Initial-to-Interface Maps for the Heat Equation on Composite Domains

TL;DR

This paper constructs a map linking initial conditions to interface values for the heat equation on composite domains, enabling a new approach to solving interface problems by converting them into boundary value problems.

Contribution

It introduces an initial-to-interface map for the heat equation on domains with multiple interfaces, providing an alternative method to traditional closed-form solutions.

Findings

The map exists for finite and infinite domains with multiple interfaces.

It facilitates transforming interface problems into boundary value problems.

Provides a new analytical tool for heat equation on composite domains.

Abstract

A map from the initial conditions to the values of the function and its first spatial derivative evaluated at the interface is constructed for the heat equation on finite and infinite domains with $n$ interfaces. The existence of this map allows changing the problem at hand from an interface problem to a boundary value problem which allows for an alternative to the approach of finding a closed-form solution to the interface problem.