Exact solutions for the generalized Klein–Gordon equation via a transformation and Exp-function method and comparison with Adomian’s method

In this paper, a suitable transformation and a so-called Exp-function method are used to obtain different types of exact solutions for the generalized Klein–Gordon equation. These exact solutions are in full agreement with the previous results obtained in Refs. [Sirendaoreji, Auxiliary equation method and new solutions of Klein–Gordon equations, Chaos, Solitons & Fractals 31 (4) (2007) 943–950; Huiqun Zhang, Extended Jacobi elliptic function expansion method and its applications, Communications in Nonlinear Science and Numerical Simulation, 12 (5) (2007) 627–635]. One of these exact solutions is compared with the approximate solutions obtained by the modified decomposition method. Accurate numerical results for a wider range of time are obtained after using different types of ADM-Padè approximation. Our results show that the Exp-function method is very effective in finding exact solutions for the problem considered while the modified decomposition method is very powerful in finding numerical solutions with good accuracy for nonlinear PDE without any need for a transformation or perturbation.