An Efficient Technique for Time-Fractional Water Dynamics Arising in Physical Systems Pertaining to Generalized Fractional Derivative Operators

This study proposed the q˜-homotopy analysis transform method (q˜-HATM) as a revolutionary mathematical method for addressing nonlinear time-fractional Boussinesq and approximate long wave dynamics models with the Caputo and Atangana–Baleanu fractional derivatives in the Caputo sense. Through a specific velocity distribution, these models play an essential role in explaining the physics of wave propagation. The q˜-HATM is a new improvement to the Elzaki transform (ET) that simplifies the computations. The presented scheme addresses computational complexity by avoiding the use of Adomian and He’s polynomials, which is a distinguishing feature of this innovative methodology over decomposition and the homotopy perturbation transform method. The convergence analysis and error analyses are carried out in the current investigation for the upcoming strategy. We provide illustrations to exemplify the suggested system’s strength and efficacy, and the error estimates are described to ensure reliability. The analytical and graphic illustrations show that the projected methodology is numerically very precise and pragmatic to analyze the solution of fractional associated dynamics that arise in physics and engineering.