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

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• 作者：Zhiqiang Li^a ; ^{z.li@chester.ac.uk} ; Yubin Yan^b ; ^{y.yan@chester.ac.uk} ; Neville J. Ford^b ; ^{njford@chester.ac.uk}
• 关键词：Fractional differential equations ; Fractional derivative ; Error estimates
• 刊名：Applied Numerical Mathematics
• 出版年：2017
• 出版时间：April 2017
• 年：2017
• 卷：114
• 期：Complete
• 页码：201-220
• 全文大小：486 K
• 卷排序：114

文摘

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(Δt2&minus;α),0<α<1O(Δt2&minus;α),0<α<1, where α is the order of the fractional derivative and Δt is the step size. We then use a similar idea to prove the error estimates of the 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(Δt3&minus;α),0<α<1O(Δt3&minus;α),0<α<1. Numerical examples are given to show that the numerical results are consistent with the theoretical results.