Abstract
Such physical processes as the diffusion in the environments with fractal geometry and the particles’ subdiffusion lead to the initial value problems for the nonlocal fractional order partial differential equations. These equations are the generalization of the classical integer order differential equations. An analytical solution for fractional order differential equation with the constant coefficients is obtained in [1] by using Laplace-Fourier transform. However, nowadays many of the practical problems are described by the models with variable coefficients. In this paper we discuss the numerical vector decomposition model which is based on a shifted version of usual Gr¨unwald finite-difference approximation [2] for the non-local fractional order operators. We prove the unconditional stability of the method for the fractional diffusion equation with Dirichlet boundary conditions. Moreover, a numerical example using a finite difference algorithm for 2D fractional order partial differential equations is also presented and compared with the exact analytical solution.
Highlights
In this paper we use the Riemann-Liouville fractional derivative x DLαf (x) = dαf (x) dxα =1 Γ(n − α) dn dxn f (ξ) dξ (x − ξ)α+1−n, (1)L where n is an integer, and the αth order is in the following interval n − 1 < α ≤ n
We prove the unconditional stability of the method for the fractional diffusion equation with Dirichlet boundary conditions
Where n is an integer, and the αth order is in the following interval n − 1 < α ≤ n
Summary
L where n is an integer, and the αth order is in the following interval n − 1 < α ≤ n. Following [3] the case L = 0 of the formula (1) is called the Riemann form and the case L = ∞ is called Liouville form for the fractional derivatives. With the boundary conditions offered below the Riemann and Liouville forms become equivalent. Grunwald-Letnikov formula for solving the one-dimensional diffusion equation is used in [2] by M.Meerschaert et al According to the author the application of this formula. This fact is the reason for the appearance of the important and interesting scientific results for the numerical methods theory to solve the fractional order differential equations [2]. Allowing for [2], in the present paper we use the rightshifted Grunwald approximation which is of the following form at 1 < α ≤ 2
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