A Comparative Study of Hybrid Analytical and Laplace Transform Approaches for Solving Partial Differential Equations in Python

This research presents a rigorous and innovative approach, the Homotopy Perturbation Method-Laplace Transform Method (HPM-LTM), implemented in Python, for the efficient solution of linear and nonlinear partial differential equations (PDEs). By combining the Homotopy Perturbation technique (HPM) with the Laplace Transform Method (LTM), our method successfully addresses the significant challenges posed by equations with nonlinear components. Through the utilization of He's polynomials, the HPM-LTM approach effectively handles nonlinear terms, resulting in accurate and reliable solutions. To demonstrate the efficacy of our method, we extensively apply it to five representative PDE scenarios, including heat and wave equations. Our comprehensive results substantiate the remarkable accuracy and reliability of the HPM-LTM approach, highlighting its superiority in comparison to conventional approaches that require restrictive assumptions or discretization, which can introduce round-off errors. Furthermore, our method overcomes the limitations imposed by numerical errors inherent in traditional HPM techniques. The robustness, effectiveness, and adaptability of our proposed approach are further validated by its successful application to a wide range of PDE problems across various fields. This research presents a significant contribution to the development of a powerful computational tool for resolving diverse PDE problems, with particular relevance to the engineering discipline.

Keywords:

Partial Differential Equations , Homotopy Perturbation Method , Laplace Transform Method , Python programming