High-Precision and Stability-Preserving Approximations to the Time-Fractional Harry Dym Model Using the Tantawy Technique

Fractional differential equations provide a flexible framework for describing evolutionary processes in complex media, where nonlocality and memory effects play central roles, and classical integer-order models are frequently inadequate to capture these behaviors. In this work, we revisit the time-fractional Harry Dym (HD) evolution equation in the Caputo sense and construct high-precision analytical approximations using the recently developed Tantawy technique (TT). The method generates a rapidly convergent fractional-power series in time without resorting to perturbative assumptions, auxiliary decomposition polynomials, linearization procedures, or integral transforms, and it remains computationally economical even at high approximation orders. Closed, compact expressions are derived up to the fifth-order approximation and can be systematically extended, yielding excellent agreement with the known exact solution of the classical/integer HD model and with approximations obtained via the new iterative method. A detailed error analysis is carried out by computing absolute and maximum residual errors over the entire computational domain, demonstrating the accuracy, stability, and robustness of the TT for the HD-type fractional nonlinear evolution equation. From a physical perspective, the proposed framework offers a reliable tool for modeling nonlinear wave structures in dispersive media with significant memory and, more generally, for treating a broad class of fractional nonlinear wave equations arising in physics and engineering.