Nonlinear Wave Dynamics in a Time-Fractional Gardner Model with Power-Law Memory

This study investigates a time-fractional Gardner equation formulated with the Liouville–Caputo derivative and its weakly singular power-law (PL) memory kernel. Approximate analytical solutions are constructed using the q-Homotopy Analysis Transform Method (q-HATM) through the fourth truncation order. For the adopted normalization H PL = 1, the terminal truncated approximation depends on the auxiliary parameters through the effective ratio λ = h/n. Fractional time-derivative h-curves and residual measures are used as finite-order sensitivity and equation-satisfaction diagnostics rather than as proofs of convergence or stability. The influence of fractional memory is examined for fractional orders from 0.6 to 1. At τ = 10, the transition-front position changes from 0.099991 in the classical case to 0.044551 at the lowest fractional order considered, whereas the transition width and maximum spatial slope vary by less than approximately 0.005% and 0.007%, respectively. In the classical limit, the sampled maximum error decreases from 1.10 × 10 -4 at the first truncation order to 1.18 × 10 -7 at the fourth. Independent validation using a fully implicit L1 finite-difference scheme gives maximum discrepancies of 3.04 × 10 -7 for the principal fractional case and 1.56 × 10 -7 for an additional test case. These results indicate that, for the parameter regime examined, powerlaw memory primarily delays wave-front propagation while largely preserving the transition shape. Future work may extend the analysis to broader parameter regimes and related nonlinear dispersive models, together with rigorous convergence and error-bound investigations.