Asymptotic behavior for the discrete in time heat equation

TL;DR

This paper analyzes the long-term behavior and decay rates of solutions to the discrete-time N-dimensional heat equation, providing convergence rates and optimal decay estimates using novel integral representations.

Contribution

It introduces a new approach based on the discrete fundamental solution and special functions to analyze asymptotic behavior, differing from continuous-time methods.

Findings

Established convergence rate to the fundamental solution

Proved optimal L^2-decay of solutions

Derived decay estimates in L^p spaces

Abstract

In this paper we investigate the asymptotic behavior and decay of the solution of the discrete in time $N$-dimensional heat equation. We give a convergence rate with which the solution tends to the discrete fundamental solution, and the asymptotic decay, both in $L^p(\R^N).$ Furthermore we prove optimal $L^2$-decay of solutions. Since the technique of energy methods is not applicable, we follow the approach of estimates based on the discrete fundamental solution which is given by an original integral representation and also by MacDonald's special functions. As a consequence, the analysis is different to the continuous in time heat equation and the calculations are rather involved.