A new solution procedure for the nonlinear telegraph equation

This paper involves a theoretical but fundamental question in the numerical computation of partial differential equations. Is it possible to construct the solution for a nonlinear telegraph equation (or a nonlinear damped wave equation) by using a hyperbolic linear solution of Klein-Gordon equation? To answer the question, firstly, an analytic solution of the linear Klein-Gordon equation is introduced here. Through the introduction, we show how the original nonlinear telegraph equation can be transformed into an equivalent nonlinear system of two integral equations of the second kind. Here, the singularities of the system's kernels are asymptotically shown to be just removable. Then, the above question may be answered by applying Banach fixed point theorem to the two (coupled) integral equations and thus showing how to construct nonlinear iterative solutions of the telegraph equation. This results in a new (functional) iterative procedure for the constructing of the (numerical) solutions of a general nonlinear telegraph equation.