Fractional Diffusion Equations and Anomalous Diffusion

Description

Anomalous diffusion has been detected in a wide variety of scenarios, from fractal media, systems with memory, transport processes in porous media, to fluctuations of financial markets, tumour growth, and complex fluids. Providing a contemporary treatment of this process, this book examines the recent literature on anomalous diffusion and covers a rich class of problems in which surface effects are important, offering detailed mathematical tools of usual and fractional calculus for a wide audience of scientists and graduate students in physics, mathematics, chemistry and engineering. Including the basic mathematical tools needed to understand the rules for operating with the fractional derivatives and fractional differential equations, this self-contained text presents the possibility of using fractional diffusion equations with anomalous diffusion phenomena to propose powerful mathematical models for a large variety of fundamental and practical problems in a fast-growing field of research. Contents Preface 1 Mathematical Preliminaries 1.1 Integral Transforms 1.2 Special Functions of Fractional Calculus 1.3 Integral Transforms of Special Functions 2 A Survey of Fractional Calculus 2.1 The Origins of Fractional Calculus 2.2 The Grunwald–Letnikov Operator 2.3 The Caputo Operator 2.4 The Riesz–Weyl Operator 2.5 Integral Transforms of Fractional Operators 2.6 A Generalised Fourier Transform 3 From Normal to Anomalous Diffusion 3.1 Historical Perspectives on Diffusion Problems 3.2 Continuous-Time Random Walk 3.3 Diffusion Equation 4 Fractional Diffusion Equations 4.1 Fractional Time Derivative: Simple Situations 4.2 Fractional Spatial Derivative: Simple Situations 4.3 Sorption and Desorption Processes 4.4 Reaction Terms 4.5 Reaction and CTRW Formalism 5 Fractional Diffusion Equations 5.1 1D and 2D Cases: Different Diffusive Regimes 5.2 3D Case: External Force and Reaction Term 5.3 Reaction on a Solid Surface: Anomalous Mass Transfer 5.4 Heterogeneous Media and Transport through a Membrane 6 Fractional Nonlinear Diffusion Equations 6.1 Nonlinear Diffusion Equations 6.2 Nonlinear Diffusion Equations: Intermittent Motion 6.3 Fractional Spatial Derivatives 6.4 d-Dimensional Fractional Diffusion Equations 7 Anomalous Diffusion 7.1 The Adsorption–Desorption Process in Anisotropic Media 7.2 Fractional Diffusion Equations in Anisotropic Media 7.3 The Comb Model 8 Fractional Schrodinger Equations 8.1 The Schrodinger Equation and Anomalous Behaviour 8.2 Time-Dependent Solutions 8.3 CTRW and the Fractional Schrodinger Equation 8.4 Memory and Nonlocal Effects 8.5 Nonlocal Effects on the Energy Spectra 9 Anomalous Diffusion and Impedance Spectroscopy 9.1 Impedance Spectroscopy: Preliminaries 9.2 The PNP Time Fractional Model 9.3 Anomalous Diffusion and Memory Effects 9.4 Anomalous Interfacial Conditions 10 The Poisson–Nernst–Planck Anomalous Models 10.1 PNPA Models and Equivalent Circuits 10.2 PNPA Models: A Framework References Index