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Communications on Pure and Applied Analysis (CPAA)
 

The Stefan problem with temperature-dependent thermal conductivity and a convective term with a convective condition at the fixed face

Pages: 1209 - 1220, Volume 9, Issue 5, September 2010      doi:10.3934/cpaa.2010.9.1209

 
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Adriana C. Briozzo - Depto. de Matemática and CONICET, FCE, Universidad Austral, Paraguay 1950, S2000FZF Rosario, Argentina (email)
María F. Natale - Depto. de Matemática , FCE, Universidad Austral, Paraguay 1950, S2000FZF Rosario, Argentina (email)
Domingo A. Tarzia - Depto. de Matemática and CONICET, FCE, Universidad Austral, Paraguay 1950, S2000FZF Rosario, Argentina (email)

Abstract: We study a one-phase Stefan problem for a semi-infinite material with temperature-dependent thermal conductivity and a convective term with a convective boundary condition at the fixed face $x=0$. We obtain sufficient conditions for data in order to have a parametric representation of the solution of the similarity type for $t \geq t_0 > 0$ with $t_0 $ an arbitrary positive time. We obtain explicit solutions through the unique solution of a Cauchy problem with the time as a parameter and we also give an algorithm in order to compute the explicit solution.

Keywords:  Stefan problem, free boundary problem, moving boundary problem, phase-change process, nonlinear thermal conductivity, fusion, solidification, similarity solution.
Mathematics Subject Classification:  Primary: 35R35, 80A22, 35C05.

Received: August 2009;      Revised: September 2009;      Published: May 2010.