A solution to the heat equation with a cubic moving boundary

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 $\phi$.