Convergence and Divergence of Approximations in terms of the Derivatives of Heat Kernel

TL;DR

This paper investigates the convergence behavior of approximations to the heat equation derived from derivatives of the heat kernel, highlighting conditions for convergence or divergence based on initial data and time.

Contribution

It provides new conditions under which these derivative-based approximations converge or diverge, and clarifies their meaning through eigenfunction comparison.

Findings

Convergence occurs when initial data is Gaussian and t > t0.

Divergence occurs when t < t0 for Gaussian initial data.

An $L^ abla$-error estimate is established and the approximation's meaning is clarified.

Abstract

We consider an approximate solution to the heat equation which consists of the derivatives of heat kernel. Some conditions in the initial value, under which the approximation converges to the solution of the heat equation or diverges when the number of terms of the approximation goes to infinity with a fixed time $t$, will be given. For example, when the initial data is a Gaussian $e^{-|{\bf x}|^2/4t_0}$, the approximation converges when $t>t_0$. But if $t<t_0$, it diverges to infinity. Also the $L^\infty$-error estimate will be given and the meaning of the approximation will be clarified by comparing with eigenfunction expansion.