ABSTRACT Modeling nonlinear long‐wave motion becomes more challenging when the underlying medium exhibits memory effects or delayed temporal responses. Classical integer‐order models cannot capture these hereditary behaviors, which are observed in many natural wave processes. To address this limitation, we investigate the time‐fractional Whitham–Broer–Kaup (TFWBK) system, where memory is introduced through the Caputo fractional derivative. A numerical scheme based on Hermite wavelet collocation is utilized to solve TFWBK. This method converts the original nonlinear fractional system into a sparse algebraic structure that is efficient to compute and stable over long simulation intervals. Numerical tests show rapid convergence and high spectral accuracy for both classical and fractional cases, with errors remaining consistently small. The performance of the method is examined through comparison with several well established approaches including the B‐spline collocation, the variational iteration method (VIM), the Adomian decomposition method (ADM), and two versions of the optimal homotopy asymptotic method (OHAM1 and OHAM2). The results indicate that the present scheme offers improved accuracy while maintaining computational economy. Overall, the wavelet‐based framework provides a reliable and memory‐aware tool for simulating nonlinear dispersive wave dynamics described by the TFWBK model.
Nayak et al. (Fri,) studied this question.