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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.

##### MSC:
 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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