THEORIES AND APPLICATIONS OF ADOMIAN DECOMPOSITION $ \mathcal{J} $-TRANSFORM METHOD WITH THEORETICAL ANALYSIS AND SIMULATION

This work focuses on using the Adomian decomposition $ \mathcal{J} $-transform method ($ \mathcal{ADJM} $) to solve both linear and nonlinear differential equations, with the aim of obtaining exact solutions for various types of differential equations, such as the Bratu equations and second-order linear partial telegraph equation. We present comprehensive proofs for new theorems associated with the $ \mathcal{J} $-transform method. This approach combines the $ \mathcal{J} $-transform method ($ \mathcal{JTM} $) and the Adomian decomposition method ($ \mathcal{ADM} $). We carry out a theoretical analysis of $ \mathcal{ADJM} $ applied to certain nonlinear differential equations and give proofs to the existence and uniqueness theorems along with error estimates. The solutions obtained are compared with exact solutions from other established methods in the literature. The study emphasizes the notable advantages of $ \mathcal{ADJM} $, highlighting its effectiveness in solving both ODEs and PDEs. In order to demonstrate the unique advantages of the employed method, we give exact solutions in the form of convergent power series with easily obtainable coefficients. Some of the symbolic and numerical calculations were executed using Mathematica software 13.