A Novel Numerical Approach to Time-Fractional Parabolic Equations with Nonsmooth Solutions

This paper is concerned with numerical solutions of time-fractional parabolic equations. Due to the Caputo time derivative being involved, the solutions of equations are usually singular near the initial time t = 0 even for a smooth setting. Based on a simple change of variable s = t^β, an equivalent s-fractional differential equation is derived and analyzed. Two types finite difference methods based on linear and quadratic approximations in the s-direction are presented, respectively, for solving the s-fractional differential equation. We show that the method based on the linear approximation provides the optimal accuracy O(N^−(2−α)) where N is the number of grid points in temporal direction. Numerical examples for both linear and nonlinear fractional equations are presented in comparison with L1 methods on uniform meshes and graded meshes, respectively. Our numerical results show clearly the accuracy and efficiency of the proposed methods.