Petrov–Galerkin finite element method for solving the time-fractional Rosenau–Hyman equation

This study explores the fractional-order Rosenau–Hyman equation, which models nonlinear wave dynamics in dispersive and memory-dependent media. A Petrov–Galerkin finite element method (PG-FEM) is employed, incorporating the Grünwald–Letnikov fractional derivative to efficiently capture long-range memory effects and non-local interactions. The proposed numerical scheme ensures high accuracy and computational efficiency, overcoming the limitations of traditional finite difference and spectral methods in handling compacton solutions and nonlinearities. Numerical simulations demonstrate that decreasing the fractional order (α < 1) ( α < 1 ) leads to broader wave dispersion and slower attenuation, mimicking physical behaviors observed in viscoelastic materials, fluid flow through porous media, and biological diffusion processes. The results further validate that fractional-order models outperform integer-order counterparts in accurately describing power-law wave decay and anomalous transport phenomena. These findings reinforce the necessity of fractional calculus in modeling wave propagation, biomechanics, and material science applications, paving the way for future research in adaptive computational techniques and experimental validation.