Efficient nonic B-spline collocation method for seventh-order nonlinear KdV equations

This paper presents an efficient computational approach based on 9th order(nonic) B-spline basis functions for the numerical solution of the seventh-order nonlinear Korteweg–de Vries (KdV) equation. The novel method takes advantage of the greater smoothness and accuracy of derivatives of nonic B-splines and is thus able to achieve accurate spatial approximations, most appropriately suited to accommodate the intricate dispersive nature inherent in higher-order wave equations. The nonlinear terms are based on Taylor series expansions, and time discretization is carried out through an implicit Crank–Nicolson integration scheme. The method proposed is validated through analysis on several classic seventh-order integrable KdV equations, viz., the Kaup–Kuperschmidt, Lax, and Sawada–Kotera equations. The numerical results exhibit very good agreement with analytical expectations and demonstrate uniformly lower maximum error norms than existing methods. These findings confirm the validity and efficiency of the nonic B-spline collocation scheme for simulating strongly dispersive wave propagation modeled by complex nonlinear partial differential equations.