Solving the heat equation with variable thermal conductivity

TL;DR

This paper presents an explicit solution method for the heat equation with spatially variable thermal conductivity using the Unified Transform Method, enabling eigenvalue computation and solution analysis for complex variable-coefficient problems.

Contribution

It introduces a novel explicit solution representation for variable-coefficient heat equations using the Fokas method, applicable to general second-order problems.

Findings

Derived explicit solution representations for variable-coefficient heat equations.

Established a method to compute eigenvalues as roots of transcendental functions.

Demonstrated the approach's applicability to broader variable-coefficient PDEs.

Abstract

We consider the heat equation with spatially variable thermal conductivity and homogeneous Dirichlet boundary conditions. Using the Method of Fokas or Unified Transform Method, we derive solution representations as the limit of solutions of constant-coefficient interface problems where the number of subdomains and interfaces becomes unbounded. This produces an explicit representation of the solution, from which we can compute the solution and determine its properties. Using this solution expression, we can find the eigenvalues of the corresponding variable-coefficient eigenvalue problem as roots of a transcendental function. We can write the eigenfunctions explicitly in terms of the eigenvalues. The heat equation is the first example of more general variable-coefficient second-order initial-boundary value problems that can be solved using this approach.