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A fast and high-order IMEX method for non-linear time-space-fractional reaction-diffusion equations

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Abstract

An efficient and high-order numerical method is presented for solving a time-space-fractional reaction-diffusion equation. Matrix transfer technique based on fourth-order compact finite differences is first used to discretize the space-fractional Laplacian operator which results in a system with a linear stiff term. Then, an implicit-explicit (IMEX) trapezoidal product-integration rule is implemented for time integration which treats the stiff linear term implicitly and non-linear non-stiff term explicitly. The stability and convergence of the method are analyzed. Due to the discontinuity of the solution derivative at \(t=0\), the numerical method is only \(1+\alpha \) order accurate in time where \(\alpha \) is the order of the time-fractional derivative. Richardson extrapolation is introduced to obtain a modified version of the method which is second order accurate in time. A fast algorithm based on discrete sine transform is also implemented to reduce the cost of computing the discretized space-fractional Laplacian operator.

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Acknowledgements

The author is grateful to the reviewers for their constructive suggestions and comments to improve the paper.

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  1. Department of Mathematics, University of Wisconsin Oshkosh, Oshkosh, WI, 54901, USA

    Kamran Kazmi

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Kazmi, K. A fast and high-order IMEX method for non-linear time-space-fractional reaction-diffusion equations. Numer Algor 95, 243–266 (2024). https://doi.org/10.1007/s11075-023-01570-5

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