An integral equation formalism for solving the nonlinear Klein–Gordon equation

In the paper, a new method for solving the nonlinear Klein–Gordon equation is proposed. To this end, the nonlinear partial differential equation of Klein–Gordon is transformed into an equivalent non-linear integral equation through a transformation. Here, a hyperbolic type of Green’s function is newly incorporated into the transformation in such a way that nonlinearities due to large wave motion are effectively taken into account. Based on the equivalent integral equation, a functional iteration procedure is constructed for solving the equation. The method proposed here is a semi-analytical one, not only fairly simple but straightforward to apply. Mathematical analysis is performed on the method’s convergence and uniqueness. An illustrative example of a solitary wave (or soliton) is presented to investigate the validity of the method, resulting in numerical solutions, appearing to be higher accurate compared to the usual 2nd order centered difference scheme in a stable manner. In fact, just a few iterations are enough for the numerical solution.