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.
Author(s): Luiz Roberto Evangelista, Ervin Kaminski Lenzi
Edition: 1
Publisher: Cambridge University Press
Year: 2018
Language: English
Pages: 360
Contents......Page 8
Preface......Page 10
1.1 Integral Transforms......Page 16
1.2 Special Functions of Fractional Calculus......Page 32
1.3 Integral Transforms of Special Functions......Page 58
2.1 The Origins of Fractional Calculus......Page 61
2.2 The Grunwald–Letnikov Operator......Page 72
2.3 The Caputo Operator......Page 76
2.4 The Riesz–Weyl Operator......Page 77
2.5 Integral Transforms of Fractional Operators......Page 78
2.6 A Generalised Fourier Transform......Page 83
3.1 Historical Perspectives on Diffusion Problems......Page 86
3.2 Continuous-Time Random Walk......Page 105
3.3 Diffusion Equation......Page 110
4.1 Fractional Time Derivative: Simple Situations......Page 116
4.2 Fractional Spatial Derivative: Simple Situations......Page 126
4.3 Sorption and Desorption Processes......Page 129
4.4 Reaction Terms......Page 139
4.5 Reaction and CTRW Formalism......Page 149
5.1 1D and 2D Cases: Different Diffusive Regimes......Page 154
5.2 3D Case: External Force and Reaction Term......Page 160
5.3 Reaction on a Solid Surface: Anomalous Mass Transfer......Page 166
5.4 Heterogeneous Media and Transport through a Membrane......Page 173
6 Fractional Nonlinear Diffusion Equations......Page 184
6.1 Nonlinear Diffusion Equations......Page 185
6.2 Nonlinear Diffusion Equations: Intermittent Motion......Page 188
6.3 Fractional Spatial Derivatives......Page 197
6.4 d-Dimensional Fractional Diffusion Equations......Page 203
7.1 The Adsorption–Desorption Process in Anisotropic Media......Page 215
7.2 Fractional Diffusion Equations in Anisotropic Media......Page 224
7.3 The Comb Model......Page 235
8.1 The Schrodinger Equation and Anomalous Behaviour......Page 249
8.2 Time-Dependent Solutions......Page 257
8.3 CTRW and the Fractional Schrodinger Equation......Page 264
8.4 Memory and Nonlocal Effects......Page 269
8.5 Nonlocal Effects on the Energy Spectra......Page 279
9.1 Impedance Spectroscopy: Preliminaries......Page 286
9.2 The PNP Time Fractional Model......Page 295
9.3 Anomalous Diffusion and Memory Effects......Page 301
9.4 Anomalous Interfacial Conditions......Page 307
10.1 PNPA Models and Equivalent Circuits......Page 321
10.2 PNPA Models: A Framework......Page 328
References......Page 338
Index......Page 356
