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

Zhang, Qian; Zhang, Jiwei; Jiang, Shidong; Zhang, Zhimin

doi:10.1090/mcom/3229

Numerical solution to a linearized time fractional KdV equation on unbounded domains
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by Qian Zhang, Jiwei Zhang, Shidong Jiang and Zhimin Zhang;

Math. Comp. 87 (2018), 693-719

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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