AN EFFICIENT TECHNIQUE VIA THE $\mathbb{J}$-TRANSFORM DECOMPOSITION METHOD: THEORETICAL ANALYSIS WITH APPLICATIONS

The nonlinear Klein–Gordon equation, which describes nonlinear wave interaction and arises from the study of quantum field theory, is one of the most notable differential equations. In this research, we solve the equation using a novel approach. It is used in many areas of mathematics, such as conformal mapping theory, physics, and algebraic geometry. In the current work, the $ \mathbb{J}- $transform Adomian decomposition method ($ \mathbb{JADM} $) is applied to provide exact solutions for a variety of nonlinear partial differential equations (PDES). We provide comprehensive proofs for novel theorems related to the $ \mathbb{J}- $transform methodology. This method is based on the $ \mathbb{J}- $transform method ($ \mathbb{JTM} $) and the Adomian decomposition method ($ \mathbb{ADM} $). The theoretical analysis of the $ \mathbb{JADM} $ is investigated and computed using easily obtained terms for some differential equations. Our results are compared with exact solutions obtained by other methods that can be found in the literature. The paper describes the important aspects of the $ \mathbb{JADM} $. The $ \mathbb{JADM} $ has demonstrated a high degree of efficiency, accuracy, and adaptability to a wide range of differential equations, both linear and nonlinear. Mathematica was used for much of the symbolic and numerical calculations $ \Omega $.