Numerical Study of Time-Fractional Cattaneo Equation via the Atangana–Baleanu–Caputo Operator

Abstract

Fractional models play a vital role in describing diffusion and heat transfer processes with memory and nonlocal effects. In this paper, we develop a novel spectral collocation method for the time fractional Cattaneo equation involving the Atangana–Baleanu–Caputo (ABC) fractional derivative. The solution is approximated using two-dimensional expansions of shifted Legendre polynomials over the space–time domain, reducing the fractional problem to a system of algebraic equations. Numerical experiments demonstrate that the proposed Legendre spectral scheme yields excellent agreement with available analytical solutions and reduces the approximation error to nearly half compared with existing numerical approaches. These results confirm that the method is both reliable and efficient, offering a powerful tool for modeling diffusion processes governed by fractional Cattaneo dynamics.