A Comparative Analysis of Laplace Transform and Adomian Decomposition Methods for Solving the 3D Heat Equation: Accuracy, Efficiency, and Flexibility

This paper presents a comparative analysis of the Laplace Transform Method and the Adomian Decomposition Method (ADM) for solving the three-dimensional heat equation. The Laplace Transform Method converts equations into the frequency domain, enabling precise solutions for linear systems but struggling with asymmetric and nonlinear cases. In contrast, ADM decomposes the solution into an infinite series computed recursively, making it suitable for complex nonlinear applications. Through comparative analysis, this paper demonstrates that the Laplace Transform Method offers high accuracy for linear cases, while ADM is more flexible and better suited for handling complex boundary conditions. Consequently, the Laplace Transform is preferable for simple linear problems, whereas ADM proves to be more effective for complex and nonlinear cases.