A class of explicit implicit alternating difference schemes for generalized time fractional Fisher equation

Xiao Qin; Xiaozhong Yang; Peng Lyu; Xiao Qin; Xiaozhong Yang; Peng Lyu

doi:10.3934/math.2021663

AIMS Mathematics

2021, Volume 6, Issue 10: 11449-11466. doi: 10.3934/math.2021663

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A class of explicit implicit alternating difference schemes for generalized time fractional Fisher equation

School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China

Received: 31 May 2021 Accepted: 04 August 2021 Published: 09 August 2021
MSC : 65M06, 65M12

The generalized time fractional Fisher equation is one of the significant models to describe the dynamics of the system. The study of effective numerical techniques for the equation has important scientific significance and application value. Based on the alternating technique, this article combines the classical explicit difference scheme and the implicit difference scheme to construct a class of explicit implicit alternating difference schemes for the generalized time fractional Fisher equation. The unconditional stability and convergence with order $ O\left({\tau }^{2-\alpha }+{h}^{2}\right) $ of the proposed schemes are analyzed. Numerical examples are performed to verify the theoretical analysis. Compared with the classical implicit difference scheme, the calculation cost of the explicit implicit alternating difference schemes is reduced by almost $ 60 $%. Numerical experiments show that the explicit implicit alternating difference schemes are also suitable for solving the time fractional Fisher equation with initial weak singularity and have an accuracy of order $ O\left({\tau }^{\alpha }+{h}^{2}\right) $, which verify that the methods proposed in this paper are efficient for solving the generalized time fractional Fisher equation.
- generalized time fractional Fisher equation,
- explicit implicit alternating difference scheme,
- stability,
- convergence,
- numerical experiments
Citation: Xiao Qin, Xiaozhong Yang, Peng Lyu. A class of explicit implicit alternating difference schemes for generalized time fractional Fisher equation[J]. AIMS Mathematics, 2021, 6(10): 11449-11466. doi: 10.3934/math.2021663

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Abstract

The generalized time fractional Fisher equation is one of the significant models to describe the dynamics of the system. The study of effective numerical techniques for the equation has important scientific significance and application value. Based on the alternating technique, this article combines the classical explicit difference scheme and the implicit difference scheme to construct a class of explicit implicit alternating difference schemes for the generalized time fractional Fisher equation. The unconditional stability and convergence with order $ O\left({\tau }^{2-\alpha }+{h}^{2}\right) $ of the proposed schemes are analyzed. Numerical examples are performed to verify the theoretical analysis. Compared with the classical implicit difference scheme, the calculation cost of the explicit implicit alternating difference schemes is reduced by almost $ 60 $%. Numerical experiments show that the explicit implicit alternating difference schemes are also suitable for solving the time fractional Fisher equation with initial weak singularity and have an accuracy of order $ O\left({\tau }^{\alpha }+{h}^{2}\right) $, which verify that the methods proposed in this paper are efficient for solving the generalized time fractional Fisher equation.

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