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

TL;DR

This paper proves that solutions of discrete semi-linear space-time fractional evolution equations converge to their continuous counterparts, with uniform convergence in time, supported by numerical simulations.

Contribution

It establishes the convergence of solutions from discrete fractional Laplacian equations to continuous ones, a novel result in fractional PDE analysis.

Findings

Solutions of discrete equations converge to continuous solutions

Convergence is uniform in time on compact sets

Numerical simulations support theoretical results

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.