High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations (III)

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• 作者：Hefeng Li^a ; Jianxiong Cao^b ; Changpin Li^a ; ^{lcp@shu.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Caputo derivative ; Advection&ndash ; diffusion equation ; Difference scheme ; Convergence
• 刊名：Journal of Computational and Applied Mathematics
• 出版年：2016
• 出版时间：June 2016
• 年：2016
• 卷：299
• 期：Complete
• 页码：159-175
• 全文大小：442 K

文摘

In this paper, a series of new high-order numerical approximations to α$\alpha$th (0<α<1)$(0<\alpha <1)$ order Caputo derivative is constructed by using r$r$th degree interpolation approximation for the integral function, where r≥4$r\ge 4$ is a positive integer. As a result, the new formulas can be viewed as the extensions of the existing jobs (Cao et al., 2015; Li et al., 2014), the convergence orders are O(τ^r+1−α)$O\left({\tau }^{r+1-\alpha }\right)$, where τ$\tau$ is the time stepsize. Two test examples are given to demonstrate the efficiency of these schemes. Then we adopt the derived schemes to solve the Caputo type advection–diffusion equation with Dirichlet boundary conditions. The local truncation error of the derived difference scheme is O(τ^r+1−α+h²)$O({\tau }^{r+1-\alpha }+h^2)$, where τ$\tau$ is the time stepsize, and h$h$ the space one. The stability and convergence of the proposed schemes for r=4$r=4$ are also considered. Without loss of generality, we only display the numerical examples for r=4,5$r=4,5$, which support the numerical algorithms.