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.

本研究论文介绍了对偶互易边界元法 (DRBEM) 在二维空间中求解正弦-戈登方程 (SGE) 的应用。最初，使用中心差分方案扩展时间导数。在控制方程中插入有限差分近似后，得到修正亥姆霍兹方程的模式。在积分方程公式中采用了修正亥姆霍兹方程的基本解。方程的非齐次项导致积分方程公式中的域积分，从而导致方法的优势丧失。DRBEM 提供通过用薄板样条近似方程的非齐次项将域积分转换为边界积分（$r^2\mathrm{输入}r$）。首先，程序的代码验证是使用测试问题完成的，然后将所提出的方法应用于涉及线和环孤子的几种情况，以显示其处理问题的能力。观察到的数值结果与文献中可用的其他数值结果非常一致。