Integral Equations for Heat Kernel in Compound Media

TL;DR

This paper derives integral equations for heat kernels in compound media using potential methods, providing explicit solutions in one dimension and establishing a basis for asymptotic analysis and applications to various interface configurations.

Contribution

It introduces a rigorous derivation of integral equations for heat kernels in compound media and solves them explicitly in one dimension, advancing the theoretical understanding of heat conduction across interfaces.

Findings

Explicit solutions for heat kernels in 1D cases

Rigorous derivation of multiple scattering expansions

Method applicable to complex interface configurations

Abstract

By making use of the potentials of the heat conduction equation the integral equations are derived which determine the heat kernel for the Laplace operator $-a^2\Delta$ in the case of compound media. In each of the media the parameter $a^2$ acquires a certain constant value. At the interface of the media the conditions are imposed which demand the continuity of the `temperature' and the `heat flows'. The integration in the equations is spread out only over the interface of the media. As a result the dimension of the initial problem is reduced by 1. The perturbation series for the integral equations derived are nothing else as the multiple scattering expansions for the relevant heat kernels. Thus a rigorous derivation of these expansions is given. In the one dimensional case the integral equations at hand are solved explicitly (Abel equations) and the exact expressions for the regarding heat kernels are obtained for diverse matching conditions. Derivation of the asymptotic expansion of the integrated heat kernel for a compound media is considered by making use of the perturbation series for the integral equations obtained. The method proposed is also applicable to the configurations when the same medium is divided, by a smooth compact surface, into internal and external regions, or when only the region inside (or outside) this surface is considered with appropriate boundary conditions.