This study aims to find the numerical solution of the Rosenau–Hyman and Fornberg–Whitham equations via the quintic B-spline collocation method. Quintic B-spline, along with finite difference and theta-weighted schemes, is used for the discretization and approximation purposes. The effectiveness and robustness of the procedure is assessed by comparing the computed results with the exact and available results in the literature using absolute and relative error norms. The stability of the proposed scheme is studied using von Neumann stability analysis. Graphical representations are drawn to analyze the behavior of the solution.

This work focuses on solving the singularly perturbed generalized Hodgkin-Huxley (HH) problem. The HH equation is numerically solved by a collocation approach using third-degree splines. The forward difference technique is utilized for time discretization, while $\theta$-weighted schemes are employed for space discretization. Solving non-linear models using discretization and quasi-linearization results in a set of linear algebraic equations, which are solved using matrices. Furthermore, Von Neumann’s (VN) stability and Spectral Radius (S.R) reveal that the suggested technique is unconditionally stable. To assess the performance and accuracy of this method, absolute error (AE), $L_2$, and $L_{\infty }$ norms are offered. The results align with the literature. Simulation results show that the proposed strategy produces accurate results.