THEORY AND APPLICATIONS OF FRACTIONAL DIFFERENTIAL EQUATIONS

THEORY AND APPLICATIONSOF FRACTIONAL DIFFERENTIALEQUATIONSANATOLYA. KILBASBelarusian State UniversityMinsk, BelarusHARI M. SRIVASTAVAUniversity of VictoriaVictoria, British Columbia, CanadaJUAN J. TRUJILLOUniversidad de La LagunaLa Laguna (Tenerife)Canary Islands, SpainELSEVIERAmsterdam - Boston - Heidelberg - London - New York - OxfordParis - San Diego - San Francisco - Singapore - Sydney - TokyoContents1 PRELIMINARIES 1.1 Spaces of Integrable, Absolutely Continuous, and Continuous Func-tions ' 1.2 Generalized Functions 1.3 Fourier Transforms 1.4 Laplace and Mellin Transforms 1.5 The Gamma Function and Related Special Functions 1.6 Hypergeometric Functions 1.7 Bessel Functions 1.8 Classical Mittag-Leffler Functions 1.9 Generalized Mittag-Leffler Functions 1.10 Functions of the Mittag-Leffler Type 1.11 Wright Functions 1.12 The ^-Function 1.13 Fixed Point Theorems 2 FRACTIONAL INTEGRALS AND FRACTIONALDERIVATIVES 1161018242732404549545867692.1 Riemann-Liouville Fractional Integrals and Fractional Deri-vatives 692.2 Liouville Fractional Integrals and Fractional Derivatives on the Half-Axis 792.3 Liouville Fractional Integrals and Fractional Derivatives on the RealAxis 872.4 Caputo Fractional Derivatives 902.5 Fractional Integrals and Fractional Derivatives of a Function withRespect to Another Function 992.6 Erdelyi-Kober Type Fractional Integrals and Fractional Deriva-tives 1052.7 Hadamard Type Fractional Integrals and Fractional Derivatives . . 1102.8 Griinwald-Letnikov Fractional Derivatives 1212.9 Partial and Mixed Fractional Integrals and FractionalDerivatives 1232.10 Riesz Fractional Integro-Differentiation 1272.11 Comments and Observations 132xii Theory and Applications of Fractional Differential Equations3 ORDINARY FRACTIONAL DIFFERENTIAL EQUATIONS.EXISTENCE AND UNIQUENESS THEOREMS 1353.1 Introduction and a Brief Overview of Results 1353.2 Equations with the Riemann-Liouville Fractional Derivative in theSpace of Summable Functions 1443.2.1 Equivalence of the Cauchy Type Problem and the VolterraIntegral Equation 1453.2.2 Existence and Uniqueness of the Solution to the Cauchy TypeProblem 1483.2.3 The Weighted Cauchy Type Problem 1513.2.4 Generalized Cauchy Type Problems 1533.2.5 Cauchy Type Problems for Linear Equations 1573.2.6 Miscellaneous Examples 1603.3 Equations with the Riemann-Liouville Fractional Derivative in theSpace of Continuous Functions. Global Solution 1623.3.1 Equivalence of the Cauchy Type Problem and the VolterraIntegral Equation 1633.3.2 Existence and Uniqueness of the Global Solution to theCauchy Type Problem 1643.3.3 The Weighted Cauchy Type Problem 1673.3.4 Generalized Cauchy Type Problems 1683.3.5 Cauchy Type Problems for Linear Equations 1703.3.6 More Exact Spaces 1713.3.7 Further Examples 1773.4 Equations with the Riemann-Liouville Fractional Derivative in theSpace of Continuous Functions. Semi-Global and Local Solutions . 1823.4.1 The Cauchy Type Problem with Initial Conditions at theEndpoint of the Interval. Semi-Global Solution 1833.4.2 The Cauchy Problem with Initial Conditions at the InnerPoint of the Interval. Preliminaries 1863.4.3 Equivalence of the Cauchy Problem and the Volterra IntegralEquation . 1893.4.4 The Cauchy Problem with Initial Conditions at the InnerPoint of the Interval. The Uniqueness of Semi-Global andLocal Solutions 1913.4.5 A Set of Examples 1963.5 Equations with the Caputo Derivative in the Space of ContinuouslyDifferentiable Functions 1983.5.1 The Cauchy Problem with Initial Conditions at the Endpointof the Interval. Global Solution 1993.5.2 The Cauchy Problems with Initial Conditions at the End andInner Points of the Interval. Semi-Global and LocalSolutions 2053.5.3 Illustrative Examples 209Contents 3.6 Equations with the Hadamard Fractional Derivative in the Space ofContinuous Functions xiii2124 METHODS FOR EXPLICITLY SOLVING FRACTIONALDIFFERENTIAL EQUATIONS 2214.1 Method of Reduction to Volterra Integral Equations 2214.1.1 The Cauchy Type Problems for Differential Equations withthe Riemann-Liouville Fractional Derivatives 2224.1.2 The Cauchy Problems for Ordinary Differential Equa-tions 2284.1.3 The Cauchy Problems for Differential Equations with theCaputo Fractional Derivatives 2304.1.4 The Cauchy Type Problems for Differential Equations withHadamard Fractional Derivatives 2344.2 Compositional Method 2384.2.1 Preliminaries 2384.2.2 Compositional Relations 2394.2.3 Homogeneous Differential Equations of Fractional Order withRiemann-Liouville Fractional Derivatives 2424.2.4 Nonhomogeneous Differential Equations of Fractional Or-der with Riemann-Liouville and Liouville Fractional Deriva-tives with a Quasi-Polynomial Free Term 2454.2.5 Differential Equations of Order 1/2 2484.2.6 Cauchy Type Problem for Nonhomogeneous DifferentialEquations with Riemann-Liouville Fractional Derivatives andwith a Quasi-Polynomial Free Term 2514.2.7 Solutions to Homogeneous Fractional Differential Equationswith Liouville Fractional Derivatives in Terms of Bessel-Type Functions \\ 2574.3 Operational Method 2604.3.1 Liouville Fractional Integration and Differentiation Opera-tors in Special Function Spaces on the Half-Axis 2614.3.2 Operational Calculus for the Liouville Fractional CalculusOperators 2634.3.3 Solutions to Cauchy Type Problems for Fractional Differen-tial Equations with Liouville Fractional Derivatives 2664.3.4 Other Results 2704.4 Numerical Treatment 2725 INTEGRAL TRANSFORM METHOD FOR EXPLICITSOLUTIONS TO FRACTIONAL DIFFERENTIALEQUATIONS 2795.1 Introduction and a Brief Survey of Results 5.2 Laplace Transform Method for Solving Ordinary Differential Equa-tions with Liouville Fractional Derivatives 279283xiv Theory and Applications of Fractional Differential Equations5.2.1 Homogeneous Equations with Constant Coefficients 2835.2.2 Nonhomogeneous Equations with Constant Coefficients . . . 2955.2.3 Equations with Nonconstant Coefficients 3035.2.4 Cauchy Type for Fractional Differential Equations 3095.3 Laplace Transform Method for Solving Ordinary Differential Equa-tions with Caputo Fractional Derivatives 3125.3.1 Homogeneous Equations with Constant Coefficients 3125.3.2 Nonhomogeneous Equations with Constant Coefficients . . . 3225.3.3 Cauchy Problems for Fractional Differential Equations . . . 3265.4 Mellin Transform Method for Solving Nonhomogeneous FractionalDifferential Equations with Liouville Derivatives 3295.4.1 General Approach to the Problems 3295.4.2 Equations with Left-Sided Fractional Derivatives 3315.4.3 Equations with Right-Sided Fractional Derivatives 3365.5 Fourier Transform Method for Solving Nonhomogeneous Differen-tial Equations with Riesz Fractional Derivatives 3415.5.1 Multi-Dimensional Equations 3415.5.2 One-Dimensional Equations 3446 PARTIAL FRACTIONAL DIFFERENTIAL EQUATIONS 3476.1 Overview of Results 3476.1.1 Partial Differential Equations of Fractional Order 3476.1.2 Fractional Partial Differential Diffusion Equations 3516.1.3 Abstract Differential Equations of Fractional Order 3596.2 Solution of Cauchy Type Problems for Fractional Diffusion-WaveEquations 3626.2.1 Cauchy Type Problems for Two-Dimensional Equations . . 3626.2.2 Cauchy Type Problems for Multi-Dimensional Equations . . 3666.3 Solution of Cauchy Problems for Fractional Diffusion-Wave Equa-tions 3736.3.1 Cauchy Problems for Two-Dimensional Equations 3746.3.2 Cauchy Problems for Multi-Dimensional Equations 3776.4 Solution of Cauchy Problems for Fractional Evolution Equations . 3806.4.1 Solution of the Simplest Problem 3806.4.2 Solution to the General Problem 3846.4.3 Solutions of Cauchy Problems via the H-Functions 3887 SEQUENTIAL LINEAR DIFFERENTIAL EQUATIONS OFFRACTIONAL ORDER 3937.1 Sequential Linear Differential Equations of Fractional Order .... 3947.2 Solution of Linear Differential Equations with Constant Coef-ficients 4007.2.1 General Solution in the Homogeneous Case 4007.2.2 General Solution in the Non-Homogeneous Case. FractionalGreen Function 403Contents 7.3 Non-Sequential Linear Differential Equations with Constant Co-efficients 7.4 Systems of Equations Associated with Riemann-Liouville and Ca-puto Derivatives 7.4.1 General Theory 7.4.2 General Solution for the Case of Constant Coefficients. Frac-tional Green Vectorial Function 7.5 Solution of Fractional Differential Equations with Variable Coef-ficients. Generalized Method of Frobenius 7.5.1 Introduction 7.5.2 Solutions Around an Ordinary Point of a Fractional Differ-ential Equation of Order a 7.5.3 Solutions Around an Ordinary Point of a Fractional Differ-ential Equation of Order 2a 7.5.4 Solution Around an a-Singular Point of a Fractional Differ-ential Equation of Order a 7.5.5 Solution Around an a-Singular Point of a Fractional Differ-ential Equation of Order 2a 7.6 Some Applications of Linear Ordinary Fractional DifferentialEquations 7.6.1 Dynamics of a Sphere Immersed in an Incompressible Vis-cous Fluid. Basset's Problem 7.6.2 Oscillatory Processes with Fractional Damping 7.6.3 Study of the Tension-Deformation Relationship of Viscoelas-tic Materials 8 FURTHER APPLICATIONS OF FRACTIONAL MODELS xv4074094094124154154184214244274334344364394498.1 Preliminary Review 4498.1.1 Historical Overview 4508.1.2 Complex Systems 4528.1.3 Fractional Integral and Fractional Derivative Operators . . 4568.2 Fractional Model for the Super-Diffusion Processes 4588.3 Dirac Equations for the Ordinary Diffusion Equation 4628.4 Applications Describing Carrier Transport in Amorphous Semicon-ductors with Multiple Trapping 463Bibliography Subject Index 469521