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

Abstract

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.

References in corpus (1)

• Analytical properties and applications of the Wright function