Nonlocal one-dimensional Fisher-Kolmogorov-Petrovskii-Piskunov equation with abnormal diffusion

Abstract

Analytical solutions are constructed for the nonlocal space fractional Fisher-Kolmogorov-Petrovskii- Piskunov equation with abnormaldiffusion. Such solutions allow us to describe quasi-steady state patterns. Special attention is given to the role of fractional derivative. Fractional diffusion equations are useful for applications in which a cloud of particles spreads faster than predicted by the classical equation. The resulting solutions spread faster than the classical solutions and may exhibit asymmetry, depending on the fractional derivative used. Results of numerical simulations and properties of analytical solutions are presented. Influence of the fractional derivative on patterns ordered in space and time is discussed.