A Split-Step Predictor-Corrector Method for Space-Fractional Reaction-Diffusion Equations with Nonhomogeneous Boundary Conditions

doi:10.1007/s42967-019-00030-z

Communications on Applied Mathematics and Computation ›› 2019, Vol. 1 ›› Issue (4): 525-544.doi: 10.1007/s42967-019-00030-z

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A Split-Step Predictor-Corrector Method for Space-Fractional Reaction-Diffusion Equations with Nonhomogeneous Boundary Conditions

Kamran Kazmi¹, Abdul Khaliq²

1 Department of Mathematics, University of Wisconsin Oshkosh, Oshkosh, WI 54901, USA;
2 Department of Mathematical Sciences and Center for Computational Sciences, Middle Tennessee State University, Murfreesboro, TN 37132, USA

收稿日期:2018-09-27 出版日期:2019-12-30 发布日期:2019-10-16
通讯作者: Abdul Khaliq, Kamran Kazmi E-mail:Abdul.Khaliq@mtsu.edu;kazmis@uwosh.edu

A Split-Step Predictor-Corrector Method for Space-Fractional Reaction-Diffusion Equations with Nonhomogeneous Boundary Conditions

Kamran Kazmi¹, Abdul Khaliq²

1 Department of Mathematics, University of Wisconsin Oshkosh, Oshkosh, WI 54901, USA;
2 Department of Mathematical Sciences and Center for Computational Sciences, Middle Tennessee State University, Murfreesboro, TN 37132, USA

Received:2018-09-27 Online:2019-12-30 Published:2019-10-16
Contact: Abdul Khaliq, Kamran Kazmi E-mail:Abdul.Khaliq@mtsu.edu;kazmis@uwosh.edu

摘要/Abstract

摘要： A split-step second-order predictor-corrector method for space-fractional reaction-diffusion equations with nonhomogeneous boundary conditions is presented and analyzed for the stability and convergence. The matrix transfer technique is used for spatial discretization of the problem. The method is shown to be unconditionally stable and second-order convergent. Numerical experiments are performed to confirm the stability and second-order convergence of the method. The split-step predictor-corrector method is also compared with an IMEX predictor-corrector method which is found to incur oscillatory behavior for some time steps. Our method is seen to produce reliable and oscillation-free results for any time step when implemented on numerical examples with nonsmooth initial data. We also present a priori reliability constraint for the IMEX predictor-corrector method to avoid unwanted oscillations and show its validity numerically.

关键词: Fractional Laplacian, Space-fractional reaction diffusion equations, Nonhomogeneous boundary conditions, Matrix transfer technique, Predictor-corrector method

Abstract: A split-step second-order predictor-corrector method for space-fractional reaction-diffusion equations with nonhomogeneous boundary conditions is presented and analyzed for the stability and convergence. The matrix transfer technique is used for spatial discretization of the problem. The method is shown to be unconditionally stable and second-order convergent. Numerical experiments are performed to confirm the stability and second-order convergence of the method. The split-step predictor-corrector method is also compared with an IMEX predictor-corrector method which is found to incur oscillatory behavior for some time steps. Our method is seen to produce reliable and oscillation-free results for any time step when implemented on numerical examples with nonsmooth initial data. We also present a priori reliability constraint for the IMEX predictor-corrector method to avoid unwanted oscillations and show its validity numerically.

Key words: Fractional Laplacian, Space-fractional reaction diffusion equations, Nonhomogeneous boundary conditions, Matrix transfer technique, Predictor-corrector method

中图分类号:

Kamran Kazmi, Abdul Khaliq. A Split-Step Predictor-Corrector Method for Space-Fractional Reaction-Diffusion Equations with Nonhomogeneous Boundary Conditions[J]. Communications on Applied Mathematics and Computation, 2019, 1(4): 525-544.

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A Split-Step Predictor-Corrector Method for Space-Fractional Reaction-Diffusion Equations with Nonhomogeneous Boundary Conditions

A Split-Step Predictor-Corrector Method for Space-Fractional Reaction-Diffusion Equations with Nonhomogeneous Boundary Conditions

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