A computational study of multi-soliton of good Boussinesq equation and Euler–Bernoulli beam model via nonpolynomial spline algorithm

A numerical method based on the nonpolynomial spline technique, incorporating half-step points in the spatial direction, is developed to address fourth-order partial differential equations, such as multi-soliton of good Boussinesq equation and the Euler–Bernoulli beam model. The good Boussinesq equation governs the phenomena of shallow water waves propagation. The interaction between single and multi-soliton in the good Boussinesq equation is illustrated across various time levels, resulting in the emergence of bell-shaped and periodic solitons under specific parameter settings. Additionally, both 3D and 2D plots are employed to visualize the numerical solutions of benchmark problems, examining the wave behavior under Gaussian initial conditions. The proposed scheme demonstrates stability when applied to the linearized good Boussinesq equation. Moreover, numerical findings underscore the efficacy of the proposed approach compared to previously established methods.