Geodesic
Investigation of mathematical models of one-phase Stefan problems with unknown nonlinear coefficients
Eurasian mathematical journal, Tome 8 (2017) no. 3, pp. 48-59 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

One-phase models of inverse Stefan problems with unknown temperature-dependent convection coefficients are considered. The final observation is considered as an additional information on the solution of the direct Stefan problem. For such inverse problems we justify the corresponding mathematical statements allowing to determine coefficients multiplying the lowest order derivatives in quasilinear parabolic equations in a one-phase domain with an unknown moving boundary. On the basis of the duality principle conditions for the uniqueness of their smooth solution are obtained. The proposed approach allows one to clarity a relationship between the uniqueness property for coefficient inverse Stefan problems and the density property of solutions of the corresponding adjoint problems. It is shown that this density property follows, in turn, from the known inverse uniqueness for linear parabolic equations. 
@article{EMJ_2017_8_3_a5,
     author = {N. L. Gol'dman},
     title = {Investigation of mathematical models of one-phase {Stefan} problems with unknown nonlinear coefficients},
     journal = {Eurasian mathematical journal},
     pages = {48--59},
     year = {2017},
     volume = {8},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EMJ_2017_8_3_a5/}
}
TY  - JOUR
AU  - N. L. Gol'dman
TI  - Investigation of mathematical models of one-phase Stefan problems with unknown nonlinear coefficients
JO  - Eurasian mathematical journal
PY  - 2017
SP  - 48
EP  - 59
VL  - 8
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/EMJ_2017_8_3_a5/
LA  - en
ID  - EMJ_2017_8_3_a5
ER  - 
%0 Journal Article
%A N. L. Gol'dman
%T Investigation of mathematical models of one-phase Stefan problems with unknown nonlinear coefficients
%J Eurasian mathematical journal
%D 2017
%P 48-59
%V 8
%N 3
%U http://geodesic.mathdoc.fr/item/EMJ_2017_8_3_a5/
%G en
%F EMJ_2017_8_3_a5
N. L. Gol'dman. Investigation of mathematical models of one-phase Stefan problems with unknown nonlinear coefficients. Eurasian mathematical journal, Tome 8 (2017) no. 3, pp. 48-59. http://geodesic.mathdoc.fr/item/EMJ_2017_8_3_a5/

[1] N. Zabaras, Y. Ruan, “A deforming finite element method analysis of inverse Stefan problem”, Int. J. Numer. Meth. Eng., 28 (1989), 295–313 | DOI | MR | Zbl

[2] Y. Rabin, A. Shitzer, “Combined solution of the inverse Stefan problem for successive freezing-thawing in nonideal biological tissues”, J. Biomech. Eng. Trans. ASME, 119 (1997), 146–152 | DOI

[3] A. El. Badia, F. Moutazaim, “A one-phase inverse Stefan problem”, Inverse Probl., 15:6 (1999), 1507–1522 | DOI | MR | Zbl

[4] H.-W. Engl, “Identification of heat transfer functions in continuous casting of steel by regularization”, Inverse and Ill-Posed Probl., 8 (2000), 677–693 | MR | Zbl

[5] B. Furenes, B. Lie, “Solidification and control of a liquid metal column”, Simulation Model. Practic and Theory., 14:8 (2006), 1112–1120 | DOI

[6] D. Slota, “Homotopy perturbation method for solving the two-phase inverse Stefan problem”, Numer. Heat Transfer Part A, 59:10 (2011), 755–768 | DOI

[7] B. T. Johansson, D. Lesnic, T. Reeve, “A method of fundamental solutions for one-dimensional inverse Stefan problem”, Appl. Math. Model., 35:9 (2011), 4367–4378 | DOI | MR | Zbl

[8] N. N. Salva, D. A. Tarzia, “Simultaneous determination of unknown coefficients through a phase-change process with temperature-dependent thermal conductivity”, J.P. J. Heat Mass Transfer, 5:1 (2011), 11–39 | MR | Zbl

[9] N. L. Gol'dman, Inverse Stefan problems, Kluwer Academic, Dordrecht, 1997 | MR | Zbl

[10] N. L. Gol'dman, “Properties of solutions of the inverse Stefan problem”, Diff. Equations, 39:1 (2003), 66–72 | DOI | MR | Zbl

[11] N. L. Gol'dman, “One-phase inverse Stefan problems with unknown nonlinear sources”, Diff. Equations, 49:6 (2013), 680–687 | DOI | MR | Zbl

[12] N. L. Gol'dman, “Properties of solutions of parabolic equations with unknown coeffcients”, Diff. Equations, 47:1 (2011), 60–68 | DOI | Zbl

[13] M. Lees, M. H. Protter, “Unique continuation for parabolic differential equations and inequalities”, Duke Math. J., 28 (1961), 369–383 | DOI | MR

[14] A. Friedman, Partial differential equations of parabolic type, Prentice Hall, Englewood Cliffs, N.J., 1964 ; Mir, M., 1968 (in Russian) | MR | Zbl | Zbl

[15] Am. Math. Soc., Providence, R.I., 1968 | Zbl