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The fundamental solution and numerical solution of the Riesz fractional advection-dispersion equation. (English) Zbl 1179.37073
The paper is concerned with the advection-dispersion equation with Riesz fractional derivative. Such fractional advection-dispersion equations may be derived either as an extension of the continuous-time random walk model or from the kinetics of chaotic dynamics. The authors obtain an expression for the fundamental solution to this equation using Laplace and Fourier transform. The finite-difference approximation for the initial value and boundary value problem for this equation is considered. A discrete form of the original equation is studied in both time and space. The Riesz fractional advection and diffusion terms are approximated so that the standard symmetric three-point difference scheme is used. Stability and convergence of the proposed numerical method are discussed. Numerical examples are presented.

37H10 Generation, random and stochastic difference and differential equations
47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
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