Eur. Phys. J. Plus (2020) 135: 707
https://doi.org/10.1140/epjp/s13360-020-00547-w

Regular Article

A meshless method for solving a class of nonlinear generalized telegraph equations with time-dependent coefficients based on radial basis functions

¹ College of Mechanics and Materials, Hohai University, 210098, Nanjing, China
² State Institution “Institute of Technical Problems of Magnetism of the National Academy of Sciences of Ukraine”, 61106, Kharkov, Ukraine
³ Nanjing Hydraulic Research Institute, 210029, Nanjing, China

^a zhengbinyilan@163.com
^c lujunhhu1981@126.com

Received: 18 April 2020
Accepted: 18 June 2020
Published online: 8 September 2020

Abstract

In this paper, we present a method based on the combination of the finite difference and the meshless techniques for solving 2D generalized telegraph equations in single and multi-connected domains. The three-layer Crank–Nicolson time-stepping scheme transforms the equation into a sequence of second-order elliptic partial differential equations with mixed derivatives and variable coefficients. The approximate solution on each time layer is sought as series over radial basis functions based approximations. These are constructed in such a way that they satisfy the homogeneous boundary conditions. Thus, any their linear combination satisfies the homogeneous boundary condition too. The coefficients of the series are enforced to satisfy the governing equation in the solution domain. Several numerical examples are presented to demonstrate stability and accuracy.