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Anomalous diffusion is a phenomenon that cannot be modeled accurately bysecond-order diffusion equations, but is better described by fractionaldiffusion models. The nonlocal nature of the fractional diffusion operatorsmakes substantially more difficult the mathematical analysis of these modelsand the establishment of suitable numerical schemes. This paper proposes andanalyzes the first finite difference method for solving {\emvariable-coefficient} fractional differential equations, with two-sidedfractional derivatives, in one-dimensional space. The proposed scheme combinesfirst-order forward and backward Euler methods for approximating the left-sidedfractional derivative when the right-sided fractional derivative isapproximated by two consecutive applications of the first-order backward Eulermethod. Our finite difference scheme reduces to the standard second-ordercentral difference scheme in the absence of fractional derivatives. Theexistence and uniqueness of the solution for the proposed scheme are proved,and truncation errors of order $h$ are demonstrated, where $h$ denotes themaximum space step size. The numerical tests illustrate the global $O(h)$accuracy of our scheme, except for nonsmooth cases which, as expected, havedeteriorated convergence rates.

A finite difference method for space fractional differential equations with variable diffusivity coefficient