+β−1 , Γ(α + β) α (Ia+;g f+ )(x) = (1.3.170) where f+ (x) = [g(x) − g(a)]β−1 . If f− (x) = [g(b) − g(x)]β−1 , then Γ(β) [g(b) − g(x)]α+β−1 . Γ(α + β) α (Ib−;g f− )(x) = (1.3.171) Moreover, if Re(α) > 0 and λ > 0, then α (I+;g eλg(t) )(x) = λ−α eλg(x) (1.3.172) α (I−;g e−λg(t) )(x) = λ−α e−λg(x) . (1.3.173) and The semigroup property also holds, i.e., if Re(α) > 0 and Re(β) > 0, then the relations α+β β α (Ia+;g Ia+;g f )(x) = (Ia+;g f )(x); β α+β α (Ib−;g Ib−;g f )(x) = (Ib−;g f )(x) (1.3.174) and β α+β α (I+;g I+;g f )(x) = (I+;g f )(x); α+β β α (I−;g I−;g f )(x) = (I−;g f )(x) (1.3.175) hold for “sufficiently good” functions f (x). Let g ′ (x) = 0 (−∞ ≤ a < x < b ≤ ∞) and Re(α) ≥ 0 (α = 0). Also let n = −[−Re(α)] and D = d/dx. The g-Riemann-Liouville and g-Liouville fractional derivatives of a function y with respect to g of order α (Re(α) ≥ 0; α = 0), corresponding to the g-Riemann-Liouville and gLiouville integrals in (1.3.164)–(1.3.165), (1.3.166)–(1.3.167), and (1.3.168)– (1.3.169), are defined by α (Da+;g y)(x) = α (Db−;g y)(x) = 1 D g ′ (x) − 1 D g ′ (x) n n−α (Ia+;g y)(x) (x < b), (1.3.176) n n−α (Ib−;g y)(x) (x < b), (1.3.177)
  62. 62. November 23, 2011 38 11:8 World Scientific Book - 9in x 6in book Fractional Calculus: Models and Numerical Methods and α (D+;g y)(x) = α (D−;g y)(x) = 1 D g ′ (x) − n 1 D ′ (x) g n−α (I+;g y)(x) (x ∈ R), (1.3.178) n n−α (I−;g y)(x) (x ∈ R), (1.3.179) respectively. When g(x) = x, (1.3.176) and (1.3.177) coincide with the RiemannLiouville fractional derivatives (1.3.3) and (1.3.4) α α α α (Da+;x y)(x) = (RL Da+ y)(x) and (Db−;x y)(x) = (RL Db− y)(x), (1.3.180) and (1.3.178) and (1.3.179) coincide with the Liouville fractional derivatives (1.3.63) and (1.3.64) α α α α (D+;x y)(x) = (L D+ y)(x) and (D−;x y)(x) = (L D− y)(x). (1.3.181) For Re(α) ≥ 0 (α = 0) and Re(β) > n − 1, the above derivatives have the properties α (Da+;g y+ )(x) = Γ(β + 1) [g(x) − g(a)]β−α Γ(β − α + 1) (1.3.182) where y+ (x) = [g(x) − g(a)]β , and α (Db−;g y− )(x) = Γ(β + 1) [g(b) − g(x)]β−α Γ(β − α + 1) (1.3.183) where y− (x) = [g(b) − g(x)]β . For Re(α) ≥ 0 (α = 0) and λ > 0, we have α (D+;g eλg(t) )(x) = λα eλg(x) (1.3.184) α (D−;g e−λg(t) )(x) = λα e−λg(x) . (1.3.185) and
  63. 63. November 23, 2011 11:8 World Scientific Book - 9in x 6in 39 Preliminaries 1.3.7 book Gr¨nwald-Letnikov fractional derivatives u In this section we give the definition of the Gr¨nwald-Letnikov fractional u derivatives and some of their properties. Such a fractional differentiation is based on a generalization of the classical differentiation of a function y(x) of order n ∈ N via differential quotients, (∆n y)(x) h . h→0 hn y (n) (x) = lim (1.3.186) Here (∆n y)(x) is a finite difference of order n ∈ N0 of a function y(x) with h a step h ∈ R and centered at the point x ∈ R defined by n (−1)k k=0 n y(x − kh) (x, h ∈ R; n ∈ N). k (1.3.187) (∆0 f )(x) = f (x). h (∆n y)(x) = h (1.3.188) and Property (1.3.186) is used to define a fractional derivative by directly replacing n ∈ N in (1.3.186) by α > 0. For this, hn is replaced by hα , while the finite difference (∆n y)(x) is replaced by the difference (∆α y)(x) of a h h fractional order α ∈ R defined by the following series ∞ (∆α y)(x) = h (−1)k k=0 α y(x − kh) (x, h ∈ R; α > 0), k (1.3.189) where α are the binomial coefficients. When h > 0, the difference k (1.3.189) is called left-sided difference, while for h < 0 it is called a rightsided difference. The series in (1.3.189) converges absolutely and uniformly for each α > 0 and for every bounded function y(x). The fractional difference (∆α y)(x) has the following semigroup property h (∆α ∆β y)(x) = (∆α+β y)(x) h h h (1.3.190) for α > 0 and β > 0. On the other hand, if α > 0 and y(x) ∈ L1 (R), then the Fourier transform (1.1.1) of ∆α is given by h (F ∆α y)(κ) = 1 − eiκh h α (F y)(κ). (1.3.191)
  64. 64. November 23, 2011 11:8 World Scientific Book - 9in x 6in 40 book Fractional Calculus: Models and Numerical Methods Following (1.3.186), the left- and right-sided Gr¨nwald-Letnikov derivau α) α) tives y+ (x) and y− (x) are defined by (∆α y)(x) h h→+0 hα (α > 0) (1.3.192) (∆α y)(x) −h h→+0 hα α) (α > 0), (1.3.193) y+ (x) = lim and α) y− (x) = lim respectively. These constructions coincide with the Marchaud fractional derivatives for y(x) ∈ Lp (R) (1 ≤ p < ∞). Then, by analogy with (1.3.192) and (1.3.193), the left- and right-sided Gr¨ nwald-Letnikov fractional derivatives of order α > 0 on a finite interval u [a, b] are defined by α) (∆α y)(x) h,a+ h→+0 hα (1.3.194) α) (∆α y)(x) h,b− , h→+0 hα (1.3.195) ya+ (x) = lim and yb− (x) = lim respectively, where [ x−a ] h (∆α y)(x) h,a+ (−1)k α k y(x − kh) (x ∈ R; α, h > 0) (1.3.196) (−1)k n k y(x − kh) (x, ∈ R; α, h > 0). (1.3.197) = k=0 and [ b−x ] h (∆α y)(x) h,b− = k=0 Such Gr¨ nwald-Letnikov fractional derivatives coincide with the Marchaud u fractional derivatives, for sufficiently well-behaved functions, and can be represented in the form α) ya+ (x) = y(x) α + α Γ(1 − α)(x − a) Γ(1 − α) x a y(x) − y(t) dt (0 < α < 1) (x − t)1+α (1.3.198) and α) yb− (x) = y(x) α + Γ(1 − α)(b − x)α Γ(1 − α) b x y(x) − y(t) dt (0 < α < 1). (t − x)1+α (1.3.199)
  65. 65. November 23, 2011 11:8 World Scientific Book - 9in x 6in Chapter 2 A Survey of Numerical Methods for the Solution of Ordinary and Partial Fractional Differential Equations When working with problems stemming from “real-world” applications, it is only rarely possible to evaluate the solution of a given fractional differential equation in closed form, and even if such an analytic solution is available, it is typically too complicated to be used in practice. Therefore it is indispensable to have a number of numerical algorithms at hand so that one is able to compute numerical solutions with a sufficient accuracy in reasonable time. Thus, this and the following chapter will be devoted to a study of such algorithms. To be precise, in this chapter we will give a survey of the standard numerical methods that should give the reader an impression of what he or she may expect from today’s state-of-the-art algorithms. In Chapter 3 we will then look at the most important algorithms more closely, giving a detailed account of their respective strengths and weaknesses. Since almost all numerical methods for fractional differential equations are in some sense based on the approximation of fractional differential or integral operators by appropriate formulas, we shall begin our presentation in Section 2.1 with a look at this problem. The subsequent sections will be devoted to numerical methods for fractional ordinary differential equations with an emphasis on initial value problems; specifically Section 2.2 will deal with what we call direct methods, i.e. methods where the numerical discretization scheme is applied directly to the differential operator appearing in the equation under consideration, whereas in Section 2.3 we look at indirect methods where we apply some analytic manipulation to the differential equation before the numerical work begins. Then, Section 2.4 gives an overview of a particularly important class of methods that are based on 41 book

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