Kooperativer Bibliotheksverbund

UID:

Format: 1 online resource (411 pages)

ISBN: 3-031-18150-6

Series Statement: PoliTO Springer series

Note: Intro -- Preface -- Contents -- About the Authors -- Symbols -- Sets -- Integral Transforms -- Special Functions of Fractional Calculus -- Fractional Operators -- 1 Integral Transforms and Special Functions -- 1.1 Integral Transforms -- 1.1.1 Fourier Transform -- 1.1.2 Laplace Transform -- 1.1.3 The Cauchy Integral Formula -- 1.2 Gamma and Related Functions -- 1.2.1 The Mellin Transform -- 1.3 Special Functions of Fractional Calculus -- 1.3.1 The Functions of the Mittag-Leffler Type -- 1.3.2 The Wright Function -- 1.3.3 The H-Function of Fox -- References -- 2 Concepts in Diffusion and Stochastic Processes -- 2.1 Molecular Diffusion -- 2.1.1 The Mean Square Displacement -- 2.2 Pioneering Studies on Diffusion Problems -- 2.3 Brownian Motion -- 2.3.1 The Works of Einstein, Smoluchowski, and Langevin -- 2.4 Generalized Langevin Equation -- 2.4.1 Derivation of a Diffusionlike Equation -- 2.5 Anomalous Diffusion: Basic Concepts -- 2.5.1 Superdiffusion -- 2.5.2 Lévy Flights -- 2.5.3 Subdiffusion -- References -- 3 Random Walks -- 3.1 Elementary Approach -- 3.1.1 Stochastic Variables -- 3.2 Random Walks: Nonlinear Fokker-Planck Equations -- 3.2.1 Nonlinear Random Walk -- 3.2.2 Generalized Random Walks -- 3.3 Continuous-Time Random Walk -- 3.3.1 Different Diffusive Regimes -- 3.3.2 Linear Reaction Dynamics -- 3.3.3 Coupled Jump-Length and Waiting-Time Distributions -- 3.3.4 Intermittent Continuous-Time Random Walk -- References -- 4 Elements of Fractional Calculus -- 4.1 Introduction -- 4.2 Early Definitions -- 4.3 The Riemann-Liouville Operators -- 4.4 The Grünwald-Letnikov Operators -- 4.5 The Caputo Operator -- 4.6 Some Space-Fractional Derivatives -- References -- 5 Fractional Anomalous Diffusion -- 5.1 A Space-Time Fractional Diffusion Equation -- 5.2 Generalized Space-Time Fractional Diffusion Equation -- 5.2.1 Composite Time Fractional Operator. , 5.2.2 Fractional Diffusion Equation: Singular Term -- 5.3 Tempered Fractional Diffusion Equation -- 5.4 Fractional Diffusion Equation in Heterogeneous Media -- 5.4.1 Free Case -- 5.4.2 External Force -- 5.5 Heterogeneous Media and Fractional Spatial Operator -- 5.6 Time Derivative Operators: A Comparison -- 5.6.1 Diffusion and Time Derivative Operators -- 5.6.2 Predicted Distributions: A Balance -- References -- 6 Adsorption Phenomena and Anomalous Behavior -- 6.1 Kinetic Equation: Normal and Fractional -- 6.1.1 Electrical Impedance in Liquid Crystals -- 6.2 Anomalous Diffusion in Complex Fluids -- 6.2.1 Diffusion and Surface Dynamics -- 6.2.2 Time-Dependent Solutions -- 6.3 Memory Kernels -- 6.3.1 Chemisorption and Physisorption Processes -- 6.3.2 Confined Systems: Periodically Varying Medium -- References -- 7 Reaction-Diffusion Problems -- 7.1 Diffusion and Kinetics -- 7.1.1 Reversible Linear Reaction -- 7.1.2 Irreversible Linear Reaction -- 7.2 Fractional Diffusion of Two Species -- 7.3 Subdiffusion and Linear Reaction -- 7.4 Hyperbolic Diffusion Equation with Reaction Terms -- 7.4.1 Diffusion-Reaction Processes -- References -- 8 Relaxation Under Geometric Constraints I: Classical Processes -- 8.1 Introduction -- 8.2 The Comb Model -- 8.3 Quenched and Annealed Disorder Mechanisms in Comb-Models -- 8.4 Generalized Fractal Structure of Backbones -- 8.5 Comb-Model and Reaction Diffusion -- 8.6 Diffusion and Reaction -- References -- 9 Relaxation Under Geometric Constraints II: Quantum Processes -- 9.1 Introduction -- 9.2 Constrained Quantum Motion in δ-Potential -- 9.3 Time Evolution and Asymptotic Behavior -- 9.4 Time-Dependent Schrödinger Equation in 3D -- 9.4.1 3D Constrained Quantum Model -- 9.4.2 Green's Function and Schrödinger Equation -- 9.4.3 Reduced Green's Functions and Fractional Derivatives. , 9.5 Time-Dependent Schrödinger Equation in Non-integer Dimensions -- 9.5.1 Marginal Probability Density Functions -- 9.5.2 Constrained Diffusion in Non-integer Dimensions -- References -- Index.

Additional Edition: Print version: Evangelista, Luiz Roberto An Introduction to Anomalous Diffusion and Relaxation Cham : Springer International Publishing AG,c2023 ISBN 9783031181498

Language: English