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

Abstract

In this article, barycentric rational interpolation and local radial basis functions (RBFs) based numerical algorithms are developed for solving multidimensional sine-Gordon (SG) equation. In the development of these algorithms, the first step is to drive a semi-discretization in time with a finite difference, and then the semi-discrete problem is analyzed for truncation errors and convergence in L 2 and H 1 spaces. After that, the semi-discrete system is fully discretized by two different functions, such as barycentric rational and local RBFs. Finally, we obtain a linear system in both the algorithms and the system is solved by a MATLAB routine. In numerical experiments, 1D and 2D SG are considered with various examples of line and ring solitons. Moreover, a comparative study of present results with available numerical ones and exact solutions is also discussed. KEYWORDS barycentric rational interpolation, differential quadrature method, local radial basis functions, sine-Gordon equation, truncation error and convergence analysis 1 INTRODUCTION Nonlinear phenomena play key prefaces in problems arising in physics, applied mathematics, and engineering. These phenomena are modeled by nonlinear evolution equations [1]. Solitonary solutions of evolution equations confer better information to understand the physical mechanism of the phenomena.

Application of DRBEM for 2D sine-Gordon equation

“…Moreover, it is observed that at t = 8, the elliptical ring soliton begins to take the form of a circular soliton, and when the time is reached t = 11.2, an almost circular ring soliton is formed. The obtained results are in good agreement with those presented in [27,38,43,47] and [50].…”

“…When t = 7, a deformation is observed on the graphs. All behaviours seen in the graphs are in agreement with those published in [27,32,38,43,47] and [49]. In addition, the effect of dissipative term is evaluated at t = 2 and t = 4 when ν = 0 and ν = 0.5 by giving the contourlines.…”

“…As time increases, it moves slowly in x 1 direction and a curl caused by inhomogeneity of the medium occurs in its straightness. These results show that a good agreement is obtained with those given in [27,32,43,47] and [49]. Also when these results are compared with the results given for ν = 0 in [38], it can be expressed that the propagation of the line soliton is slowing down.…”

“…The circular ring solitons for the case χ(x 1 , x 2 ) = 1 and ν = 0.05 are analysed with the initial conditions as results are good agreement with the results given in [27,32,38,43,47] and [49].…”

“…Dehghan and Shokri used radial basis functions for the solution of twodimensional SGEs in [32], Cheng and Liew used meshfree reproducing kernel particle Ritz method in [33] and Li et al used element-free Galerkin method in [34], LingDe Su used localized method of approximate particular solutions in [35]. Other presented numerical studies for the SGE were given by Argyris et al in [36] using finite element method, by Sheng et al in [37] using split cosine scheme, by Bratsos in [38] using method of lines, by Han and Zhang in [39] using operator splitting method, by Suarez in [40] using Chebyshev split-step scheme, by Adak and Natarajan in [41] using virtual element method, by Guo et al in [42] using local Kriging meshless method and by Jiwari in [43] using barycentric rational interpolation and local radial basis functions based numerical algorithms, etc.…”