Numerical approach to fractional blow-up equations with Atangana-Baleanu derivative in Riemann-Liouville sense

Abstract

In this paper, we consider a numerical approach for fourth-order time fractional partial differential equation. This equation is obtained from the classical reaction-diffusion equation by replacing the first-order time derivative with the Atangana-Baleanu fractional derivative in Riemann-Liouville sense with the Mittag-Leffler law kernel, and the first, second, and fourth order space derivatives with the fourth-order central difference schemes. We also suggest the Fourier spectral method as an alternate approach to finite difference. We employ Plais Fourier method to study the question of finite-time singularity formation in the one-dimensional problem on a periodic domain. Our bifurcation analysis result shows the relationship between the blow-up and stability of the steady periodic solutions. Numerical experiments are given to validate the effectiveness of the proposed methods.