Analytical Solution of the Klein-Gordon coupled Equations via the Differential Transform Method

The basic equations in physical applications are the Klein-Gordon equation, as it gives a description of numerical particles in the fields of quantum and quantum mechanics, through the use of the differential transformation method and combining it with the Idomian method, in order to provide an integrated method for solving this equation, and it has been widely used to solve differential equations. Partial, to get rid of nonlinear terms and complex terms by integrating these two methods, which led to reducing the error and thus increasing the efficiency and accuracy of the solution using these numerical methods and the theoretical analysis of the method.. Researchers can learn more about quantum field dynamics using our method, which makes it easier to study a wide range of physical phenomena including scalar particles. In addition to improving the computational toolset in relativistic quantum mechanics, this work offers a useful foundation for investigating basic interactions in the field of particle physics and other fields. Conclusions: Adomain decomposition method (ADM) and the Differential Transform Method (DTM) together provide a powerful approach to solving the Klein-Gordon system equation. The effectiveness of this integrated method in efficiently and precisely solving complex differential equations controlling scalar particle dynamics has been proven by our research. The nonlinearities and boundaries of the Klein-Gordon system equation have proven to be formidable obstacles, but we have overcome them by combining the two approaches, and the findings are solid and trustworthy. Scalar particle behaviour in relativity quantum mechanics can be better understood with the help of this comprehensive methodology, which also improves computational efficiency. The DTM technique is a powerful and flexible tool for academics in many fields, including applied mathematics, particle physics, and quantum field theory, as our results show. The future looks bright for tackling ever more complex systems and improving our understanding of basic physical processes as we continue to investigate and enhance this integrated approach.