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Volume 11, Issue 4
Spectral Deferred Correction Methods for Fractional Differential Equations

Chunwan Lv, Mejdi Azaiez & Chuanju Xu

Numer. Math. Theor. Meth. Appl., 11 (2018), pp. 729-751.

Published online: 2018-06

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  • Abstract

In this paper, we propose and analyze a spectral deferred correction method for the fractional differential equation of order α. The proposed method is based on a well-known finite difference method of $(2−α)$-order, see [Sun and Wu, Appl. Numer. Math., 56(2), 2006] and [Lin and Xu, J. Comput. Phys., 225(2), 2007], for prediction of the numerical solution, which is then corrected through a spectral deferred correction method. In order to derive the convergence rate of the prediction-correction iteration, we first derive an error estimate for the $(2−α)$-order finite difference method on some non-uniform meshes. Then the convergence rate of orders $\mathcal{O}(τ^{(2−α)(p+1)})$ and  $\mathcal{O}(τ^{(2−α)+p})$ of the overall scheme is demonstrated numerically for the uniform mesh and the Gauss-Lobatto mesh respectively, where $τ$ is the maximal time step size and $p$ is the number of correction steps. The performed numerical test confirms the efficiency of the proposed method.

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@Article{NMTMA-11-729, author = {Chunwan Lv, Mejdi Azaiez and Chuanju Xu}, title = {Spectral Deferred Correction Methods for Fractional Differential Equations}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2018}, volume = {11}, number = {4}, pages = {729--751}, abstract = {

In this paper, we propose and analyze a spectral deferred correction method for the fractional differential equation of order α. The proposed method is based on a well-known finite difference method of $(2−α)$-order, see [Sun and Wu, Appl. Numer. Math., 56(2), 2006] and [Lin and Xu, J. Comput. Phys., 225(2), 2007], for prediction of the numerical solution, which is then corrected through a spectral deferred correction method. In order to derive the convergence rate of the prediction-correction iteration, we first derive an error estimate for the $(2−α)$-order finite difference method on some non-uniform meshes. Then the convergence rate of orders $\mathcal{O}(τ^{(2−α)(p+1)})$ and  $\mathcal{O}(τ^{(2−α)+p})$ of the overall scheme is demonstrated numerically for the uniform mesh and the Gauss-Lobatto mesh respectively, where $τ$ is the maximal time step size and $p$ is the number of correction steps. The performed numerical test confirms the efficiency of the proposed method.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2018.s03}, url = {http://global-sci.org/intro/article_detail/nmtma/12469.html} }
TY - JOUR T1 - Spectral Deferred Correction Methods for Fractional Differential Equations AU - Chunwan Lv, Mejdi Azaiez & Chuanju Xu JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 729 EP - 751 PY - 2018 DA - 2018/06 SN - 11 DO - http://doi.org/10.4208/nmtma.2018.s03 UR - https://global-sci.org/intro/article_detail/nmtma/12469.html KW - AB -

In this paper, we propose and analyze a spectral deferred correction method for the fractional differential equation of order α. The proposed method is based on a well-known finite difference method of $(2−α)$-order, see [Sun and Wu, Appl. Numer. Math., 56(2), 2006] and [Lin and Xu, J. Comput. Phys., 225(2), 2007], for prediction of the numerical solution, which is then corrected through a spectral deferred correction method. In order to derive the convergence rate of the prediction-correction iteration, we first derive an error estimate for the $(2−α)$-order finite difference method on some non-uniform meshes. Then the convergence rate of orders $\mathcal{O}(τ^{(2−α)(p+1)})$ and  $\mathcal{O}(τ^{(2−α)+p})$ of the overall scheme is demonstrated numerically for the uniform mesh and the Gauss-Lobatto mesh respectively, where $τ$ is the maximal time step size and $p$ is the number of correction steps. The performed numerical test confirms the efficiency of the proposed method.

Chunwan Lv, Mejdi Azaiez and Chuanju Xu. (2018). Spectral Deferred Correction Methods for Fractional Differential Equations. Numerical Mathematics: Theory, Methods and Applications. 11 (4). 729-751. doi:10.4208/nmtma.2018.s03
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