Analytical solution of nonlinear Klein-Gordon equations with cubic nonlinearity by extended Adomian decomposition method

Abstract

In this paper, we are dedicated to acquire, for the first time, an analytically continuous result of the nonlinear Klein-Gordon equations (NKGE) with cubic nonlinearity via Adomian decomposition method (ADM) using multivariate Taylor’s theorem. These class of equations are nonlinear partial differential equations with initial or boundary conditions being hyperbolic or trigonometric functions. Thus leading to large solution series on application of ADM during the invertible process in the integral equations. Which is often analysed at finite discrete points. To overcome this, we extend the series solution by traditional ADM with the multivariate Taylor’s theorem. The process resulted to a simple solution series that was easier to understand and analysed continuously guaranteeing excellent convergence rate. We demonstrate our findings using four examples that were further depicted pictorially using Maple symbolic software.