The Klein-Gorden Equation's Numerical Solution Using Nonstandard Finite Difference

Abstract

We provide a numerical method in this study to solve the one-dimensional 
nonlinear Klein-Gorden problem. We provide a detailed description of the 
mathematical formulation process. The technique is based on nonstandard finite 
differences and is explicit at three levels. The step sizes' denominator function is 
nonlinear. We have provided a stability study of the approach and shown its 
unconditional stability when applied to the one-dimensional nonlinear Klein-Gorden 
problem. We use the suggested strategy to two cases to demonstrate its utility.