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The study employs sophisticated numerical and analytical methods to examine contemporary approaches to the improved Boussinesq equation. Initially, we concentrate on the exact solutions derived from modifying the Kudryashov and improved generalized Riccati equation mapping approaches, which delineate diverse soliton solutions that facilitate further generalization. These methods make it possible to deepen the understanding of the equation of physical nature as well as provide a dependable means of conducting parametric studies of nonlinear systems. We also created a mesh as the solving tool to the equation to test how the equation would respond under different operating conditions. Numerical solutions are demonstrated with analytical solutions that are generated using Taylor expansion methods whereby, analytical solutions follow the Von Neumann method for stability analysis so as to minimize the capability and confirm the two approaches of solutions. It has been shown in this work that the two methods of solving nonlinear equation gave appropriate results and are suitable for other nonlinear evolutionary systems.

Analytical and numerical investigation of exact wave patterns and chaotic dynamics in the extended improved Boussinesq equation