Modified Cubic Spline Approach for the Numerical Solution of the Fractional Order Generalized Wave Type Equation

The present study examines the numerical simulation of the generalized fractional-order wave-type Regularized Long Wave (RLW) equation, an advanced model for characterizing nonlinear dispersive wave phenomena in shallow water dynamics, plasma physics, and other fluid-like media. The incorporation of fractional-order derivatives enriches the RLW framework by capturing memory effects and nonlocal interaction features that classical integer-order formulations cannot adequately represent. The proposed model is of great significance for simulating the behavior of complex wave phenomena arising in real-world systems, such as anomalous diffusion and nonlocal elasticity.

For the numerical solution of the wave-type RLW model, the spatial domain is discretized using modified cubic B-spline basis functions to obtain smooth and accurate approximations. At the same time, the temporal fractional derivative is handled through a finite difference scheme. The method is assessed under various mesh sizes and time-step combinations to analyze its convergence properties, error behavior, and computational efficiency. The study further demonstrates how variations in the fractional order affect both the qualitative and quantitative characteristics of the resulting wave profiles. The obtained numerical results will provide deep insight into the underlying mechanisms that control wave propagation in nonlocal environments, which classical integer-order models cannot reveal. Overall, this combined spline-difference framework shows strong promise for efficiently solving fractional nonlinear dispersive wave equations.