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 of 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.