Numerical approximation of the 3rd order pseudo-parabolic equation using collocation technique

This study presents a novel numerical approach for approximating the solution of third-order pseudo-parabolic partial differential equations (PDEs), which exhibit both parabolic and hyperbolic characteristics. The proposed method employs a cubic trigonometric tension B-spline collocation technique for spatial discretization, offering greater flexibility and accuracy compared to traditional spline methods. For time discretization, the finite difference method (FDM) is used, ensuring computational efficiency. Unlike many existing methods, our approach is tailored to handle the complexity of third-order equations while maintaining stability and accuracy over large-scale problems. The method’s unconditional stability is confirmed through a detailed von Neumann stability analysis, making it particularly robust for long-term simulations. Two illustrative examples are presented to demonstrate the method’s superior accuracy and flexibility in handling complex boundary conditions, as well as its ability to manage large-scale problems without requiring restrictive time steps. Compared to the existing methods, the combination of trigonometric tension B-splines with FDM proves to be a powerful and reliable tool for solving higher-order pseudo-parabolic equations.