Approximate Localized Waves and Residual Diagnostics for a Damped Nonplanar Time–Fractional Korteweg–de Vries Equation

Abstract

This paper studies a damped nonplanar time-fractional Korteweg–de Vries equation with Caputo memory, quadratic convection, dispersion, linear damping, and geometric attenuation. A family of localized analytical predictors is constructed by combining classical KdV profiles with
a fractional time deformation and a Mittag–Leffler/nonplanar attenuation law. Since the Caputo operator does not satisfy a classical product rule and the nonplanar coefficient is singular at the initial time, the resulting profiles are not claimed to be exact solutions. Instead, their consistency is assessed by explicit residual diagnostics on truncated time intervals. The Caputo derivative is evaluated by a graded-mesh L1 history formula, and Chebyshev reconstruction is also discussed as a complementary diagnostic tool. Representative planar, damped, and cylindrical benchmarks show that the residual remains small and coherent in planar regimes, while nonplanar geometry amplifies the early-time mismatch but becomes more controlled at later times. The approach provides a transparent residual-based framework for studying memory, damping, and geometric effects in fractional KdV dynamics.