A higher order numerical method for time fractional partial differential equations with nonsmooth data

Gao et al. [11] (2014) introduced a numerical scheme to approximate the Caputo fractional derivative with the convergence rate O(k3−α), 0<α<1 by directly approximating the integer-order derivative with some finite difference quotients in the definition of the Caputo fractional derivative, see also Lv and Xu [20] (2016), where k is the time step size. Under the assumption that the solution of the time fractional partial differential equation is sufficiently smooth, Lv and Xu [20] (2016) proved by using energy method that the corresponding numerical method for solving time fractional partial differential equation has the convergence rate O(k3−α), 0<α<1 uniformly with respect to the time variable t. However, in general the solution of the time fractional partial differential equation has low regularity and in this case the numerical method fails to have the convergence rate O(k3−α), 0<α<1 uniformly with respect to the time variable t. In this paper, we first obtain a similar approximation scheme to the Riemann-Liouville fractional derivative with the convergence rate O(k3−α), 0<α<1 as in Gao et al. [11] (2014) by approximating the Hadamard finite-part integral with the piecewise quadratic interpolation polynomials. Based on this scheme, we introduce a time discretization scheme to approximate the time fractional partial differential equation and show by using Laplace transform methods that the time discretization scheme has the convergence rate O(k3−α), 0<α<1 for any fixed tn>0 for smooth and nonsmooth data in both homogeneous and inhomogeneous cases. Numerical examples are given to show that the theoretical results are consistent with the numerical results.