A Modified Time-Fractional Diffusion Equation and Its Finite Difference Method: Regularity and Error Analysis

Abstract

Abstract The time-fractional diffusion partial differential equations (tFPDEs) (of order 0 < α < 1) properly model the anomalous diffusive transport or memory effects. Recent work [23] showed that the first-order time derivatives of their solutions have a singularity of O(tα−1) near the initial time t = 0, which makes the error estimates of their numerical approximations in the literature that were proved under full regularity assumptions of the true solutions inappropriate. A sharp error estimate was proved for a finite difference method (FDM) with a graded partition for a one-dimensional tFPDE without artificial regularity assumptions on true solutions, [23]. Motivated by the derivation of the tFPDE from stochastic continuous time random walk (CTRW), we present a modified tFPDE and prove that it has full regularity on the entire time interval (including t = 0) and that its FDM on a uniform time partition has an optimal-order convergence rate only under the assumptions of the regularity of the initial condition and right-hand source term. Numerical experiments show that with the same initial data, the solutions of the modified tFPDE and the classical tFPDE converge to each other as time increases, but the solution of the former does not have the singularity as that to the classical tFPDE near time t = 0.