We develope a numerical method based on B-spline collocation method to solve linear KLEIN-GORDON EQUATION. The proposed scheme is unconditionally stable. The results of numerical experiments have been compared with the exact solution to show the efficiency of the method computationally. Easy and economical implementation is the strength of this approach.

In this paper, we use a geometric approach based on the concepts of variational principle and moving frames to obtain the conservation laws related to the one-dimensional nonlinear KLEIN-GORDON EQUATION. Noether’, s First Theorem guarantees conservation laws, provided that the Lagrangian is invariant under a Lie group action. So, for calculating conservation laws of the KLEIN-GORDON EQUATION, we first present a Lagrangian whose Euler-Lagrange EQUATION is the KLEIN-GORDON EQUATION, and then according to Gon¸, calves and Mansfield’, s method, we obtain the space of conservation laws in terms of vectors of invariants and the adjoint representation of a moving frame, for that Lagrangian, which is invariant under a hyperbolic group action.

In this paper, the Lie approximate symmetry analysis is applied to investigate new exact solutions of the perturbed nonlinear KLEIN-GORDON EQUATION. The nonlinear KLEIN-GORDON EQUATION is used to model many nonlinear phenomena. The tanh-coth method, is employed to solve some of the obtained reduced ordinary differential EQUATIONs. We construct new analytical solutions with small parameter which is effectively obtained by the proposed method.