Fractional Differential Equations: An Approach via Fractional Derivatives: 206 (Applied Mathematical Sciences, 206)

Description

This graduate textbook provides a self-contained introduction to modern mathematical theory on fractional differential equations. It addresses both ordinary and partial differential equations with a focus on detailed solution theory, especially regularity theory under realistic assumptions on the problem data. The text includes an extensive bibliography, application-driven modeling, extensive exercises, and graphic illustrations throughout to complement its comprehensive presentation of the field. It is recommended for graduate students and researchers in applied and computational mathematics, particularly applied analysis, numerical analysis and inverse problems. Preface Contents Acronyms Part I Preliminaries 1 Continuous Time Random Walk 1.1 Random Walk on a Lattice 1.2 Continuous Time Random Walk 1.3 Simulating Continuous Time Random Walk 2 Fractional Calculus 2.1 Gamma Function 2.2 Riemann-Liouville Fractional Integral 2.3 Fractional Derivatives 2.3.1 Riemann-Liouville fractional derivative 2.3.2 Djrbashian-Caputo fractional derivative 2.3.3 Grünwald-Letnikov fractional derivative 3 Mittag-Leffler and Wright Functions 3.1 Mittag-Leffler Function 3.1.1 Basic analytic properties 3.1.2 Mittag-Leffler function Eα,1(-x) 3.2 Wright Function 3.2.1 Basic analytic properties 3.2.2 Wright function Wρ,µ(-x) 3.3 Numerical Algorithms 3.3.1 Mittag-Leffler function Eα,β(z) 3.3.2 Wright function Wρ,µ(x) Part II Fractional Ordinary Differential Equations 4 Cauchy Problem for Fractional ODEs 4.1 Gronwall's Inequalities 4.2 ODEs with a Riemann-Liouville Fractional Derivative 4.3 ODEs with a Djrbashian-Caputo Fractional Derivative 5 Boundary Value Problem for Fractional ODEs 5.1 Green's Function 5.1.1 Riemann-Liouville case 5.1.2 Djrbashian-Caputo case 5.2 Variational Formulation 5.2.1 One-sided fractional derivatives 5.2.2 Two-sided mixed fractional derivatives 5.3 Fractional Sturm-Liouville Problem 5.3.1 Riemann-Liouville case 5.3.2 Djrbashian-Caputo case Part III Time-Fractional Diffusion 6 Subdiffusion: Hilbert Space Theory 6.1 Existence and Uniqueness in an Abstract Hilbert Space 6.2 Linear Problems with Time-Independent Coefficients 6.2.1 Solution representation 6.2.2 Existence, uniqueness and regularity 6.3 Linear Problems with Time-Dependent Coefficients 6.4 Nonlinear Subdiffusion 6.4.1 Lipschitz nonlinearity 6.4.2 Allen-Cahn equation 6.4.3 Compressible Navier-Stokes problem 6.5 Maximum Principles 6.6 Inverse Problems 6.6.1 Backward subdiffusion 6.6.2 Inverse source problems 6.6.3 Determining fractional order 6.6.4 Inverse potential problem 6.7 Numerical Methods 6.7.1 Convolution quadrature 6.7.2 Piecewise polynomial interpolation 7 Subdiffusion: Hölder Space Theory 7.1 Fundamental Solutions 7.1.1 Fundamental solutions 7.1.2 Fractional θ-functions 7.2 Hölder Regularity in One Dimension 7.2.1 Subdiffusion in mathbbR 7.2.2 Subdiffusion in mathbbR+ 7.2.3 Subdiffusion on bounded intervals 7.3 Hölder Regularity in Multi-Dimension 7.3.1 Subdiffusion in mathbbRd 7.3.2 Subdiffusion in mathbbRd+ 7.3.3 Subdiffusion on bounded domains A Mathematical Preliminaries A.1 AC Spaces and Hölder Spaces A.1.1 AC spaces A.1.2 Hölder spaces A.2 Sobolev Spaces A.2.1 Lebesgue spaces A.2.2 Sobolev spaces A.2.3 Fractional Sobolev spaces A.2.4 s(Ω) spaces A.2.5 Bochner spaces A.3 Integral Transforms A.3.1 Laplace transform A.3.2 Fourier transform A.4 Fixed Point Theorems References References Index