Application of the Aboodh Adomian Decomposition Method to Klein-Gordon and Sine-Gordon Equations

Abstract

In this study, an efficient method is presented for the analysis of the Klein-Gordon (KG) and Sine-Gordon (SG) equations with initial value problems. KG and SG equations are hyperbolic partial differential equations that possess the capability to model phenomena in both quantum and classical mechanics, as well as solitons and condensed matter physics. KG equation represents a relativistic wave equation while SG equation represents the d’Alembert operator with a nonlinear sine term of the dependent variable. The proposed method is based on applying the coupling of Aboodh transformation and Adomian decomposition method (ADM) to partial differential equations and this study is limited to KG and SG equations. The non-linear term is replaced by Adomian polynomials for the index n. The elements of the dependent variable are substituted within the recurrence relation by their respective Aboodh transform components corresponding to the same index. Consequently, the nonlinear problem is addressed in a direct manner, devoid of any linearization or discretization processes. Illustrations are presented to demonstrate the efficacy and veracity of the method. A comparison of the findings with the precise solution indicates that the method proved to be efficient because the results are in closed agreement with the exact solution (errors = 0 with just 5–6 terms). The study concludes that this method can be applied to a variety of linear and nonlinear partial differential equation because Aboodh Adomian Decomposition Method (AADM) provides accurate numerical solutions for linear and nonlinear problems, and can be extended to solve other problems arising in applied science.