Numerical Solution of Sine-Gordon Equation with the Local Kriging Meshless Method

Abstract

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.

Barycentric rational interpolation and local radial basis functions based numerical algorithms for multidimensional sine‐Gordon equation

“…Due to a wide range of applicability of SG equation in the fields of physics and electronics of SG equation, various numerical schemes or methods have been constructed to simulate the equation, and this is still an active area of research in the community of researchers and scientists. Various numerical schemes such as finite difference schemes [9], two‐level dissipationless Maxwell–Bloch systems [10], generalized leapfrog method [6], finite element methods [7], a split cosine scheme [11], a three‐time level fourth‐order explicit finite‐difference scheme [12], method of lines [13], a modified predictor–corrector scheme [14], dual reciprocity boundary element method [15], a numerical method based on radial basis functions (RBFs) [16], boundary element method [4], meshless local Petrov–Galerkin method [17], a local weak meshless technique based on the radial point interpolation method [18], a method based on collocation and RBFs [19], meshless local boundary integral equation method [20], interpolated coefficient finite element method [21], differential quadrature methods [22–25], and space–time spectral collocation method [26] have been developed for solving the SG equation. Recently, localized methods of approximate particular solutions [27] and structure‐preserving algorithms [28] have been sublimed for simulation of the SG equation.…”

Solutions of Two-Dimensional Nonlinear Sine-Gordon Equation via Triple Laplace Transform Coupled with Iterative Method

“…In 2020, [13] developed 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.…”

Analytical Solution of Two-Dimensional Sine-Gordon Equation

“…The nonlinear sine-Gordon equation (SGE), a type of hyperbolic partial differential equation, is often used to describe and simulate the physical phenomena in a variety of fields of engineering and science, such as nonlinear waves, propagation of fluxions, and dislocation of metals, for details see [10] and the references therein. Because the sine-Gordon equation has many kinds of soliton solutions, it has attracted wide spread attention [11].…”

“…In the early 1970s, it was first realized that the sine-Gordon equation led to kink and antikink (so-called solitons) [13]. As one of the crucial equations in nonlinear science, the sine-Gordon equation has been constantly investigated and solved numerically and analytically in recent years [10,[14][15][16][17][18]. Different scholars employed different methods to solve the one-dimensional sine-Gordon equation, for example, the Adomian decomposition method (ADM) [19][20][21][22][23], the EXP function method [24], the homotopy perturbation method (HPM) [25][26][27], the homotopy analysis method (HAM) [28], the variable separated ODE method [29,30], and the variational iteration method (VIM) [31,32].…”