Application of DRBEM for 2D sine-Gordon equation

Abstract

This research paper introduces an application of dual reciprocity boundary element method (DRBEM) for the solution of sine-Gordon equation (SGE) in two-space dimension. Initially, the time derivatives are expanded using central difference schemes. After inserting the finite difference approximations into the governing equation, the pattern of the modified Helmholtz equation is obtained. The fundamental solution of modified Helmholtz equation is employed in the integral equation formulation. The inhomogeneous terms of the equation cause a domain integral in the integral equation formulation which leads to loss of advantage of the method. The DRBEM provides to transform the domain integral into the boundary integral by approximating the inhomogeneous term of the equation with thin plate spline ( $ r^2\ln r $ ). First, code validation of the procedure is done using a test problem and then proposed method is applied for several cases involving line and ring solitons to display its capacity to treat the problem. Presented numerical results are observed to be in good agreement with other numerical results available in the literature.

Keywords

• drbem
• sine-gordon equation
• modified helmholtz equation