A Direct Meshless Method for Solving Two-Dimensional Second-Order Hyperbolic Telegraph Equations

Fuzhang Wang,Enran Hou,Imtiaz Ahmad

doi:10.1155/2020/8832197

Abstract

In this paper, a direct meshless method (DMM), which is based on the radial basis function, is developed to the numerical solution of the two-dimensional second-order hyperbolic telegraph equations. Since these hyperbolic telegraph equations are time dependent, we present two schemes for the basis functions from radial and nonradial 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 hyperbolic telegraph equations can be solved in a direct way. Numerical experiments performed with the proposed numerical scheme for several two-dimensional second-order hyperbolic telegraph equations are presented with some discussions, which show that the DMM solutions are converging very fast in comparison with the various existing numerical methods.

Highlights

It is well-known that it is difficult to get the analytical solutions for relatively complex problems. us, the approximate numerical approximations to the telegraph equation are a better choice
Meshless methods have witnessed the research boom in science and engineering [12,13,14]. Dehghan and his coworkers proposed several meshless methods to investigate the solution of secondorder two-space dimensional linear hyperbolic telegraph equation which include the implicit collocation method [15], the thin plate splines radial basis functions [16], the meshless local Petrov–Galerkin method [17], and the boundary knot method [18]
Based on the abovementioned investigations, we propose a direct meshless method with one-level approximation, which is based on the radial basis functions, for the two-dimensional second-order hyperbolic telegraph equations

Summary

Introduction

It is well-known that it is difficult to get the analytical solutions for relatively complex problems. us, the approximate numerical approximations to the telegraph equation are a better choice. Meshless methods have witnessed the research boom in science and engineering [12,13,14] For these branches, Dehghan and his coworkers proposed several meshless methods to investigate the solution of secondorder two-space dimensional linear hyperbolic telegraph equation which include the implicit collocation method [15], the thin plate splines radial basis functions [16], the meshless local Petrov–Galerkin method [17], and the boundary knot method [18]. Based on the abovementioned investigations, we propose a direct meshless method with one-level approximation, which is based on the radial basis functions, for the two-dimensional second-order hyperbolic telegraph equations. In order to overcome the two-level strategy, we propose a direct collocation scheme by using space-time radial and nonradial basis functions

Objectives

Methods

Conclusion