Similarity solution for the heat equation

Summary

A similarity solution of the heat equation is a solution whose profile keeps its shape in time. The profile only stretches like $\sqrt{t}$ and changes height like a power of $t$. Up to a power of $t$, it depends on $x$ and $t$ only through the similarity variable $\eta = x / (2\sqrt{\alpha t})$. The heat equation then becomes an ordinary differential equation in $\eta$. The heat kernel and the error function profiles of a suddenly heated half-space are the main examples. It is the case of the heat equation of a general similarity solution, and it also gives the classical Neumann solution of the Stefan problem. See the definition.