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Numerical solution to a linearized time fractional KdV equation on unbounded domains


Authors: Qian Zhang, Jiwei Zhang, Shidong Jiang and Zhimin Zhang
Journal: Math. Comp. 87 (2018), 693-719
MSC (2010): Primary 33F05, 35Q40, 35Q55, 34A08, 35R11, 26A33
DOI: https://doi.org/10.1090/mcom/3229
Published electronically: July 13, 2017
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Abstract: An efficient numerical scheme is developed to solve a linearized time fractional KdV equation on unbounded spatial domains. First, the exact absorbing boundary conditions (ABCs) are derived which reduces the pure initial value problem into an equivalent initial boundary value problem on a finite interval that contains the compact support of the initial data and the inhomogeneous term. Second, the stability of the reduced initial-boundary value problem is studied in detail. Third, an efficient unconditionally stable finite difference scheme is constructed to solve the initial-boundary value problem where the nonlocal fractional derivative is evaluated via a sum-of-exponentials approximation for the convolution kernel. As compared with the direct method, the resulting algorithm reduces the storage requirement from $ O(MN)$ to $ O(M\log ^dN)$ and the overall computational cost from $ O(MN^2)$ to $ O(MN\log ^dN)$ with $ M$ the total number of spatial grid points and $ N$ the total number of time steps. Here $ d=1$ if the final time $ T$ is much greater than $ 1$ and $ d=2$ if $ T\approx 1$. Numerical examples are given to demonstrate the performance of the proposed numerical method.


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Additional Information

Qian Zhang
Affiliation: Beijing Computational Science Research Center, Beijing 100193, China
Email: qianzhang@csrc.ac.cn

Jiwei Zhang
Affiliation: Beijing Computational Science Research Center, Beijing 100193, China
Email: jwzhang@csrc.ac.cn

Shidong Jiang
Affiliation: Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey 07102
Email: shidong.jiang@njit.edu

Zhimin Zhang
Affiliation: Beijing Computational Science Research Center, Beijing 100193, China — and — Department of Mathematics, Wayne State University, Detroit, Michigan 48202
Email: zmzhang@csrc.ac.cn, zzhang@math.wayne.edu

DOI: https://doi.org/10.1090/mcom/3229
Received by editor(s): May 10, 2016
Received by editor(s) in revised form: September 24, 2016, and October 13, 2016
Published electronically: July 13, 2017
Additional Notes: The research of the first author was supported in part by the National Natural Science Foundation of China under grants 91430216 and U1530401.
The research of the second author was supported in part by the National Natural Science Foundation of China under grants 91430216 and U1530401.
The research of the third author was supported in part by NSF under grant DMS-1418918
The research of the fourth author was supported in part by the National Natural Science Foundation of China under grants 11471031, 91430216, U1530401, and by NSF under grant DMS-1419040.
Article copyright: © Copyright 2017 American Mathematical Society

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