Error estimates of a high order numerical method for solving linear fractional differential equations

Abstract

In this paper, we first introduce an alternative proof of the error estimates of the numerical methods for solving linear fractional differential equations proposed in Diethelm [6] where a first-degree compound quadrature formula was used to approximate the Hadamard finite-part integral and the convergence order of the proposed numerical method is O(∆t 2−α ), 0 < α < 1, where α is the order of the fractional derivative and ∆t is the step size. We then use the similar idea to prove the error estimates of a high order numerical method for solving linear fractional differential equations proposed in Yan et al. [37], where a second-degree compound quadrature formula was used to approximate the Hadamard finite-part integral and we show that the convergence order of the numerical method is O(∆t 3−α ), 0 < α < 1. The numerical examples are given to show that the numerical results are consistent with the theoretical results.

High‐order algorithms for Riesz derivative and their applications (V)

“…For Riesz derivative ∂ α u(x,t) ∂|x| α , we choose the fourth-order fractional-compact numerical differential formula (11). It follows from (17) that…”

“…As a result, the analytic solutions of most fractional differential equations cannot be obtained explicitly. Hence, developing numerical methods for these equations are becoming more and more necessary and important, one can refers to the papers [5][6][7][8][9][10][11][12][13] and reference therein. Generally speaking, the first step of solving fractional differential equations is to approximate the fractional derivatives.…”

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

“…They obtained an asymptotic expansion of the error, but there are no error estimates proved in [33]. Recently, Li et al [19] gave the detailed and thorough error estimates for the numerical method in [33] for solving the linear fractional differential equation.…”

“…Ford et al [6] applied the Diethelm's method for solving time fractional partial differential equation and proved that the convergence order is O(τ 2−α ) if u ∈ C 2 [0, T ]. Higher order Diethelm's schemes are also available in the literature, see [5,7,19,33], etc.…”

“…The time discretization is based on the numerical method in [33] and the space discretization is based on the standard finite element method. The error estimates with convergence order O(τ 3−α + h 2 ) are proved in detail by using the argument developed in [23] (see also [19]). The assumption of the regularity for the exact solution in our paper…”