Numerical solution of Klein–Gordon and sine-Gordon equations by meshless method of lines

We investigate numerical solution of the one dimensional nonlinear Klein–Gordon and two-dimensional sine-Gordon equations by meshless method of lines using radial basis functions. Results are compared with some earlier work showing the efficiency of the applied method. Salient feature of this method is that it does not require a mesh in the problem domain. Multiquadric and Gaussian are used as radial basis functions, which use a shape parameter. Choice of the shape parameter is still an open problem. We explore optimal value of the shape parameter without applying any extra treatment. For multiquadric, eigenvalue stability is studied without enforcing the boundary conditions whereas for Gaussian, the boundary conditions are enforced.