By Yuriy Povstenko
This e-book systematically provides recommendations to the linear time-fractional diffusion-wave equation. It introduces the vital remodel procedure and discusses the homes of the Mittag-Leffler, Wright, and Mainardi capabilities that seem within the strategies. The time-nonlocal dependence among the flux and the gradient of the transported volume with the “long-tail” strength kernel leads to the time-fractional diffusion-wave equation with the Caputo fractional spinoff. Time-nonlocal generalizations of classical Fourier’s, Fick’s and Darcy’s legislation are thought of and other kinds of boundary stipulations for this equation are mentioned (Dirichlet, Neumann, Robin, ideal contact). The ebook offers strategies to the fractional diffusion-wave equation with one, and 3 house variables in Cartesian, cylindrical and round coordinates.
The respective sections of the publication can be utilized for college classes on fractional calculus, warmth and mass move, delivery tactics in porous media and fractals for graduate and postgraduate scholars. the quantity also will function a precious reference consultant for experts operating in utilized arithmetic, physics, geophysics and the engineering sciences.
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139) 26 Chapter 2. 2. Mittag-Leffler function 27 d z β−1 E1,β (z) = z β−2 E1,β−1 (z) . 140) dz The essential role of the Mittag-Leffler functions in fractional calculus results from the formula for the inverse Laplace transform [56, 77, 143]: L−1 sα−β sα + b = tβ−1 Eα,β (−btα ). 143) = tα−1 Eα,α (−btα ). 144) L−1 sα 1 +b The series representation of the Mittag-Leffler functions is inconvenient for numerical calculation. The integral representations of these functions suitable for such calculation were obtained in [52, 56].
188). 29). Subdiffusion with α = 1/2 q0 GΦ = √ 2aπt3/4 ∞ √ u exp −u2 − 0 x2 √ 8a tu du. 36) Subdiffusion with α = 2/3 31/3 q0 GΦ = √ 2/3 Ai 2 at |x| √ 31/3 at1/3 . 37) Fast diffusion with α = 4/3 32/3 q0 GΦ = √ 1/3 Ai 2 at x2 34/3 at4/3 exp − 2 x3 27 a3/2 t2 . 38) were obtained in [67]. 39) on the similarity variable is presented in Figs. 4, respectively. 22). 34) presented in Figs. 34) in different ways. 34) is valid for 0 < α ≤ 2. 1. 4: The fundamental solution to the source problem 48 Chapter 4.
1. 54) ξk2 and d2 f (x) dx2 F + + = −ξk2 f (ξk ) ξk ξk2 + H 2 + 2H L ξk ξk2 + H 2 + − 2H L df (x) + Hf (x) dx x=0 df (x) ξk2 + H 2 + Hf (x) cos(Lξk ) ξk2 − H 2 dx . 55) x=L We have restricted ourselves to the case of the same H in the Robin boundary conditions at x = 0 and x = L. The general case of different coefficients H1 and H2 is considered in [48]. 10 Finite sin-Fourier transform for a sphere This type of finite sin-Fourier transform is convenient for central symmetric problems for a sphere 0 ≤ r ≤ R.