On the Jaulent-Miodek System for Fluid Mechanics Using Combination of Adomian Decomposition and Padé Techniques

Solving nonlinear partial differential equations (PDEs) is crucial in various scientific and engineering domains.The Adomian Decomposition Method (ADM) has emerged as a promising technique for tackling such problems.However, its effectiveness diminishes over extended time intervals due to divergence issues.This limitation hampers its practical applicability in real-world scenarios where stable and accurate numerical solutions are essential.To address the divergence problem associated with ADM, this research explores the combination of the Adomian Decomposition Method (ADM) with the Padé technique -a method known for its accuracy and efficiency.This combination's purpose is to mitigate ADM's shortcomings, particularly when dealing with extended time intervals.Experimental analysis was conducted across varying time intervals to compare the performance of the combined technique with traditional ADM.Mathematica software was used to obtain all calculations, including the creation of tables and figures.Results from the experiments demonstrate the superiority of the combined technique in producing accurate results regardless of the time interval used.Furthermore, the combined method improves accuracy and ensures result stability over long time intervals, creating new possibilities for its use in scientific and engineering fields.This research contributes to the field by offering a solution to the divergence issue associated with ADM, thereby enhancing its applicability in solving nonlinear PDEs.While acknowledging limitations such as reliance on numerical simulations, the study highlights the practical implications of its findings, including more accurate predictions and modeling in complex systems, with potential social implications in decision-making and problem-solving contexts.