This paper develops a local Kriging meshless solution to the nonlinear 2 + 1-dimensional sine-Gordon equation. The meshless shape function is constructed by Kriging interpolation method to have Kronecker delta function property for the two-dimensional field function, which leads to convenient implementation of imposing essential boundary conditions. Based on the local Petrov–Galerkin formulation and the center difference method for time discretization, a system of nonlinear discrete equations is obtained. The numerical examples are presented and the numerical solutions are found to be in good agreement with the results in the literature to validate the ability of the present meshless method to handle the 2 + 1-dimensional sine-Gordon equation related problems.

中文翻译：

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Sine-Gordon方程的局部Kriging无网格法数值解

本文开发了非线性2 +1维正弦-Gordon方程的局部Kriging无网格解。通过Kriging插值方法构造无网格形状函数，使其具有二维场函数的Kronecker delta函数属性，从而可以轻松实施强加的基本边界条件。基于局部Petrov-Galerkin公式和时间离散的中心差方法，获得了非线性离散方程组。给出了数值例子，发现数值解与文献中的结果吻合良好，验证了本无网格方法处理2 +1维正弦-Gordon方程相关问题的能力。