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Special functions of fractional calculus. Applications to diffusion and random search processes. (English) Zbl 1514.33001

Singapore: World Scientific (ISBN 978-981-12-5294-5/hbk; 978-981-12-5296-9/ebook). xvii, 273 p. (2023).
Mathematics is the art of giving subjects misleading names, e.g., fractional calculus (which deals with integrals and differentials of arbitrary order) etc. There is no particular class of special functions related to fractional calculus. All special functions are related to fractional calculus. This book has given a brief introduction to special functions, integral transforms & fractional calculus and then focuses on applications to diffusion and random search processes.
The first chapter on mathematical background includes integral transforms (Laplace, Fourier and Mellin), asymptotic expansions, and special functions. Definitions of gamma, beta functions are given with their simple properties. Mittag-Leffler functions of one and two variables are defined with some particular cases and properties. Riemann-Liouville (R-L) fractional integrals and derivatives are discussed.
Fox H-function and related functions are given in chapter two. Elements of random walk theory are tackled in chapter three. This includes Markov processes, continuous time random walk (CTRW) and fractional diffusion equations. Stochastic equations, fractional Fokker-Planck equation & Langevin equation are considered. Continuous time random walk on combs is subject matter of chapter four. Heterogeneous diffusion processes, turbulent diffusion and geometric Brownian motion are given in chapter five.
Diffusion processes with stochastic resetting are given in chapter six. Numerical simulation, Langevin equation approach is discussed. Diffusion-advection equation with resetting, diffusion in comb with confining branches with resetting are given. Random search, one dimension search, random search on comb-like structures, Brownian search with drift on comb, Turbulent diffusion search on comb etc. are treated.
Diffusion on fractal tartan is subject matter of chapter eight. Grid comb model, finite number of backbones, fractal grid, infinite number of backbones are given. Fractal mesh, fractal structures of fingers etc. are considered. Finite velocity diffusion, Cattaneo equation on a comb structure and persistent random are discussed.
There are a number of appendices: fractional calculus, stochastic differential equations, large deviation principle, fractals and fractal dimension, Implementation of Wolfram Mathematica. References are given at the end of each chapter.
A detailed account of special functions in fractional calculus can be found in the following recent publications of [V. Kiryakova, Fract. Calc. Appl. Anal. 11, No. 2, 203–220 (2008; Zbl 1153.26003); “A guide to special functions in fractional calculus”, Mathematics 9, No. 1, 106 (2021; doi:10.3390/math9010106); “Unified approach to fractional calculus images of special functions”, Mathematics 8, No. 12, 2260 (2020; doi:10.3390/math8122260)]

MSC:

33-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to special functions
26-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to real functions
33Cxx Hypergeometric functions
26A33 Fractional derivatives and integrals
60G50 Sums of independent random variables; random walks
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics

Citations:

Zbl 1153.26003
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