This work introduces the closed form approach to generate solitary wave solutions to the modified equal width (MEW) model. This model generalizes the equal width (EW) equation, which characterizes wave propagation in shallow water. It aims to better represent nonlinear and dispersive effects by altering the original model's terms. Because of its simplicity, reliability, and efficiency, the proposed technique has the potential to be applied to a range of nonlinear partial differential equations (NPDEs) in practical research. We also employ the finite difference method to provide the numerical solution for the MEW model. A comparison with the analytical solution we arrived at demonstrates the method's accuracy. This work shows that the numerical method stays stable and accurate despite alterations in time stepping, wave speed, and spatial discretization. This also allows further exploration of nonlinear models that accurately depict significant physical processes in our everyday existence.