Numerical Solutions and Pattern Formation Process in Fractional Diffusion-Like Equations

Abstract

Nowadays, a lot of researchers have challenged the use of classical diffusion equation to model real life situations. To circumvent some of the up-roaring challenges, time and space fractional derivatives have been proposed as alternative to model some anomalous diffusion or related processes where a particle plume spreads at inconsistent rate with the classical Brownian motion model. In this work, we shall consider the general diffusion equations with fractional order derivatives which describe the diffusion in complex systems. Fractional diffusion equation is obtained by allowing the exponent order \(\alpha \) to vary in the intervals (0, 1) and (1, 2) which correspond to subdiffusion and superdiffusion special cases. For the numerical approximations, we propose to use the newly correct version of the Adams-Bashforth scheme which takes into account the nonlinearity of the kernels such as the Mittag-Leffler law for the Atangana-Baleanu case, the power law for the Riemann-Liouville and Caputo derivatives. The efficiency and accuracy of the numerical schemes based on these operators will be justified by reporting their norm infinity and norm relative errors. The complexity of the dynamics in the equations will be discussed theoretically by examining their local and global stability analysis. Our numerical experiment results are expected to give a new direction into pattern formation process in fractional diffusion-like scenarios.