Numerical solution of the linear time fractional Klein-Gordon equation using transform based localized RBF method and quadrature

Xiangmei Li; Kamran; Absar Ul Haq; Xiujun Zhang; Xiangmei Li; Kamran; Absar Ul Haq; Xiujun Zhang

doi:10.3934/math.2020339

AIMS Mathematics

2020, Volume 5, Issue 5: 5287-5308. doi: 10.3934/math.2020339

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Research article

Numerical solution of the linear time fractional Klein-Gordon equation using transform based localized RBF method and quadrature

1.
School of Cybersecurity, Chengdu University of Information Technology, Chengdu, 610225, China
2.
Department of Mathematics, Islamia College Peshawar, Khyber Pakhtoon Khwa, Pakistan
3.
Department of Basic Science and Humanities, University of Engineering and Technology, Lahore (Narowal Campus), Pakistan
4.
Key Laboratory of Pattern Recognition and Intelligent Information Processing, Institutions of Higher Education of Sichuan Province, Chengdu University, Chengdu, 610106, China

Received: 28 March 2020 Accepted: 10 June 2020 Published: 19 June 2020
MSC : 26A33, 65M12, 65R10, 81Q05

This work aims to approximate the solution of the linear time-fractional Klein-Gordon equations in Caputo's sense. The Laplace transform is applied to linear time fractional Klein-Gordon equation to eliminate the time variable and avoid the time stepping procedure. Application of the Laplace transform avoids the time instability issues which commonly occurs in time stepping methods and reduces the computational cost. The transform problem is then solved using local RBFs and finally the solution is obtained by the inverse Laplace transform. The solution is represented as an integral along a smooth curve in the complex plane which is then approximated by quadrature rule. The proposed method is capable of solving linear time fractional partial differential equations. The stability and convergence of the method are discussed. The efficiency of the method is demonstrated with the help of numerical experiments.
- Klein-Gordon equations,
- Caputo’s time fractional derivative,
- Laplace transform,
- local RBF method,
- contour integration,
- quadrature rule
Citation: Xiangmei Li, Kamran, Absar Ul Haq, Xiujun Zhang. Numerical solution of the linear time fractional Klein-Gordon equation using transform based localized RBF method and quadrature[J]. AIMS Mathematics, 2020, 5(5): 5287-5308. doi: 10.3934/math.2020339

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Abstract

This work aims to approximate the solution of the linear time-fractional Klein-Gordon equations in Caputo's sense. The Laplace transform is applied to linear time fractional Klein-Gordon equation to eliminate the time variable and avoid the time stepping procedure. Application of the Laplace transform avoids the time instability issues which commonly occurs in time stepping methods and reduces the computational cost. The transform problem is then solved using local RBFs and finally the solution is obtained by the inverse Laplace transform. The solution is represented as an integral along a smooth curve in the complex plane which is then approximated by quadrature rule. The proposed method is capable of solving linear time fractional partial differential equations. The stability and convergence of the method are discussed. The efficiency of the method is demonstrated with the help of numerical experiments.

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