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This paper applies two computational techniques for constructing novel solitary wave solutions of the ill-posed Boussinesq dynamic wave (IPB) equation. Jacques Hadamard has formulated this model for studying the dynamic behavior of waves in shallow water under gravity. Extended simple equation (ESE) method and novel Riccati expansion (NRE) method have been applied to the investigated model's converted nonlinear ordinary differential equation through the wave transformation. As a result of this research, many solitary wave solutions have been obtained and represented in different figures in two-dimensional, three-dimensional, and density plots. The explanation of the methods used shows their dynamics and effectiveness in dealing with certain nonlinear evolution equations.

In this paper, the nonlinear fractional Lotka–Volterra model is analyzed and numerically studied. This research is based on applying the three latest analytical schemes and three other numerical schemes to construct rich wave solutions. In different forms, many novel solitary wave solutions are built and presented in two-dimensional, three-dimensional, and contour plots. The numerical method conditions are evaluated through the obtained analytical solutions, and the accuracy of the analytical solutions is studied. Many numerical solutions are constructed based on the employed schemes. Additionally, the analytical, semi-analytical, numerical, and absolute values of error between the values of obtained solutions are calculated with the different values of the given variable in the solutions. Furthermore, the match between the obtained analytical solution and the numerical solution has been explained through some two-dimensional distributed radar charts. The contribution of this article is demonstrated by comparing the obtained solution with the recently published results of the same model.

This paper investigates the analytical, semianalytical, and numerical solutions of the 2+1–dimensional integrable Schwarz–Korteweg–de Vries (SKdV) equation. The extended simplest equation method, the sech-tanh method, the Adomian decomposition method, and cubic spline scheme are employed to obtain distinct formulas of solitary waves that are employed to calculate the initial and boundary conditions. Consequently, the numerical solutions of this model can be investigated. Moreover, their stability properties are also analyzed. The solutions obtained by means of these techniques are compared to unravel relations between them and their characteristics illustrated under the suitable choice of the parameter values.

This study, using the extended simplest method of equation, examines the explicit movement solutions of both the Schwarzian Korteweg-de Vries (SKdV) and (2 + 1)-Ablowitz-Kaup-Newell-Segur (AKNS.) equation. These models show the movement of the waves in optical fiber mathematically. The SKdV equation explains the movement of the isolated waves in diverse fields and on the site in a small space microsection. Some solutions obtained have been developed to show the physical and dynamic behaviors of these solutions in the obtained wave.