A numerical algorithm based on ultraspherical wavelets for solution of linear and nonlinear Klein-Go

In this paper, Galerkin method based on the Ultraspherical wavelets expansion together with operational matrix of integration is developed to solve linear and nonlinear Klein Gordon (KG) equations with the given initial and boundary conditions. Firstly, we present the ultraspherical wavelets, then the corresponding operational matrix of integration is presented. To transform the given PDE into a system of linear-nonlinear algebraic equations which can be efficiently solved by suitable solvers, we utilize the operational matrix of integration and both properties of Ultraspherical wavelets. The applicability of the method is shown by two test problems and acquired results show that the method is good accuracy and efficiency.