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September  2010, 9(5): 1209-1220. doi: 10.3934/cpaa.2010.9.1209

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

Adriana C. Briozzo 1, , María F. Natale 2, and  Domingo A. Tarzia 1,

1. 

Depto. de Matemática and CONICET, FCE, Universidad Austral, Paraguay 1950, S2000FZF Rosario, Argentina, Argentina
2. 

Depto. de Matemática , FCE, Universidad Austral, Paraguay 1950, S2000FZF Rosario, Argentina

Received  August 2009 Revised  September 2009 Published  May 2010

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: phase-change process, Stefan problem, fusion, similarity solution., solidification, free boundary problem, moving boundary problem, nonlinear thermal conductivity.
Mathematics Subject Classification: Primary: 35R35, 80A22, 35C0.
Citation: Adriana C. Briozzo, María F. Natale, Domingo A. Tarzia. The Stefan problem with temperature-dependent thermal conductivity and a convective term with a convective condition at the fixed face. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1209-1220. doi: 10.3934/cpaa.2010.9.1209
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