Damped time-fractional KdV-type waves: a residual-minimization approach with Chebyshev–Lobatto corrections

Fractional non-linear wave equations provide refined models for dispersive media with simultaneous damping and hereditary memory, including a wide class of plasma and fluid systems. Motivated by the need for physically interpretable approximations in such settings, this work analyzes a damped time-fractional Korteweg-de Vries (KdV)-type equation in the Caputo sense. The model couples non-linear steepening, dispersion, and collisional damping with a non-local temporal memory kernel, making exact solutions rare and fully discrete schemes difficult to interpret analytically. To address this gap, we develop a residual-minimization framework that starts from a transformed solitary wave solution of the associated unforced integer-order damped KdV equation and uses it as a theorem-based analytical seed. The seed already incorporates collisional damping, but it does not satisfy the fractional evolution. Therefore, we introduce a low-dimensional cubic Chebyshev–Lobatto (CL) correction with three free parameters, constructed explicitly in terms of the interpolation basis and its mapping to the final ansatz coefficients. These parameters are determined by minimizing a strong space–time residual functional of the original fractional model on a finite domain. The resulting approximation remains localized, maintains a direct connection to the classical KdV profile, and, at the same time, adapts to the memory-induced modulation of amplitude and shape. The quality of the optimized profile is quantified through global and instantaneous residual indicators, endpoint residual diagnostics, and an independent Caputo-L1/Fourier pseudospectral benchmark equipped with a grid-refinement protocol. The computations show how the fractional order and damping coefficient jointly control attenuation, apparent stationarity, and splitting of the non-linear structures. They illustrate the practical range of the proposed residual-based methodology for damped fractional dispersive waves on finite time intervals.