Abstract
Two approaches resulting in two different generalizations of the space-time-fractional advection-diffusion equation are discussed. The Caputo time-fractional derivative and Riesz fractional Laplacian are used. The fundamental solutions to the corresponding Cauchy and source problems in the case of one spatial variable are studied using the Laplace transform with respect to time and the Fourier transform with respect to the spatial coordinate. The numerical results are illustrated graphically.
Highlights
The one-dimensional Riesz derivative can be defined by its Fourier transform rule [77]:
We have considered two approaches to deriving the space-time fractional advection-diffusion equation
In the case of one spatial dimension, we have studied the fundamental solutions to the Cauchy and source problems for the obtained equations
Summary
The Fokker–Planck equation with constant coefficients has the form [1,2,3,4,5]:. = a∆w − v · grad w. Effective implicit numerical methods for the solution of the space-time fractional Fokker–Planck equation and fractional advection diffusion equation were proposed in [68,69]; their stability and convergence were studied in [62] (see [70,71]). We discuss two possibilities of obtaining the space-time fractional generalization of the advection-diffusion equation. The properties of the fundamental solution to the Cauchy problem for the space-time fractional advection-diffusion equation in the case of the first approach were investigated in [74]. The explicit representation of the fundamental solution for the space fractional advection-diffusion equation (α = 1) was obtained.
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