Advanced Techniques in Solving Coupled Burgers' Equations: Homotopy Analysis Method (HAM)

Abstract: The study applies the Homotopy Analysis method (HAM) to solve coupled 1D non-linear Burgers' equations, demonstrating its effectiveness in transforming complex non-linear problems into simpler linear forms. By constructing a homotopy that transitions from an initial approximation to the exact solution, the method efficiently handles the non-linear dynamics of the equations. Solutions are expressed as power series, and higher-order deformation equations are solved iteratively, incorporating non-linear effects. Graphical analyses, including 3D surface plots and time evolution graphs, illustrate the dynamic behavior of the solutions, such as wave propagation and diffusion. The results underscore HAM's robustness in solving non-linear differential equations, though the study suggests future exploration of hybrid methods to address challenges in strongly non-linear or chaotic systems.

Keywords: Non-linear coupled Burgers' equations, Homotopy Analysis Method, source terms, semi-analytical technique, 3D visualizations