Finite element approximation of fractional hyperbolic integro-differential equation

Zhengang Zhao; Yunying Zheng; Xianglin Zeng; Zhengang Zhao; Yunying Zheng; Xianglin Zeng

doi:10.3934/math.2022841

AIMS Mathematics

2022, Volume 7, Issue 8: 15348-15369. doi: 10.3934/math.2022841

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Finite element approximation of fractional hyperbolic integro-differential equation

1.
Department of Fundamental Courses, Shanghai Customs College, Shanghai 201204, China
2.
School of Mathematical Sciences, Huaibei Normal University, Huaibei 235000, China
3.
Academic Adminstration, Shanghai Customs College, Shanghai 201204, China

Received: 02 April 2022 Revised: 25 May 2022 Accepted: 30 May 2022 Published: 17 June 2022
MSC : 35R11, 76M10

In this article, we propose a Galerkin finite element method for numerically solving a type of fractional hyperbolic integro-differential equation, which can be considered as the generalization of the classical hyperbolic Volterra integro-differential equation. Along with Galerkin finite element method in spatial direction, we apply a second order symmetric difference method in time. Next we discuss the regularity analysis of the weak solution and convergence analysis of the semi-discrete scheme. Then we further study the stability analysis and the error estimation of the fully discrete problems, according to the properties of fractional Ritz-Volterra projection, Ritz projection and so on. Numerical examples with comparisons among the proposed schemes verify our theoretical analyses.
Citation: Zhengang Zhao, Yunying Zheng, Xianglin Zeng. Finite element approximation of fractional hyperbolic integro-differential equation[J]. AIMS Mathematics, 2022, 7(8): 15348-15369. doi: 10.3934/math.2022841

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Abstract

In this article, we propose a Galerkin finite element method for numerically solving a type of fractional hyperbolic integro-differential equation, which can be considered as the generalization of the classical hyperbolic Volterra integro-differential equation. Along with Galerkin finite element method in spatial direction, we apply a second order symmetric difference method in time. Next we discuss the regularity analysis of the weak solution and convergence analysis of the semi-discrete scheme. Then we further study the stability analysis and the error estimation of the fully discrete problems, according to the properties of fractional Ritz-Volterra projection, Ritz projection and so on. Numerical examples with comparisons among the proposed schemes verify our theoretical analyses.

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