A Finite Diference Method for Space Fractional Diferential Equations with Variable Difusivity Coefcient

doi:10.1007/s42967-020-00066-6

Abstract

Abstract: Anomalous difusion is a phenomenon that cannot be modeled accurately by second-order difusion equations, but is better described by fractional difusion models. The nonlocal nature of the fractional difusion operators makes substantially more difcult the mathematical analysis of these models and the establishment of suitable numerical schemes. This paper proposes and analyzes the frst fnite diference method for solving variable-coefcient onedimensional (steady state) fractional diferential equations (DEs) with two-sided fractional derivatives (FDs). The proposed scheme combines frst-order forward and backward Euler methods for approximating the left-sided FD when the right-sided FD is approximated by two consecutive applications of the frst-order backward Euler method. Our scheme reduces to the standard second-order central diference in the absence of FDs. The existence and uniqueness of the numerical solution are proved, and truncation errors of order h are demonstrated (h denotes the maximum space step size). The numerical tests illustrate the global O(h) accuracy, except for nonsmooth cases which, as expected, have deteriorated convergence rates.

Key words: Two-sided fractional derivatives, Variable coefcients, Finite diferences

CLC Number:

K. A. Mustapha, K. M. Furati, O. M. Knio, O. P. Le Maître. A Finite Diference Method for Space Fractional Diferential Equations with Variable Difusivity Coefcient[J]. Communications on Applied Mathematics and Computation, 2020, 2(4): 671-688.

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