Applications of the general similarity solution of the heat equation to boundary-value problems

Abstract

It is shown that the initial-value problem for the well-known fundamental solution of the heat equation for an infinite bar is invariant under a three-parameter Lie group. This leads to the solution in an elegant fashion. An inverse Stefan problem for the melting of a finite bar is considered. Analytical solutions are obtained for a two-parameter class of moving boundaries, extending the previous work of R. W. Sanders and D. Langford. A new solution expressible in terms of a Fourier series is derived for a phase change boundary moving at a constant velocity.