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.

Recently, the successive approximation method (SAM) has attracted the attention of many authors due to its simplicity, ease of use, and great results. However, the results obtained by SAM start to diverge when the time interval is increased. To address this issue, this paper develops an improved version of the SAM for solving non-linear Partial Differential Equations (PDEs) numerically. Here, the initial condition of the differential equations has been combined with the SAM to obtain stable and more accurate results. The test that was conduct included the original SAM and the improved one on the system of strongly non-linear PDEs. Experimental results revealed that the proposed technique gives better and more accurate numerical solutions regardless of the time interval used.

In this paper, we propose two new techniques for solving system of nonlinear partial differential equations numerically, which we first combine Laplace transformation method into a successive approximation method. Second, we combine Padé [2,2] technique into the first proposed technique. To test the efficiency of our techniques, Jaulent-Miodek system was used, which contains partial differential equations and has strongly nonlinear terms. Experimental results revealed that the first proposed technique gives better results when the interval of t is small in terms of error approximation in tabular and graphical manners. Moreover, the results also demonstrated that the second proposed technique gives better results regardless of the given interval of t in terms of the least square errors.

In this paper, Modified Variational Iteration Method (MVIM) and Homotopy Analysis Method (HAM) are used to find approximate solutions for the Variable-Coefficient Variant Boussinesq System the (VCVB) system is able to describe the nonlinear and dispersive long gravity waves in shallow water traveling in two horizontal directions with varying depth, as an example we took the Boussinesq-Burgers (B-B) system, (B-B) system arise in the study of fluid flow and describing the long-wave propagation of shallow water waves. The solutions of these equations helpful for the coastal and civil engineering’s