Theory and applications of fractional differential equations

This monograph provides the most recent and up-to-date developments on fractional differential and fractional integro-differential equations involving many different potentially useful operators of fractional calculus.The subject of fractional calculus and its applications (that is, calculus of integrals and derivatives of any arbitrary real or complex order) has gained considerable popularity and importance during the past three decades or so, due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering.Some of the areas of present-day applications of fractional models include Fluid Flow, Solute Transport or Dynamical Processes in Self-Similar and Porous Structures, Diffusive Transport akin to Diffusion, Material Viscoelastic Theory, Electromagnetic Theory, Dynamics of Earthquakes, Control Theory of Dynamical Systems, Optics and Signal Processing, Bio-Sciences, Economics, Geology, Astrophysics, Probability and Statistics, Chemical Physics, and so on.In the above-mentioned areas, there are phenomena with estrange kinetics which have a microscopic complex behaviour, and their macroscopic dynamics can not be characterized by classical derivative models.The fractional modelling is an emergent tool which use fractional differential equations including derivatives of fractional order, that is, we can speak about a derivative of order 1/3, or square root of 2, and so on. Some of such fractional models can have solutions which are non-differentiable but continuous functions, such as Weierstrass type functions. Such kinds of properties are, obviously, impossible for the ordinary models.What are the useful properties of these fractional operators which help in the modelling of so many anomalous processes? From the point of view of the authors and from known experimental results, most of the processes associated with complex systems have non-local dynamics involving long-memory in time, and the fractional integral and fractional derivative operators do have some of those characteristics.This book is written primarily for the graduate students and researchers in many different disciplines in the mathematical, physical, engineering and so many others sciences, who are interested not only in learning about the various mathematical tools and techniques used in the theory and widespread applications of fractional differential equations, but also in further investigations which emerge naturally from (or which are motivated substantially by) the physical situations modelled mathematically in the book.This monograph consists of a total of eight chapters and a very extensive bibliography. The main objective of it is to complement the contents of the other books dedicated to the study and the applications of fractional differential equations. The aim of the book is to present, in a systematic manner, results including the existence and uniqueness of solutions for the Cauchy type problems involving nonlinear ordinary fractional differential equations, explicit solutions of linear differential equations and of the corresponding initial-value problems through different methods, closed-form solutions of ordinary and partial differential equations, and a theory of the so-called sequential linear fractional differential equations including a generalization of the classical Frobenius method, and also to include an interesting set of applications of the developed theory.Key features:- It is mainly application oriented.- It contains a complete theory of Fractional Differential Equations.- It can be used as a postgraduate-level textbook in many different disciplines within science and engineering.- It contains an up-to-date bibliography.- It provides problems and directions for further investigations.- Fractional Modelling is an emergent tool with demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering.- It contains many examples.- and so on!

1 Preliminaries	p. 1
1.1 Spaces of Integrable, Absolutely Continuous, and Continuous Functions	p. 1
1.2 Generalized Functions	p. 6
1.3 Fourier Transforms	p. 10
1.4 Laplace and Mellin Transforms	p. 18
1.5 The Gamma Function and Related Special Functions	p. 24
1.6 Hypergeometric Functions	p. 27
1.7 Bessel Functions	p. 32
1.8 Classical Mittag-Leffler Functions	p. 40
1.9 Generalized Mittag-Leffler Functions	p. 45
1.10 Functions of the Mittag-Leffler Type	p. 49
1.11 Wright Functions	p. 54
1.12 The H-Function	p. 58
1.13 Fixed Point Theorems	p. 67
2 Fractional Integrals and Fractional Derivatives	p. 69
2.1 Riemann-Liouville Fractional Integrals and Fractional Derivatives	p. 69
2.2 Liouville Fractional Integrals and Fractional Derivatives on the Half-Axis	p. 79
2.3 Liouville Fractional Integrals and Fractional Derivatives on the Real Axis	p. 87
2.4 Caputo Fractional Derivatives	p. 90
2.5 Fractional Integrals and Fractional Derivatives of a Function with Respect to Another Function	p. 99
2.6 Erdelyi-Kober Type Fractional Integrals and Fractional Derivatives	p. 105
2.7 Hadamard Type Fractional Integrals and Fractional Derivatives	p. 110
2.8 Grunwald-Letnikov Fractional Derivatives	p. 121
2.9 Partial and Mixed Fractional Integrals and Fractional Derivatives	p. 123
2.10 Riesz Fractional Integro-Differentiation	p. 127
2.11 Comments and Observations	p. 132
3 Ordinary Fractional Differential Equations. Existence and Uniqueness Theorems	p. 135
3.1 Introduction and a Brief Overview of Results	p. 135
3.2 Equations with the Riemann-Liouville Fractional Derivative in the Space of Summable Functions	p. 144
3.2.1 Equivalence of the Cauchy Type Problem and the Volterra Integral Equation	p. 145
3.2.2 Existence and Uniqueness of the Solution to the Cauchy Type Problem	p. 148
3.2.3 The Weighted Cauchy Type Problem	p. 151
3.2.4 Generalized Cauchy Type Problems	p. 153
3.2.5 Cauchy Type Problems for Linear Equations	p. 157
3.2.6 Miscellaneous Examples	p. 160
3.3 Equations with the Riemann-Liouville Fractional Derivative in the Space of Continuous Functions. Global Solution	p. 162
3.3.1 Equivalence of the Cauchy Type Problem and the Volterra Integral Equation	p. 163
3.3.2 Existence and Uniqueness of the Global Solution to the Cauchy Type Problem	p. 164
3.3.3 The Weighted Cauchy Type Problem	p. 167
3.3.4 Generalized Cauchy Type Problems	p. 168
3.3.5 Cauchy Type Problems for Linear Equations	p. 170
3.3.6 More Exact Spaces	p. 171
3.3.7 Further Examples	p. 177
3.4 Equations with the Riemann-Liouville Fractional Derivative in the Space of Continuous Functions. Semi-Global and Local Solutions	p. 182
3.4.1 The Cauchy Type Problem with Initial Conditions at the Endpoint of the Interval. Semi-Global Solution	p. 183
3.4.2 The Cauchy Problem with Initial Conditions at the Inner Point of the Interval. Preliminaries	p. 186
3.4.3 Equivalence of the Cauchy Problem and the Volterra Integral Equation	p. 189
3.4.4 The Cauchy Problem with Initial Conditions at the Inner Point of the Interval. The Uniqueness of Semi-Global and Local Solutions	p. 191
3.4.5 A Set of Examples	p. 196
3.5 Equations with the Caputo Derivative in the Space of Continuously Differentiable Functions	p. 198
3.5.1 The Cauchy Problem with Initial Conditions at the Endpoint of the Interval. Global Solution	p. 199
3.5.2 The Cauchy Problems with Initial Conditions at the End and Inner Points of the Interval. Semi-Global and Local Solutions	p. 205
3.5.3 Illustrative Examples	p. 209
3.6 Equations with the Hadamard Fractional Derivative in the Space of Continuous Functions	p. 212
4 Methods for Explicitly Solving Fractional Differential Equations	p. 221
4.1 Method of Reduction to Volterra Integral Equations	p. 221
4.1.1 The Cauchy Type Problems for Differential Equations with the Riemann-Liouville Fractional Derivatives	p. 222
4.1.2 The Cauchy Problems for Ordinary Differential Equations	p. 228
4.1.3 The Cauchy Problems for Differential Equations with the Caputo Fractional Derivatives	p. 230
4.1.4 The Cauchy Type Problems for Differential Equations with Hadamard Fractional Derivatives	p. 234
4.2 Compositional Method	p. 238
4.2.1 Preliminaries	p. 238
4.2.2 Compositional Relations	p. 239
4.2.3 Homogeneous Differential Equations of Fractional Order with Riemann-Liouville Fractional Derivatives	p. 242
4.2.4 Nonhomogeneous Differential Equations of Fractional Order with Riemann-Liouville and Liouville Fractional Derivatives with a Quasi-Polynomial Free Term	p. 245
4.2.5 Differential Equations of Order 1/2	p. 248
4.2.6 Cauchy Type Problem for Nonhomogeneous Differential Equations with Riemann-Liouville Fractional Derivatives and with a Quasi-Polynomial Free Term	p. 251
4.2.7 Solutions to Homogeneous Fractional Differential Equations with Liouville Fractional Derivatives in Terms of Bessel-Type Functions	p. 257
4.3 Operational Method	p. 260
4.3.1 Liouville Fractional Integration and Differentiation Operators in Special Function Spaces on the Half-Axis	p. 261
4.3.2 Operational Calculus for the Liouville Fractional Calculus Operators	p. 263
4.3.3 Solutions to Cauchy Type Problems for Fractional Differential Equations with Liouville Fractional Derivatives	p. 266
4.3.4 Other Results	p. 270
4.4 Numerical Treatment	p. 272
5 Integral Transform Method for Explicit Solutions to Fractional Differential Equations	p. 279
5.1 Introduction and a Brief Survey of Results	p. 279
5.2 Laplace Transform Method for Solving Ordinary Differential Equations with Liouville Fractional Derivatives	p. 283
5.2.1 Homogeneous Equations with Constant Coefficients	p. 283
5.2.2 Nonhomogeneous Equations with Constant Coefficients	p. 295
5.2.3 Equations with Nonconstant Coefficients	p. 303
5.2.4 Cauchy Type for Fractional Differential Equations	p. 309
5.3 Laplace Transform Method for Solving Ordinary Differential Equations with Caputo Fractional Derivatives	p. 312
5.3.1 Homogeneous Equations with Constant Coefficients	p. 312
5.3.2 Nonhomogeneous Equations with Constant Coefficients	p. 322
5.3.3 Cauchy Problems for Fractional Differential Equations	p. 326
5.4 Mellin Transform Method for Solving Nonhomogeneous Fractional Differential Equations with Liouville Derivatives	p. 329
5.4.1 General Approach to the Problems	p. 329
5.4.2 Equations with Left-Sided Fractional Derivatives	p. 331
5.4.3 Equations with Right-Sided Fractional Derivatives	p. 336
5.5 Fourier Transform Method for Solving Nonhomogeneous Differential Equations with Riesz Fractional Derivatives	p. 341
5.5.1 Multi-Dimensional Equations	p. 341
5.5.2 One-Dimensional Equations	p. 344
6 Partial Fractional Differential Equations	p. 347
6.1 Overview of Results	p. 347
6.1.1 Partial Differential Equations of Fractional Order	p. 347
6.1.2 Fractional Partial Differential Diffusion Equations	p. 351
6.1.3 Abstract Differential Equations of Fractional Order	p. 359
6.2 Solution of Cauchy Type Problems for Fractional Diffusion-Wave Equations	p. 362
6.2.1 Cauchy Type Problems for Two-Dimensional Equations	p. 362
6.2.2 Cauchy Type Problems for Multi-Dimensional Equations	p. 366
6.3 Solution of Cauchy Problems for Fractional Diffusion-Wave Equations	p. 373
6.3.1 Cauchy Problems for Two-Dimensional Equations	p. 374
6.3.2 Cauchy Problems for Multi-Dimensional Equations	p. 377
6.4 Solution of Cauchy Problems for Fractional Evolution Equations	p. 380
6.4.1 Solution of the Simplest Problem	p. 380
6.4.2 Solution to the General Problem	p. 384
6.4.3 Solutions of Cauchy Problems via the H-Functions	p. 388
7 Sequential Linear Differential Equations of Fractional Order	p. 393
7.1 Sequential Linear Differential Equations of Fractional Order	p. 394
7.2 Solution of Linear Differential Equations with Constant Coefficients	p. 400
7.2.1 General Solution in the Homogeneous Case	p. 400
7.2.2 General Solution in the Non-Homogeneous Case. Fractional Green Function	p. 403
7.3 Non-Sequential Linear Differential Equations with Constant Coefficients	p. 407
7.4 Systems of Equations Associated with Riemann-Liouville and Caputo Derivatives	p. 409
7.4.1 General Theory	p. 409
7.4.2 General Solution for the Case of Constant Coefficients. Fractional Green Vectorial Function	p. 412
7.5 Solution of Fractional Differential Equations with Variable Coefficients. Generalized Method of Frobenius	p. 415
7.5.1 Introduction	p. 415
7.5.2 Solutions Around an Ordinary Point of a Fractional Differential Equation of Order [alpha]	p. 418
7.5.3 Solutions Around an Ordinary Point of a Fractional Differential Equation of Order 2[alpha]	p. 421
7.5.4 Solution Around an [alpha]-Singular Point of a Fractional Differential Equation of Order [alpha]	p. 424
7.5.5 Solution Around an [alpha]-Singular Point of a Fractional Differential Equation of Order 2[alpha]	p. 427
7.6 Some Applications of Linear Ordinary Fractional Differential Equations	p. 433
7.6.1 Dynamics of a Sphere Immersed in an Incompressible Viscous Fluid. Basset's Problem	p. 434
7.6.2 Oscillatory Processes with Fractional Damping	p. 436
7.6.3 Study of the Tension-Deformation Relationship of Viscoelastic Materials	p. 439
8 Further Applications of Fractional Models	p. 449
8.1 Preliminary Review	p. 449
8.1.1 Historical Overview	p. 450
8.1.2 Complex Systems	p. 452
8.1.3 Fractional Integral and Fractional Derivative Operators	p. 456
8.2 Fractional Model for the Super-Diffusion Processes	p. 458
8.3 Dirac Equations for the Ordinary Diffusion Equation	p. 462
8.4 Applications Describing Carrier Transport in Amorphous Semiconductors with Multiple Trapping	p. 463
Bibliography	p. 469
Subject Index	p. 521

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