An implicit RBF meshless approach for solving the time fractional nonlinear sine-Gordon and Klein–Gordon equations

Abstract

In this paper, we propose a numerical method for the solution of time fractional nonlinear sine-Gordon equation that appears extensively in classical lattice dynamics in the continuum media limit and Klein–Gordon equation which arises in physics. In this method we first approximate the time fractional derivative of the mentioned equations by a scheme of order O(τ3−α),1<α<2 then we will use the Kansa approach to approximate the spatial derivatives. We solve the two-dimensional version of these equations using the method presented in this paper on different domains such as rectangular and non-rectangular domains. Also, we prove the unconditional stability and convergence of the time discrete scheme. We show that convergence order of the time discrete scheme is O(τ). We solve these fractional PDEs on different non-rectangular domains. The aim of this paper is to show that the meshless method based on the radial basis functions and collocation approach is also suitable for the treatment of the nonlinear time fractional PDEs. The results of numerical experiments are compared with analytical solutions to confirm the accuracy and efficiency of the presented scheme.