Numerical solution of a free boundary problem from heat transfer by Legendre Wavelets

In this paper we reduce a free boundary problem from heat transfer to a weakly Singular Volterra integral equation of the first kind. Since the solution of the partial differential Equation and associated integral equation is not continuous in general and the first kind integral equation is ill posed, we apply the Legendre wavelets to solve the integral equation. Numerical implementation of the method is illustrated by benchmark problems originated from heat transfer.