Mathematics

CLC number: O 242

The Space-Time Meshless Methods for the Solution of One-Dimensional Klein-Gordon Equations

¹ General Education Center, Zhengzhou Business University, Gongyi 451200, Henan, China
² School of Mathematics and Statistics, Xuzhou University of Technology, Xuzhou 221018, Jiangsu, China
³ School of Computer Science & Technology, Huaibei Normal University, Huaibei 235000, Anhui, China
⁴ School of Information and Control Engineering, China University of Mining and Technology, Xuzhou 221116, Jiangsu, China

^† To whom correspondence should be addressed. E-mail: wangfuzhang1984@163.com

Received: 21 March 2022

Abstract

A simple direct space-time meshless scheme, based on the radial or non-radial basis function, is proposed for the one-dimensional Klein-Gordon equations. Since these equations are time-dependent, it is worthwhile to present two schemes for the basis functions from radial and non-radial aspects. The first scheme is fulfilled by considering time variable as normal space variable, to construct an "isotropic" space-time radial basis function. The other scheme considered a realistic relationship between space variable and time variable which is not radial. The time-dependent variable is treated regularly during the whole solution process and the Klein-Gordon equations can be solved in a direct way. Numerical results show that the proposed meshless schemes are simple, accurate, stable, easy-to-program and efficient for the Klein-Gordon equations.