Convergence of solutions of discrete semi-linear space-time fractional evolution equations

Let $(-\Delta)_c^s$ be the realization of the fractional Laplace operator on the space of continuous functions $C_0(\mathbb{R})$, and let $(-\Delta_h)^s$ denote the discrete fractional Laplacian on $C_0(\mathbb{Z}_h)$, where $0<s<1$ and $\mathbb{Z}_h:=\{hj:\; j\in\mathbb{Z}\}$ is a mesh of fixed size $h>0$. We show that solutions of fractional order semi-linear Cauchy problems associated with the discrete operator $(-\Delta_h)^s$ on $C_0(\mathbb{Z}_h)$ converge to solutions of the corresponding Cauchy problems associated with the continuous operator $(-\Delta)_c^s$. In addition, we obtain that the convergence is uniform in $t$ in compact subsets of $[0,\infty)$. We also provide numerical simulations that support our theoretical results.