A moving Kriging‐based MLPG method for nonlinear Klein–Gordon equation

Abstract

In this paper, the meshless local Petrov–Galerkin approximation is proposed to solve the 2‐D nonlinear Klein–Gordon equation. We used the moving Kriging interpolation instead of the MLS approximation to construct the meshless local Petrov–Galerkin shape functions. These shape functions possess the Kronecker delta function property. The Heaviside step function is used as a test function over the local sub‐domains. Here, no mesh is needed neither for integration of the local weak form nor for construction of the shape functions. So the present method is a truly meshless method. We employ a time‐stepping method to deal with the time derivative and a predictor–corrector scheme to eliminate the nonlinearity. Several examples are performed and compared with analytical solutions and with the results reported in the extant literature to illustrate the accuracy and efficiency of the presented method. Copyright © 2016 John Wiley & Sons, Ltd.

Generalized Lagrange Jacobi‐Gauss‐Lobatto vs Jacobi‐Gauss‐Lobatto collocation approximations for solving (2 + 1)‐dimensional Sine‐Gordon equations

“…18 For damped equations, however, such numerical methods will be discussed later in this section. Recently, some meshless methods [19][20][21][22][23][24] are being used to obtain the numerical solutions of the SG equation. Recently, some meshless methods [19][20][21][22][23][24] are being used to obtain the numerical solutions of the SG equation.…”

“…It is known to all that soliton solutions appear in the 1D and 2D SG equations, particularly solitons solutions to the 1D SG equation studied numerically and theoretically. Recently, some meshless methods [19][20][21][22][23][24] are being used to obtain the numerical solutions of the SG equation. In a very intuitive and easy way to express, Francisco et al 25 implemented operational matrices to SG equation.…”

Meshless Singular Boundary Method for Nonlinear Sine-Gordon Equation

“…For comparison, the errors of the RBF method [21], the explicit method [6], and the moving Kriging-based meshless local Petrov-Galerkin (MK-MLPG) method [25] obtained by using = 0.001 and ℎ = 0.25 are also given. Comparing the errors of these methods confirms the good accuracy of the SBM.…”

“…Up to now, a lot of meshless methods have been developed. Recently, some meshless methods [21][22][23][24][25][26][27][28] have been used to obtain the numerical solutions of the sine-Gordon equation.…”

An efficient meshless method based on the moving Kriging interpolation for two‐dimensional variable‐order time fractional mobile/immobile advection‐diffusion model

“…Meshfree techniques 33‐41 are very interesting and impressive for solving PDEs since these methods include simple projecting, variety in solving metamorphosis, and have capability to improve non‐smooth solutions. The meshless local Petrov–Galerkin (MLPG) method is an efficient meshfree method to solve PDEs with complicated domains.…”