Efficient Techniques for Solving System of (2+1) D & (3+1) D PDES Using Rangaig Transforms Based Ham

Abstract

In this study, we explore the application of Rangaig transforms combined with the Homotopy Analysis Method (HAM) for solving systems of (2+1)D and (3+1)D nonlinear partial differential equations. Nonlinear PDEs frequently arise in various scientific and engineering fields, modeling complex phenomena such as fluid dynamics, heat transfer, and wave propagation. Traditional numerical and analytical methods often encounter limitations when addressing such systems, particularly in higher dimensions. The Rangaig transform, a powerful integral transform, is integrated with HAM to efficiently construct approximate analytical solutions to these complex systems. The proposed methodology leverages the strengths of both techniques: the Rangaig transform simplifies the PDEs by converting them into a more tractable form, while HAM provides a flexible framework for obtaining convergent series solutions without restrictive assumptions on small parameters. We demonstrate the effectiveness of this combined approach through several benchmark problems, showcasing its accuracy, efficiency, and convergence properties. The results indicate that the Rangaig-HAM hybrid approach not only accelerates the solution process but also extends the applicability of HAM to more challenging nonlinear PDE systems in multidimensional spaces. This study contributes to advancing analytical techniques for solving high-dimensional nonlinear PDEs and provides a robust tool for researchers and engineers dealing with complex physical systems.