ERROR ESTIMATES OF HIGH-ORDER NUMERICAL METHODS FOR SOLVING TIME FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS

Error estimates of some high-order numerical methods for solving time fractional partial differential equations are studied in this paper. We first provide the detailed error estimate of a high-order numerical method proposed recently by Li et al. [21] for solving time fractional partial differential equation. We prove that this method has the convergence order O(tau(3-alpha)) for all alpha is an element of(0, 1) when the first and second derivatives of the solution are vanish at t = 0, where tau is the time step size and alpha is the fractional order in the Caputo sense. We then introduce a new time discretization method for solving time fractional partial differential equations, which has no requirements for the initial values as imposed in Li et al. [21]. We show that this new method also has the convergence order O(tau(3-alpha)) for all alpha is an element of(0, 1). The proofs of the error estimates are based on the energy method developed recently by Lv and Xu [26]. We also consider the space discretization by using the finite element method. Error estimates with convergence order O(tau(3-alpha) + h(2)) are proved in the fully discrete case, where h is the space step size. Numerical examples in both one-and two-dimensional cases are given to show that the numerical results are consistent with the theoretical results.