The fractional non-polynomial spline method: Precision and modeling improvements

This research introduces the fractional non-polynomial spline method as a novel scheme for solving the time-fractional Korteweg-de Vries (KdV) equation. The study focuses on numerical analysis and algorithm development for simulation purposes. The proposed method involves the construction of a fractional non-polynomial spline to estimate the equation's solution, offering improved precision and modeling capabilities compared to existing approaches. To assess the stability of the proposed approach, the von Neumann method is employed, demonstrating its unconditional stability within a specific parameter range. To validate the effectiveness of our numerical analysis and simulation algorithm, contour, 2D, and 3D graphs are utilized to compare the solution obtained through our method with an analytical solution. Through rigorous comparative analysis with previous works, the superiority of our approach in terms of accuracy and efficiency is demonstrated. Norm error calculations, specifically the (L2 and L infinity) error norms, provide a quantitative assessment of the accuracy and reliability of our proposed scheme.