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An introduction to fractional diffusion. (English) Zbl 1221.60047

Dewar, Robert L. (ed.) et al., Complex physical, biophysical and econophysical systems. Proceedings of the 22nd Canberra international physics summer school, The Australian National University, Canberra, Australia, December 8–19, 2008. Hackensack, NJ: World Scientific (ISBN 978-981-4277-31-0/hbk). World Scientific Lecture Notes in Complex Systems 9, 37-90 (2010).
In this well-written exposition, the authors present in a lucid way the essentials of the theory and simulation of fractional diffusion. A list of the section headings may exhibit the contents of this work.
Chapter 2.1, Mathematical models of diffusion, has the following subsections: Brownian motion and the Langevin equation; Random walks and the central limit theorem; Fick’s law and the diffusion equation; Master equations and the Fokker-Planck equation; the Chapman-Kolmogorov equation and Markov processes.
Chapter 2.2, Fractional diffusion, is divided into the subsections: Diffusion on fractals; Fractional Brownian motion; Continuous time random walks and power laws; Simulating random walks for fractional diffusion; Fractional Fokker-Planck equations; Fractional diffusion based models; Power laws and fractional diffusion.
An appendix is devoted to the required types of fractional integration and differentiation, to the effects of Fourier and Laplace transforms on these and to Mittag-Leffler and Fox \(H\) functions. The paper closes with a useful list of 58 references, ranging from the year 1828 (R. Brown on his microscopical observations – the empirical origin) to the year 2008 (M. M. Meerschaert and H. P. Scheffler on triangular arrays – highly theoretical), and, in the text, there appear many historical comments on the origins of ideas, methods and results, comments suited to seduce the reader to look into the sources and study them.
Reviewer’s remark: The reader wanting to apply formulas given in the paper is advised to check them. Formula (2.166), unfortunately, is wrong. In the denominator on the right hand side \(\Gamma(\alpha)\) must be replaced by \(\Gamma(1-\alpha)\), and the exponent \(1-\alpha\) must be replaced by \(\alpha\).
For the entire collection see [Zbl 1197.00054].

MSC:

60G15 Gaussian processes
45K05 Integro-partial differential equations
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
33E12 Mittag-Leffler functions and generalizations
60G50 Sums of independent random variables; random walks
26A33 Fractional derivatives and integrals
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