The use of radial basis functions (RBFs) collocation and RBF-QR methods for solving the coupled nonlinear sine-Gordon equations

Abstract

Radial basis function (RBF) approximation is an extremely powerful tool for solving various types of partial differential equations, since the method is meshless and can be spectrally accurate. A perceived practical obstacle is that the interpolation matrix becomes increasingly ill-conditioned as the RBF shape parameter becomes small, corresponding to flat RBFs. In this paper, the authors solve the one and two-dimensional time-dependent coupled sine-Gordon equations using RBFs collocation and RBF-QR methods and show how one can overcome the ill-conditioning of coefficient matrix for the small shape parameters using RBF-QR method. The main aim of the current paper is to show that the meshless techniques based on the collocation methods are also suitable for solving the system of coupled nonlinear equations especially sine-Gordon equation. Several test problems are employed and results of numerical experiments are presented and also are compared with analytical solutions. The obtained results confirm the acceptable accuracy of the new methods.