A solution to the heat equation with a cubic moving boundary

TL;DR

This paper presents a method to solve the heat equation with a cubic moving boundary by convolving the fundamental solution with a function solving a third-order ODE, providing a new approach to boundary problems.

Contribution

The paper introduces a novel convolution-based procedure linking heat equation solutions with cubic moving boundaries through a third-order ODE.

Findings

Derived explicit solution for heat equation with cubic boundary

Established connection between boundary solutions and third-order ODE

Proposed a general convolution method for moving boundary problems

Abstract

In this work we find a solution to problem of the heat equation which is annihiliated at a cubic boundary $f$. The solution turns out to be the convolution between the fundamental solution of the heat equation and a function $\phi$ which solves a third order ODE. However we believe that the main contribution is the procedure itself, which links in a rather straightforward way, solutions of the heat equation $v$ with moving boundaries $f$ through the convolution of the heat kernel with $\mathbb{C}^p$ funtions $\phi$.