scholarly journals High-Order Numerical Methods for Solving Time Fractional Partial Differential Equations

2016 ◽  
Vol 71 (2) ◽  
pp. 785-803 ◽  
Author(s):  
Zhiqiang Li ◽  
Zongqi Liang ◽  
Yubin Yan
Keyword(s):  
Partial Differential Equations ◽  
Numerical Methods ◽  
Differential Equations ◽  
High Order ◽  
Fractional Partial Differential Equations ◽  
Partial Differential

scholarly journals Error estimates of high-order numerical methods for solving time fractional partial differential equations

2018 ◽  
Vol 21 (3) ◽  
pp. 746-774 ◽  
Author(s):  
Zhiqiang Li ◽  
Yubin Yan
Keyword(s):  
Partial Differential Equations ◽  
Numerical Methods ◽  
Differential Equations ◽  
Error Estimates ◽  
High Order ◽  
Convergence Order ◽  
Fractional Partial Differential Equations ◽  
Step Size ◽  
Partial Differential ◽  
Method Error

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(τ3−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(τ3−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(τ3−α + h2) 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.

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