Explicit solutions for a one-phase Stefan problem with temperature-dependent thermal conductivity
Bollettino della Unione matematica italiana, Série 8, 9B (2006) no. 1, pp. 79-99.

Voir la notice de l'article dans Biblioteca Digitale Italiana di Matematica

Si studia un problema di Stefan a una fase per un materiale semi-infinito con un coefficiente di conduttività termica dipendente dalla temperatura e con una condizione di temperatura costante o un flusso di calore del tipo $-q_0/\sqrt{t}$ ($q_0 > 0$) sulla faccia fissa $x=0$ . Si ottengono, in entrambi i casi, condizioni sufficienti per i dati in modo da avere una rappresentazione parametrica della soluzione di tipo similarità per $t\geq t_0>0$ con $t_0$ un tempo positivo arbitrario. Queste soluzioni esplicite sono ottenute attraverso l’unica soluzione di una equazione integrale dove il tempo è un parametro. 
@article{BUMI_2006_8_9B_1_a4,
     author = {Natale, Mar{\'\i}a F. and Tarzia, Domingo A.},
     title = {Explicit solutions for a one-phase {Stefan} problem with temperature-dependent thermal conductivity},
     journal = {Bollettino della Unione matematica italiana},
     pages = {79--99},
     publisher = {mathdoc},
     volume = {Ser. 8, 9B},
     number = {1},
     year = {2006},
     zbl = {1118.80005},
     mrnumber = {1702131},
     language = {it},
     url = {https://geodesic-test.mathdoc.fr/item/BUMI_2006_8_9B_1_a4/}
}
TY  - JOUR
AU  - Natale, María F.
AU  - Tarzia, Domingo A.
TI  - Explicit solutions for a one-phase Stefan problem with temperature-dependent thermal conductivity
JO  - Bollettino della Unione matematica italiana
PY  - 2006
SP  - 79
EP  - 99
VL  - 9B
IS  - 1
PB  - mathdoc
UR  - https://geodesic-test.mathdoc.fr/item/BUMI_2006_8_9B_1_a4/
LA  - it
ID  - BUMI_2006_8_9B_1_a4
ER  - 
%0 Journal Article
%A Natale, María F.
%A Tarzia, Domingo A.
%T Explicit solutions for a one-phase Stefan problem with temperature-dependent thermal conductivity
%J Bollettino della Unione matematica italiana
%D 2006
%P 79-99
%V 9B
%N 1
%I mathdoc
%U https://geodesic-test.mathdoc.fr/item/BUMI_2006_8_9B_1_a4/
%G it
%F BUMI_2006_8_9B_1_a4
Natale, María F.; Tarzia, Domingo A. Explicit solutions for a one-phase Stefan problem with temperature-dependent thermal conductivity. Bollettino della Unione matematica italiana, Série 8, 9B (2006) no. 1, pp. 79-99. https://geodesic-test.mathdoc.fr/item/BUMI_2006_8_9B_1_a4/

[1] V. Alexiades - A. D. Solomon, Mathematical modeling of melting and freezing processes, Hemisphere - Taylor & Francis, Washington (1983).

[2] I. ATHANASOPOULOS - G. MAKRAKIS - J. F. RODRIGUES (EDS.), Free Boundary Problems: Theory and Applications, CRC Press, Boca Raton (1999). | MR

[3] J. R. Barber, An asymptotic solution for short-time transient heat conductionbetween two similar contacting bodies, Int. J. Heat Mass Transfer, 32, No 5 (1989),943-949.

[4] D. A. Barry - G. C. Sander, Exact solutions for water infiltration with an arbitrary surface flux or nonlinear solute adsorption, Water Resources Research, 27, No 10 (1991), 2667-2680.

[5] G. Bluman - S. Kumei, On the remarkable nonlinear diffusion equation, J. Math. Phys., 21 (1980), 1019-1023. | fulltext (doi) | MR | Zbl

[6] A. C. Briozzo - M. F. Natale - D. A. Tarzia, Determination of unknown thermal coefficients for Storm’s-type materials through a phase-change process, Int. J.Non-Linear Mech., 34 (1999), 324-340. | fulltext (doi) | MR

[7] P. Broadbridge, Non-integrability of non-linear diffusion-convection equationsin two spatial dimensions, J. Phys. A: Math. Gen., 19 (1986), 1245-1257. | MR | Zbl

[8] P. Broadbridge, Integrable forms of the one-dimensional flow equation for unsaturated heterogeneous porous media, J. Math. Phys., 29 (1988), 622-627. | fulltext (doi) | MR | Zbl

[9] J. R. Cannon, The one-dimensional heat equation, Addison - Wesley, Menlo Park (1984). | fulltext (doi) | MR | Zbl

[10] H. S. Carslaw - J. C. Jaeger, Conduction of heat in solids, Oxford University Press, London (1959). | MR | Zbl

[11] J. M. CHADAM - H. RASMUSSEN H. (EDS.), Free boundary problems involving solids, Pitman Research Notes in Mathematics Series 281, Longman, Essex (1993). | MR

[12] M. N. Coelho Pinheiro, Liquid phase mass transfer coefficients for bubbles growing in a pressure field: a simplified analysis, Int. Comm. Heat Mass Transfer,27, No 1 (2000), 99-108.

[13] J. Crank, Free and moving boundary problems, Clarendon Press, Oxford (1984). | MR | Zbl

[14] J. I. DIAZ - M. A. HERRERO - A. LIÑAN - J. L. VAZQUEZ (Eds.), Free boundary problems: theory and applications, Pitman Research Notes in Mathematics Series323, Longman, Essex (1995).

[15] A. Fasano - M. Primicerio (Eds.), Nonlinear diffusion problems, Lecture Notes in Math., N. 1224, Springer Verlag, Berlin (1986). | fulltext (doi) | MR

[16] A. S. Fokas - Y. C. Yortsos, On the exactly solvable equation $S_t = [(\beta S + \gamma)^{-2} S_x]_x + \alpha(\beta S + g)^{-2} Sx$ occurring in two-phase flow in porous media, SIAM J. Appl. Math., 42, No 2 (1982), 318-331. | fulltext (doi) | MR

[17] N. KENMOCHI (Ed.), Free Boundary Problems: Theory and Applications, I,II, Gakuto International Series: Mathematical Sciences and Applications, Vol. 13, 14, Gakkotosho, Tokyo (2000). | MR

[18] J. H. Knight - J. R. Philip, Exact solutions in nonlinear diffusion, J. Engrg. Math., 8 (1974), 219-227. | fulltext (doi) | MR | Zbl

[19] G. Lamé - B. P. Clapeyron, Memoire sur la solidification par refroidissementd’un globe liquide, Annales Chimie Physique, 47 (1831), 250-256.

[20] V. J. Lunardini, Heat transfer with freezing and thawing, Elsevier, Amsterdam (1991)

[21] A. Munier - J. R. Burgan - J. Gutierrez - E. Fijalkow - M. R. Feix, Group transformations and the nonlinear heat diffusion equation, SIAM J. Appl. Math., 40,No 2 (1981), 191-207. | fulltext (doi) | MR | Zbl

[22] M. F. Natale - D. A. Tarzia, Explicit solutions to the two-phase Stefan problemfor Storm-type materials, J. Phys. A: Math. Gen., 33 (2000), 395-404. | fulltext (doi) | MR | Zbl

[23] M. F. Natale - D. A. Tarzia, Explicit solutions to the one-phase Stefan problemwith temperature-dependent thermal conductivity and a convective term, Int. J.Engng. Sci., 41 (2003), 1685-1698. | fulltext (doi) | MR | Zbl

[24] R. Philip, General method of exact solution of the concentration-dependent diffusion equation, Australian J. Physics, 13 (1960), 1-12. | MR | Zbl

[25] A. D. Polyanin - V. V. Dil’Man, The method of the «carry over» of integral transforms in non-linear mass and heat transfer problems, Int. J. Heat Mass Transfer, 33, No 1 (1990), 175-181

[26] C. Rogers, Application of a reciprocal transformation to a two-phase Stefan problem, J. Phys. A: Math. Gen., 18 (1985), 105-109. | MR

[27] C. Rogers, On a class of moving boundary problems in non-linear heat condition: Application of a Bäcklund transformation, Int. J. Non-Linear Mech., 21 (1986), 249-256. | fulltext (doi) | MR | Zbl

[28] C. Rogers - P. Broadbridge, On a nonlinear moving boundary problem with heterogeneity: application of reciprocal transformation, Journal of Applied Mathematics and Physics (ZAMP), 39 (1988), 122-129. | fulltext (doi) | MR | Zbl

[29] G. C. Sander - I. F. Cunning - W. L. Hogarth - J. Y. Parlange, Exact solution fornonlinear nonhysteretic redistribution in vertical soli of finite depth, Water Resources Research, 27 (1991), 1529-1536.

[30] D. A. Tarzia, An inequality for the coefficient $\sigma$ of the free boundary $s(t) = 2\sigma\sqrt{t}$ of the Neumann solution for the two-phase Stefan problem, Quart. Appl. Math., 39(1981), 491-497. | fulltext (doi) | MR

[31] D. A. Tarzia, A bibliography on moving - free boundary problems for the heat-diffusion equation. The Stefan and related problems, MAT-Serie A #2 (2000) (with 5869 titles on the subject, 300 pages). See www.austral.edu.ar/MAT-SerieA/2(2000)/ | MR

[32] P. Tritscher - P. Broadbridge, A similarity solution of a multiphase Stefan problem incorporating general non-linear heat conduction, Int. J. Heat Mass Transfer, 37, No 14 (1994), 2113-2121. | Zbl