Two linearized finite difference schemes for time fractional nonlinear diffusion-wave equations with fourth order derivative

Emadidin Gahalla Mohmed Elmahdi; Jianfei Huang; Emadidin Gahalla Mohmed Elmahdi; Jianfei Huang

doi:10.3934/math.2021373

AIMS Mathematics

2021, Volume 6, Issue 6: 6356-6376. doi: 10.3934/math.2021373

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Research article

Two linearized finite difference schemes for time fractional nonlinear diffusion-wave equations with fourth order derivative

Emadidin Gahalla Mohmed Elmahdi ^1,2,
Jianfei Huang ^{1
,
,}

1.
College of Mathematical Sciences, Yangzhou University, Yangzhou 225002, China
2.
Faculty of Education, University of Khartoum, Khartoum P. O. Box 321, Sudan

Received: 25 January 2021 Accepted: 08 April 2021 Published: 12 April 2021
MSC : 65M06, 65M12

In this paper, we present a linearized finite difference scheme and a compact finite difference scheme for the time fractional nonlinear diffusion-wave equations with space fourth order derivative based on their equivalent partial integro-differential equations. The finite difference scheme is constructed by using the Crank-Nicolson method combined with the midpoint formula, the weighted and shifted Gr$ \ddot{\text{u}} $nwald difference formula and the second order convolution quadrature formula to deal with the temporal discretizations. Meanwhile, the classical central difference formula and fourth order Stephenson scheme are used in the spatial direction. Then, the compact finite difference scheme is developed by using the fourth order compact difference formula for the spatial direction. The proposed schemes can deal with the nonlinear terms in a flexible way while meeting weak smoothness requirements in time. Under the relatively weak smoothness conditions, the stability and convergence of the proposed schemes are strictly proved by using the discrete energy method. Finally, some numerical experiments are presented to support our theoretical results.
- fractional nonlinear diffusion-wave equations,
- linearized schemes,
- fourth order derivative,
- stability,
- convergence
Citation: Emadidin Gahalla Mohmed Elmahdi, Jianfei Huang. Two linearized finite difference schemes for time fractional nonlinear diffusion-wave equations with fourth order derivative[J]. AIMS Mathematics, 2021, 6(6): 6356-6376. doi: 10.3934/math.2021373

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Abstract

In this paper, we present a linearized finite difference scheme and a compact finite difference scheme for the time fractional nonlinear diffusion-wave equations with space fourth order derivative based on their equivalent partial integro-differential equations. The finite difference scheme is constructed by using the Crank-Nicolson method combined with the midpoint formula, the weighted and shifted Gr$ \ddot{\text{u}} $nwald difference formula and the second order convolution quadrature formula to deal with the temporal discretizations. Meanwhile, the classical central difference formula and fourth order Stephenson scheme are used in the spatial direction. Then, the compact finite difference scheme is developed by using the fourth order compact difference formula for the spatial direction. The proposed schemes can deal with the nonlinear terms in a flexible way while meeting weak smoothness requirements in time. Under the relatively weak smoothness conditions, the stability and convergence of the proposed schemes are strictly proved by using the discrete energy method. Finally, some numerical experiments are presented to support our theoretical results.

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