Eur. Phys. J. Plus (2018) 133: 290
https://doi.org/10.1140/epjp/i2018-12104-1

Regular Article

An effective semi-analytical method for solving telegraph equation with variable coefficients

¹ International Center for Simulation Software in Engineering and Sciences, College of Mechanics and Materials, Hohai University, 211100, Nanjing, China
² State Institution “Institute of Technical Problems of Magnetism of the National Academy of Sciences of Ukraine”, Industrialnaya St.,19, 61106, Kharkov, Ukraine
³ Nanjing Hydraulic Research Institute, 210029, Nanjing, China
⁴ State Key Laboratory of Hydrology Water Resources and Hydraulic Engineering, 210098, Nanjing, China

^* e-mail: lujunhhu1981@126.com

Received: 8 March 2018
Accepted: 14 June 2018
Published online: 31 July 2018

Abstract

In this paper, a new meshless numerical method is proposed for solving the second-order hyperbolic telegraph equations with coefficients variable in space and time. The temporal discretization is presented by the two-level time-stepping Crank-Nicolson scheme (CNS) with the meshless semi-analytical technique proposed by Reutskiy and Lin (Int. J. Numer. Methods Eng. 112, 2004 (2017)) for spatial discretization. For the spatial discretization, the approximation is given to satisfy the boundary conditions in advance with any free parameters. Then the approximation is substituted back to the governing equations where the unknowns are determined by the collocation approach. The efficacy of the proposed approach has been confirmed by numerical experiments and comparisons with the results obtained by other techniques. Numerical results show that the proposed method is of high accuracy and efficacy for solving a wide class of telegraph equations. The method is very simple and does not require any specific technique in handling the singularity at the endpoints of the solution domain.