A Finite Difference Method for Space Fractional Differential Equations with Variable Diffusivity Coefficient

Anomalous diffusion is a phenomenon that cannot be modeled accurately by second-order diffusion equations, but is better described by fractional diffusion models. The nonlocal nature of the fractional diffusion operators makes substantially more difficult the mathematical analysis of these models and the establishment of suitable numerical schemes. This paper proposes and analyzes the first finite difference method for solving variable-coefficient one-dimensional (steady state) fractional differential equations (DEs) with two-sided fractional derivatives (FDs). The proposed scheme combines first-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 first-order backward Euler method. Our scheme reduces to the standard second-order central difference 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.