This book is concerned with the Fractional Calculus and its application to Anomalous Diffusion. Diffusion means to scatter or pour out. In physics it usually refers to phenomenon at the molecular level. It describes the transport of matter from places of higher concentrations to places of lower concentrations. The standard diffusion equation is a partial differential equation that can be derived from the Langevin equation, a stochastic equation. If the diffusion equation is solved for the case of external forces the solution shows an anomalous form. The anomalous form can be explored using the fractional calculus. This task is commenced in Chapter 4. At first glance at the contents Chapters 4 and 5 have the same title. This is not a misprint; the sub-title is missing from each chapter heading in the Contents page. The book contains ten chapters as follows.
Chapter 1 – Mathematical Preliminaries. This chapter is concerned with reminding the reader of various integral transforms and special functions that are especially relevant to fractional calculus. In particular, the Mittag-Leffler functions, Wright functions and the H-function of Fox.
Chapter 2 – A Survey of Fractional Calculus. The first section introduces the idea of the fractional calculus by considering the way in which it developed historically. It started with Leibnitz followed by Euler, Laplace through to Harold Thayer Davis who introduced the notation used in the book. The remainder of the chapter considers the Grunwald-Letnikov operator, the Caputo operator, the Riesz-Weyl operator and then Integral Transforms of the operators and a generalised Fourier Transform.
Chapter 3 – From Normal to Anomalous Diffusion. This chapter starts by considering historical perspectives on diffusion problems. It then introduces the model of describing the diffusion process with continuous time random walks. The Diffusion equation is introduced in the final section, where the anomalous form of the solution of the equation when there are external forces provides the rationale for the following chapter.
Chapter 4 – Fractional Diffusion Equations: Elementary Applications. The chapter, being the first involving the presence of a partial differential equation with a fractional derivative, introduces the fractional aspect in the simplest possible manner. The involvement of fractional calculus gradually becomes more complicated and wide-spread. The chapter contains these sections: Fractional Time Derivative: Simple Situations, Fractional Space Derivative: Simple Situations, Sorption and Desorption Processes, Reaction Terms, Reaction and CTRW Formalism. The first two sections are concerned with anomalous diffusion with no reaction terms. Allowing special fractional derivatives leads to the use of Levy distributions, which predict power-tail behaviour rather than exponential. Allowing the presence of reaction terms that may be at the boundary of the system, or in the bulk leads to a consideration of chemical reactions during the diffusion.
Chapter 5 – Fractional Diffusion Equations: Surface Effects. This chapter is concerned with the influence of surfaces or membranes on the dispersive process. Initially in one dimension, later in two. The way in which surface effects couple with bulk dynamics is considered later in the chapter. This chapter covers the following: 1 and 2D Cases: Different Diffusive Regimes, 3D Case: External Forces and Reaction Term, Reaction on a Solid Surface: Anomalous Mass Transfer, Heterogeneous Media and Transport through a Membrane.
Chapter 6 – Fractional Nonlinear Diffusion Equations. Here nonlinear terms are allowed in the diffusion equation, initially in one dimension, later in more general situations. The chapter contains the following sections: Nonlinear Diffusion Equations, Nonlinear Diffusion Equations: Intermittent Motion, Fractional Spatial Derivatives, d-Dimensional Fractional Diffusion Equations.
Chapter 7 – Anomalous Diffusion: Anisotropic Case. Here the treatment is further extended to include suspended or dispersed particles. There is a further complication in that, because absorption and desorption occurs at the interface between there is a need to impose conservation of particles. There is another complication such that the distribution governed by the equation is not separable in space and time variables as it usually is. The Comb model is a simplified description of highly disordered systems. Sections include: The Adsorption-Desorption Process in Anisotropic Media, Fractional Diffusion Equations in Anisotropic Media, The Comb Model.
Chapter 8 – Fractional Schrödinger Equations. The Schrödinger equation is investigated with nonlocal terms and connections to anomalous diffusion. Using fractional derivatives is an elegant method to include nonlocal and non-Markovian effects. The chapter contains the following sections: The Schrödinger Equation and Anomalous Behaviour, Time-Dependent Solutions, CTRW and the Fractional Schrödinger Equation, Memory and Nonlocal Effects and Nonlocal Effects on the Energy Spectra.
Chapter 9 – Anomalous Diffusion and Impedance Spectroscopy. Here the results of applying fractional diffusion equations to the electrochemical impedance technique are used to investigate properties of condensed matter samples. This is, at present, a pioneering technique. Basic aspects of impedance spectroscopy are considered with normal derivatives, the Poisson-Nernst-Planck (PNP) model is introduced then fractional derivatives are applied and the model reformulated into the PNPA model. The following sections are covered: Impedance Spectroscopy: Preliminaries, The PNP Time Fractional Model, Anomalous Diffusion and Memory Effects and Anomalous Interfacial Conditions.
Chapter 10 – The Poisson-Nernst-Planck Anomalous Models. The paths towards PNPA models are considered to connect anomalous diffusion to impedance Spectroscopy, initially analysing the conceptual links between PNPA models and electrical circuits containing constant-phase elements.
This is a very advanced book. It introduces the fractional calculus quite briefly, as a tool to analyse the complicated phenomena involved in anomalous diffusion. This is still a relatively recent field of research, particularly using fractional calculus. The aim of the book is to disseminate recent results in the field. I would recommend it to people with a specialised interest in anomalous diffusion.