Quasi Interpolation of radial basis functions-pseudospectral method for solving nonlinear Klein–Gordon and sine-Gordon equations

We propose a new approach for solving nonlinear Klein–Gordon and sine-Gordon equations based on radial basis function-pseudospectralmethod (RBF-PS). The proposed numerical method is based on quasiinterpolation of radial basis function differentiation matrices for thediscretization of spatial derivatives combined with Runge–Kutta time stepping method in order to deal with the temporal part of the problem.The method does not require any linearization technique; in addition, a new technique is introduced to force approximations to satisfy exactlythe boundary conditions. The introduced scheme is tested for a number of one- and two-dimensional nonlinear problems. Numerical results andcomparisons with reported results in the literature are given to validate the presented method, and the reported results show the applicabilityand versatility of the proposed method.

Keywords:

Meshless method , Pseudospectral method , Radial basis functions , Klein-Gordon equation , sine-Gordon equation , Runge-Kutta fourth order method , Multiquadric quasi-interpolation