Space-time fractional modeling of memory and non-local effects in non-linear shallow water hydrodynamics

Abstract

This study develops a generalized analytical framework for Space-Time Fractional Shallow Water Equations (ST-FSWE) to characterize the combined influence of memory effects and non-local dynamics in open-channel hydrodynamics. Formulated in the Caputo sense, the coupled nonlinear system yields a convergent recursive algebraic scheme via the Dual-Fractional Differential Transform Method (DF-DTM). The proposed framework is examined using three benchmark configurations: the Fractional Parabolic Bowl, the Fractional Sine Wave, and the Fractional Gaussian Hill. For the Parabolic Bowl, the model recovers the classical integer-order behavior in the limiting case α  =  β  = 1. Across the three benchmark problems, the numerical investigations demonstrate that the temporal fractional order governs the memory-dependent evolution of the analytical flow solutions, whereas the spatial fractional order modifies their non-local spatial behavior. Furthermore, the Gaussian Hill benchmark reveals that the finite-order analytical approximation exhibits truncation-order dependence during the later stages of the evolution owing to the cumulative contribution of the retained higher-order correction terms. Overall, the proposed formulation provides a mathematically consistent analytical framework for investigating memory-dependent and non-local shallow water flow behavior while preserving consistency with the classical shallow water equations in the integer-order limit.