The Numerical Approximation of Blow-Up Times for Fractional Reaction-Diffusion Equations

Abstract

We investigate the numerical estimation of blow-up phenomena of the space fractional reaction-diffusion equation \[ \partial_t u +(-\Delta)^{\alpha/2}u=f(u), \quad x \in \Omega, t>0 \] with non-negative initial and Dirichlet boundary conditions. First, we consider the full discretization of the fractional equation using the already existing novel and accurate finite difference method for the fractional operator. Next, we implement an auxiliary function $H$ to the blow-up. The numerical blow-up times are computed for the fractional reaction-diffusion equation with the reaction term $f(u)=u^2$ and $f(u)=e^u$. Convergence results are proven. Moreover, the numerical blow-up time computed for the fractional reaction-diffusion equation with $\alpha \to 2$ is compared with the numerical blow-up time for the classical reaction-diffusion equation with $\alpha=2$, and consistent results are obtained.