A New Uniform Hyperbolic Polynomial B-Spline Computational Scheme for 3rd-Order Pseudo-Parabolic Problem

This work introduces a new UHPB-spline computation approach to approximate the pseudo-parabolic problem of order three, which exposes parabolic as well as hyperbolic physical appearance. The approach is used for the discretization of spatial derivatives, which contributes to better accuracy and flexibility. For the discretization of the time derivative, the FDM is employed, endorsing computational proficiency. The anticipated approach has significantly enhanced accuracy. Contrasting various existing approaches, the proposed process is tailored for handling the difficulty in the third-order problem while preserving accuracy and stability over extensive problems. A detailed von Neumann stability analysis confirms its unconditional stability, which makes it robust, especially for long-term simulations, while the numerical ROC demonstrates the second-order convergence both in space and in time. Two expressive examples are considered to determine the accuracy and usefulness of the projected process. Compared to existing techniques, the combination of UHPB-spline functions together with the Crank–Nicolson method and FDM is evidence of an influential and consistent tool to solve pseudo-parabolic problems of higher orders.