Error Estimates of High-Order Numerical Methods for Solving Time Fractional Partial Differential Equations

Zhiqiang Li,Yubin Yan

doi:10.1515/fca-2018-0039

Abstract

Abstract 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(τ 3−α ) for all α ∈ (0, 1) when the first and second derivatives of the solution are vanish at t = 0, where τ is the time step size and α 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(τ 3−α ) for all α ∈ (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(τ 3−α + 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.

Highlights

Consider the following time fractional partial differential equation C 0 Dtαu(x, t) − ∆u(x, t) = f (x, t), x ∈Ω, t [0, T ], u(x, 0) = u0, x ∈ Ω, u(x, t) = 0, x ∈ ∂Ω, t ∈ [0, T ], where 0 < α < 1 and ∆ denotes the Laplacian and Ω ⊂ Rd, d = 1, 2, 3 is a bounded and regular domain.Here f is a given function andDtαv(t) denotes the Caputo fractional order derivative defined by
Where P2j(s) denotes the quadratic interpolation polynomials defined on the nodes s = tj−1, tj, tj+1 as above
To apply this scheme to solve the time fractional partial differential equation (1.1)-(1.3), we have to obtain the approximate value of the solution at t1 with the required accuracy by using other numerical methods, we use this scheme to approximate the solutions at tk with k = 2, 3, . . . , N

Summary

Introduction

Where P2j(s) denotes the quadratic interpolation polynomials defined on the nodes s = tj−1, tj, tj+1 as above To apply this scheme to solve the time fractional partial differential equation (1.1)-(1.3), we have to obtain the approximate value of the solution at t1 with the required accuracy by using other numerical methods, we use this scheme to approximate the solutions at tk with k = 2, 3, . By C we denote a positive constant independent of the functions and parameters concerned, but not necessarily the same at different occurrences

Time discretization

The fully Discretization scheme

A new time discretization scheme

Numerical simulations