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RESEARCH ARTICLE

Numerical analysis for Klein-Gordon equation with time-space fractional derivatives

orcid.org/0000-0001-6841-7070

Department of Mathematics, Guizhou University, Guiyang, China

Computational Mathematics Research Center, Guizhou University of Finance and Economics, Guiyang, China

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JinRong Wang ,

Corresponding Author

wjr9668@126.com

orcid.org/0000-0002-6642-1946

Department of Mathematics, Guizhou University, Guiyang, China

School of Mathematical Sciences, Qufu Normal University, Qufu, China

Correspondence

JinRong Wang, Department of Mathematics, Guizhou University, Guiyang, 550025, China.

Email: wjr9668@126.com

Communicated by: T. Monovasilis

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Yong Zhou ,

orcid.org/0000-0002-4099-8077

Department of Mathematics, Xiangtan University, Xiangtan, China

Faculty of Information Technology, Macau University of Science and Technology, Macau, China

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First published: 23 December 2019

Citations: 1

Abstract

We present and analyze two numerical schemes for solving a nonlinear Klein-Gordon equation with time-space fractional derivatives. Numerical methods are base on finite difference scheme in fractional derivative and Fourier-spectral method in spatial variable. It is proved that the linearized method is conditionally stable while the nonlinearized one is unconditionally stable. In addition, the error estimate shows that the linearized method is in the order of $O (Δ t + N^{β - r})$ , and the nonlinearized method converge with the order $O ({Δ t}^{3 - α} + N^{β - r})$ , where $Δ t$ , $N$ , $β$ , and $r$ are, respectively, step of time, polynomial degree, the fractional derivative in space, and regularity of $u$ . Some numerical experiments are performed to demonstrate the theoretical results.

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