ISSN 15154904
SERIE A: CONFERENCIAS, SEMINARIOS Y TRABAJOS DE MATEMÁTICA
No. 2
A BIBLIOGRAPHY ON MOVINGFREE BOUNDARY
PROBLEMS FOR THE HEATDIFFUSION EQUATION.
THE STEFAN AND RELATED PROBLEMS
 Domingo Alberto TARZIA
Departamento de Matemática – CONICET,
Facultad de Ciencias Empresariales, Universidad Austral,
Paraguay 1950, S2000FZF Rosario, ARGENTINA.
Domingo.Tarzia@fce.austral.edu.ar
Rosario, octubre de 1999
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ABSTRACT
We present a bibliography on moving and free boundary problems for the heatdiffusion equation, particularly regarding the Stefan and related problems.
It contains 5869 titles referring to 588 scientific Journals, 122 books, 88 symposia (having at least 3 contributions on the subject), 30 collections, 59 thesis and 247 technical reports.
It tries to give a comprehensive account of the western existing mathematicalphysicalengineering literature on this research field.
RESUMEN
Se presenta una bibliografía sobre problemas de frontera móvil y libre para la ecuación del calordifusión, en particular sobre el problema de Stefan y problemas relacionados.
Contiene 5869 títulos en 588 revistas científicas, 122 libros, 88 simposios (teniendo al menos 3 contribuciones en el tema), 30 colecciones, 59 tesis y 247 reportes técnicos o prepublicaciones.
Se da un informe amplio de la bibliografía matemática, física e ingenierías existente en occidente sobre este tema de investigación.
Primary Mathematics Subject Classification Number (*): 35R35, 80A22
Secondary Mathematics Subject Classification Number (*): 35B40, 35C05, 35C15, 35Kxx, 35R30, 46N20, 49J20, 65Mxx, 65Nxx, 76R50, 76S05, 76T05, 93C20.
(*) Following the 1991 Mathematics Subject Classification compiled by Mathematical Reviews and Zentralblatt fur Mathematik.
Primary key words: Enthalpy formulation or method, Filtration, Free boundary problems, Freezing, Melting, Moving boundary problems, Mushy region, Phasechange problem, Solidification, Stefan problem.
Secondary key words: Continuous mechanics, Diffusion process, Functional analysis, Heat conduction, Mathematical methods, Numerical methods, Partial differential equations, Variational inequalities, Weak solutions.
Palabras claves primarias: Método o formulación en entalpía, Filtración, Problemas de frontera libre, Congelación, Derretimiento, Problemas de frontera móvil, Región pastosa, Problema de cambio de fase, Solidificación, Problema de Stefan.
Palabras claves secundarias: Mecánica del continuo, Procesos difusivos, Análisis funcional, Conducción del calor, Métodos matemáticos, Métodos numéricos, Ecuaciones diferenciales a derivadas parciales, Inecuaciones variacionales, Soluciones débiles.
El manuscrito fue recibido y aceptado en octubre de 1999.
I. INTRODUCTION
This personal bibliography on moving and free boundary problems (MFBP) for the heatdiffusion equation (HDE) contains about 5900 references to works appeared on approximately 884 different kinds of publications. It tries to give a comprehensive account of the western existing mathematicalphysicalengineering literature on this research field.
Almost all the material on the subject, published after the historical and first paper of LaméClapeyron (1831), has been collected. Sources include scientific journals, symposium or conference proceedings, technical reports and books.
References quoted in Tarzia (198l, 1984, 1988) (773, 936 and 2528 references in the years 1981, 1984 and 1988 respectively) are included in the present bibliography.
This issue is a document to be uptodated by addition of new references with the purpose of making a database and maybe a classification similar to the one done in the 1981 bibliography.
All the papers are directly related to some aspects of the MFBP for the HDE, particularly regarding the phasechange process known in the literature as Stefan problem (we remark that a more appropriate name would be LaméClapeyronStefan problem). They are concerned with theoretical, numerical and experimental methods and also with various possible applications.
Together with the term “Stefan problem”, the term phasechange problem, melting or freezing problem, fusion or solidification problem, moving or free boundary problem, Stefanlike problem are used, according to the particular field being studied.
The author’s purpose in writing this bibliography is to provide usable information in the field of MFBP for the HDE both for the theoretical and the applied aspects.
The collection of titles began in 1977. As a result of the systematic organization of the material accumulated, a first bibliography (with 773 references) on MFBP for the HDE appeared in 1981. These 644 papers (of the quoted 773 titles) were classified into three main branches (theoretical, numerical and experimental) each containing several subsections, according to the following plan:
I. Moving boundaryproblems for the heat equation
I.1. Onedimensional case
I.2. Multidimensional case
I.3. Physical applications
I.4. Application to free boundary problems
II. Free boundary problems for the heat equation
II.1. Free boundary problems of Stefan type
II.1.1. Onedimensional case
II.1.1.1. Onephase problem (theoretical, numerical methods and applications)
II.1.1.2. Twophase problem (theoretical, numerical methods and applications)
II.1.2. Multidimensional case
II.1.2.1. Onephase problem (theoretical, numerical methods and applications)
II.1.2.2. Twophase problem (theoretical, numerical methods and applications)
II.1.3. Other generalities
II.1.3.1. Free boundary problems in a gaseous state
II.1.3.2. Experimental works
II.1.3.3. Solidliquid interphase
II.1.3.4. Other applications
II.2. Free boundary problems not of stefan type
II.2.1. Diffusionconsumption of oxygen in absorbing tissue
II.2.2. Flow of two immiscible fluids in a porous medium
II.2.3. Movement of a compressible fluid through a porous medium
II.2.4. Impact of a viscoplastic bar on a rigid obstacle
II.2.5. Chemical reactions between two substances
II.2.6. Other free boundary problems for the heat equation
II.2.6.1. Of an implicit type
II.2.6.2. Of an explicit type
Our aim is now to continue this work on the same line, including a new collection of titles.
II. SOME GENERAL REMARKS.
NOTE 1:
To avoid confusion between the terms “free boundary” and “moving boundary, we think it is advisable to point out the difference between them, especially since both terms are used indiscriminately in the literature (see e.g., the International Symposiums or Conferences on this subject).
On the other hand, in Cryer (1978) the author discusses the relationship between moving boundary problems (parabolic and timedependent) and free boundary problems (elliptic and steady state). Because of this definition, approximately 1% of the references (namely 53) in his bibliography on free boundary problems [Cryer (1977)] is concerned with heat conduction and diffusion (see 1.6).
Our definition follows the one frequently used e.g. in the Italian literature (FasanoPrimicerio’s group). In general, the problems given for the heat or diffusion equation are classified in the following way:
boundary problems:
 fixed
 moving
 free (implicit type or explicit type)
The fixed boundary problems (FiBP) for the heat equation are those studied in the domain (x , x ) x (0,T), i.e., the classical problems analyzed in any basic course of partial differential equations, such as:
I) u t – u xx = f (x, t), x1 < x < x2, 0 < t < T,
(FiBP) II) u (x, 0) = h (x) x1 <= x <= x2,
III) u (x1, t) = f1(t) or ux (x1, t ) =f1(t), 0 < t < T,
IV) u (x2, t) = f2(t) or ux (x2, t ) =f2(t), 0 < t < T,
which are not included in our bibliography and analysis.
The moving boundary problems (MBP) for the heat equation are those studied e.g., in the domain {(x, t) / s1 (t) < x < s2(t), 0 < t < T} with s1 (t)< s2 (t), functions given in (0,T), i.e., the spatial domain of the unknown function varies with time because of a law of movement, known a priori, such as
I) ut – u xx = f (x, t), s1 (t) < x < s2(t), 0 < t < T,
II) u (x, 0) = h(x), s1(0) <= x <= s2(0),
(MBP) III) u (s1(t), t) =f1(t) or u x (s1(t), t) = f1(t), 0 < t < T,
IV) u (s2(t), t) = f2(t) or u x (s2 (t), t) = f2 (t), 0 < t < T,
Moreover, the domain can be of the form {(x, t) /x < s(t), 0 < t < T} or {(x, t) / s(t)< x, 0 < t < T}. All these problems were originally studied at the beginning of the XX century by Gevrey, Goursat, Holmgrem, Levi, etc., and of course there exists an enormous bibliography on this subject. We refer e.g. to J.R. Cannon’s book (1984).
The free boundary problems (FBP) for the heat equation are those in which the spatial domain of the unknown function varies with time because of a law of movement not known a priori. The fact of not knowing the boundary or part of it, determines, of course, the mathematical need to impose new condition on the unknown function, which will depend on the physical problems studied. In general, the new condition to be imposed on the unknown function is deduced from the principle of conservation of energy across the boundary. Thus it follows that this boundary is the complementary unknown of the problem, and is called free boundary of the problem under analysis.
One of the most important FBP for the HDE is the socalled Stefan problem (LaméClapeyron (1981), Stefan (19891991)). Its mathematical formulation, in a dimensionless form, is given by: Find T > 0, u = u (x, t) and x = s(t) such that they satisfy, e.g., the following conditions
I) u t – u xx = 0 , 0 < x < s(t), 0 < t < T,
II) s ( 0 ) = b,
III) u (x, 0) = h (x), 0 <= x <= b,
(FBP) IV) u (s(t), t) = 0, 0 < t < T,
V) u x (s (t), t) = s (t), 0 < t < T,
VI) u(0, t) = f(t) or u x (0, t) = f(t), 0 < t < T.
The condition (FBPv) is called the Stefan condition.
Many free boundary problems for the heat equation can be divided into two classes, the explicit and the implicit types, according to whether the speed of the free boundary appears explicitly, in the conditions imposed on this boundary. That is to say, if the free boundary is given by x = s (t), then the problem will be of an explicit (implicit) type if derivative of s(t) appears (does not appear) in the condition imposed on x = s(t). An example of a (FBP) of explicit type is the classical Stefan problem and an example of the implicit type is the diffusionconsumption of oxygen in a living tissue (CrankGupta (1972)). The free boundary conditions for the last problem are given by:
u (s (t), t) = 0, 0 < t < T,
u x (s (t), t) = 0, 0 < t < T.
In general, free boundary problems of explicit and implicit types are related to each other [Schatz (1969), Fasano (1974)].
NOTE 2:
Among the FBP for the HDE we may have Stefan problem, diffusionconsumption of oxygen in a living tissue, noncatalytic gassolid diffusionreaction problem, penetration of solvents into glassy polymers, continuous casting problem and other solidification processes, ground freezing, ablation by melting, welding two steel plates, the shape of laser meltpools, electromechanical machining, HeleShaw flow, solidification of binary alloys, storage of solar energy, porous media, fresh and salt groundwater, supercooling and superhealting effects, freezing of foodstuffs, etc.
In order to have an idea of the importance of the methods and applications related to FMBP for the HDE, we may mention :
I) Conferences, meetings or seminars completely devoted to the subject: [AlbretchCollatzHoffmann (1982), AntonsevHoffmannKhludnev (1992), BossavitDamlamianFrémond (1985), ChadamRasmussen (1993), Delfour (1992), DiazHerreroLiñanVazquez (1995), FasanoPrimicerio (1983), FriedmanSpruck (1993), Hoffmann (1977), HoffmannSprekels (1990), Magenes (1980), Neittaanmaki (1991), NiezgodkaPawlow (1985), NiezgodkaStrzelecki (1996), OckendonHodgkins (1975), Rodrigues (1989), WilsonSolomonBoggs (1978), WrobelBrebbia (1991, 1993), WrobelBrebbiaNowak (1990, 1992, 1994), WrobelSarlerBrebbia (1995)]. Moreover, didactic seminars were given in [FasanoPrimicerio (1986), Tarzia (1984, 1987, 1989, 1993, 1995)].
II) Books or booklets exclusively devoted to this subject: [AlexiadesSolomon (1993), AnantharamanSuryanarayana (1987), Carey (1992), Crank (1984), Datzeff (1970), ElliottOckendon (1982), Fasano (1987, 1989), Gurtin (1993), Hill (1987), KurzFisher (1986, 1992), Lock (1996), Lunardini (1991), Meirmanov (1992), PrimicerioGianni (1989), Rubinstein (1971), ShyyUdaykumarRaoSmith (1996), Tarzia (1987), Ughi (1991), Visintin (1996), Vuik (1993), YamaguchiNogi (1977), ZerroukatChatwin (1994)].
III) Books that devote several chapters or sections to the subject: [Arpaci (1966), AzizNa (1984), BaiocchiCapelo (1978), Barbu (1984), Barenblatt (1979, 1993, 1996), BensoussanLions (1978), Biot (1970), BoleyWeiner (1960), BuckmasterLudford (1983), CaffarelliCabre (1995), Cannon (1984), CarslawJaeger (1959), Chalmers (1984), Chipot (1984), Colton (1976, 1988), Crank (1956), Diaz (1985), DuvautLions (1972), EckertDrake (1959, 1972), Fife (1988), Flemings (1974), Friedman (1964, 1982, 1988), FromentBischoff (1979), Goursat (1927), Grisvard (1985), Hill (1982), Jerome (1983), Kenig (1994), KinderlehrerStampacchia (1980), LadyzenskajaSolonnikovUral’ceva (1968), Levi (1960), Lions (1969, 1976), Luikov (1968), Lunardini (1981), Mellor (1978), Meyer (1973), Mori (1986), Naumann (1984), NeittaanmakiTiba (1994), OckendonTayler (1983), Ozisik (1980), Rodrigues (1987), Rosenberger (1979), RubinsteinRubinstein (1998), Shukla (1990), Smirnov (1964), SokolowskiZolesio (1992), SzekelyEvansSohn (1976), SzekelyThemelis (1971), Tarzia (1981), Tayler (1986), Tiba (1990), TikhonovSamarskii (1963), Weber (1901), WilmottDewynneHowison (1993, 1995), Woodruff (1973), etc.].
IV) Review papers on the subject both from the theoretical and/or numerical point of view: Aronson (1986), Bankoff (1964), Barbu (1985), Biloni (1978), Broadbridge (1992), ChapmanHowisonOckendon (1992), Crank (1981), Danilyuk (1985), Duvaut (1976), Eckert (several times on heat transfer literature), Fasano (1987, 1998), Fox (1975), Friedman (1979), Furzeland (1980), GarguichevichSanziel (1984), Guilding (1982), Goodman (1964), HoffmannNiezgodka (1983), Luikov (1971), Magenes (1976, 1981, 1998), Meyer (1978, 1983), MuehlbauerSunderland (1965), Niezgodka (1984), Nochetto (1984), Peletier (1973), Philip (1969), Primicerio (1973, 1981), Quilghini (1975), Rodrigues (1987, 1989), Rubinstein (1980), Saguez (1981), Sestini (1960), Stakgold (1986), Tarzia (1984, 1986), Verdi (1987, 1998), Villa (1984), Visintin (1998), etc.].
V) A long bibliography on the subject can be found in BroadbridgeWhite (1988), Danilyuk (1985), Goodman (1964), Kenmochi (1981), Nitsche (1980), Philip (1969), Primicerio (1981), Senf (1984), Tarzia (1981,1984, 1988), Verdi (1998), Visintin (1998), WilsonSolomonTrent (1979), and in other books and review papers cited above].
NOTE 3:
I) Active research during the past three decades has produced a variety of methods for finding theoretical, numerical and experimental results as well as important practical applications.
The number of titles on MFBP for the HDE has been exponentially increasing during the past years; an analysis of this fact is given in
Table I
1831


18891891


19011930


19311950


19511960


19611970


19711980


19811990


19911998/9


Tabla II: The 5869 titles of our bibliography have been published in the following form.
Scientific journals  
Papers published in a Symposium (edited as a book or a collection)  
Technical reports  
Books which have at least a chapter or a section on MFBP for the HDE  
Edition of a Symposium which has at least 3 articles on MFBP for the HDE  
Thesis  
Tabla III
Scientific journals  
Books which include a chapter or some sections related to MFBP for the HDE  
Technical reports  
Proceedings of symposia not specifically on MFBP but which include at least 3 papers on MFBP for the HDE  
Proceedings of conferences on MFBP for the HDE  
Collection  
Books on MFBP for the HDE  
Tabla IV: English, French, Italian, Portuguese and Spanish are the scientific languages better known by the author. This may have affected the selection of references. Original references in Russian are not, in general, included in this work (See point (VI) in Note 4).
The 5869 titles of our bibliography have been written in the following languages.
Language

Number of titles written in

There exists translation in English

Number of titles availables in English

English  
Russian  
Spanish  
French  
German  
Italian  
Japanese  
Chinese  
Portuguese  
Rumanian  
Polish  
Finish  
We remark that 92% of our bibliography is available in the English language.
Tabla V: The 32 physicalengineering scientific journals which have published a substantial number of titles (with more than 13 papers)
in the present bibliography 

Int. J. Heat Mass Transfer  
J. Heat Transfer  
Int. Comm. Heat Mass Transfer (Letters Heat Mass Transfer)  
J. Crystal Growth  
Int. J. Numer. Meth. Eng.  
Numer. Heat Transfer  
Chem. Eng. Sci  
Heat TransferSoviet Research  
Int. J. Eng. Sci.  
J. Comput. Physics  
Soviet Phys. Dokl.  
Water Resour. Res.  
J. Appl. Phys.  
Metallurg. Tech.  
Metallurg. Mater. Trans.  
Latin Amer. Appl. Res. (Latin Amer. J. Heat Mass Transfer, Latin Amer. J. Chem. Eng. Appl. Chem.)  
AICh. E Journal  
Comp. Meth. Appl. Mech. Eng.  
AIAA Journal  
Appl. Sci. Res.  
A. I. Ch. E. Symp. Series  
Adv. Water Resour.  
J. Fluid Mech.  
Physica (*)  
Trans.Metallurg.Soc.AIME  
Metallurg. Trans.  
J. Appl.. Mech.  
Mech. Res. Comm.  
Soviet Phys. Tech. Phys.  
Nuclear Sci. Eng.  
Soil Sci.  
Soil Sci. Soc. Amer. J.  
They have published 35% of the 4039 titles that appeared on scientific journals. We remark that 10 titles (of the 17 (*)) published on Physica are due to the edition of a Symposium (Vol. 12D (1984)).
Tabla VI: The 36 mathematical scientific Journals which have published a substantial number of titles (with more than 13 papers)
SIAM J. Math. Anal.  
Quart. Appl. Math.  
IMA J. Appl. Math.(J. Inst. Math. Appl.)  
Nonlinear Anal. Th. Meth. Appl.  
J. Math. Anal. Appl.  
Euro. J. Appl. Math.  
SIAM J. Appl. Math.(J. SIAM)  
J. Diff. Eq.  
Arch. Rat. Mech. Anal.  
Boll. Un. Mat. Italiana  
USSR Comput. Maths. Math. Phys.  
Soviet Math. Dokl.  
Indiana Univ. Math.J. (J. Math. Mech.)  
C. R. Acad. Sc. Paris  
SIAM J. Numer. Anal.  
Ann. Mat. Pura Appl.  
Control Cybernet.  
Trans. Amer. Math. Soc.  
Comm. Partial Diff. Eq.  
Math. Mech. Appl. Sci.  
Z. Angew. Math. Mech.  
Quart. J. Mech. Appl. Math.  
J. Inst. Math. Appl.  
Diff. Eq.  
Ann. Sc. Norm. Sup. Pisa  
Meccanica  
IMA J. Numer. Anal.  
. Eng. Math.  
Numer. Math.  
Proc. Royal Soc. Edinburgh  
Comm. Pure Appl. Math.  
J. Appl. Math. Mech.  
Adv. Math. Sci. Appl.  
Appl. Anal.  
Rocky Mountain J. Math.  
Proc. Royal Soc. London  
They have published 33% of the 4039 titles that appeared on scientific journals. We remark that 11 titles (of the 76 (**)) published on IMA J. Applied Mathematics are due to the edition of a Conference (Vol. 35 (1985)) and that 15 titles (of the 35 (***)) published on Control and Cybernetics are due to the edition of a Conference (Vol. 14, No. 13 (1985))
Tabla VII: The collections which have published a substantial number of titles.
Pitman Research Notes in Math. Series  
Int. Series Numer. Math. (ISNM)  
Ground Freezing  
Permafrost  
Cuadern. Inst. Mat. B. Levi (Rosario)  
Moving Boundary Problems (Comp. Mech.)  
Computational Heat Transfer  
Mecánica Computacional (AMCA, Santa Fe)  
Numerical Heat Transfer  
Free Univ. Berlin  
Gakuto Int. Series: Math. Sciences Appl.  
Heat Transfer  
ECMI – European Conf. Industrial Math.  
Solidification  
Congreso Latinoam.Transf. Calor y Materia  
Lecture Notes in Math.  
Lecture Notes in Control & Inform. Sciences  
Drying 
8

They have published 94% of the 991 titles that appeared on 30 collections. We remark that there exists at least 27 others collections which have published a few numbers of papers on the subject and this fact has not been considered for the last stastistics.
We remark that 266 (over 286) titles published in the collection Pitman Res. Notes in Math. Series (Addison Wesley – Longman) (see Table IX) refers only to the papers contained in the following eleven volumes # : 78, 79, 120, 121, 185, 186, 280, 281, 282, 323, 363 corresponding to the Proceedings of the International Conferences on Free Boundary Problems.
Tabla VIII: The technical reports including a substantial number of titles are (We remark that the technical reports published in a scientific journal, proceeding or collection, etc. were not considered here)
Publ. Ist. Analisi Numerica, Pavia  
Math. Res. Center, Univ. Winsconsin  
ORNL, Oak Ridge  
INRIA, Rocquencourt  
Sem. Mat., Univ. Brescia  
CNEA, Buenos Aires  
Dip. Mat.”F. Enriques”, Univ. Milano  
ICTP, Trieste  
Dip. Mat.”U.Dini”, Univ.Firenze  
IMA, Univ. Minnesota  
Conf. Sem. Mat., Univ. Bari  
CMAF, Univ. Lisboa  
Chiba Univ.  
IAMUBA, Univ. Buenos Aires  
Univ. Bonn  
Dept. Math., Univ.ParisSud, Orsay  
They have published 75 % of 247 titles that appeared as a technical report.
Tabla IX: There are 210 titles in our bibliography (122 books and 88 proceedings which have at least three papers on the subject) which have been published by a Publisher. Among them, we may cite.
Addison WesleyLongman (ex Pitman)  
Springer Verlag  
John Wiley  
Academic Press  
Birkhäuser Verlag  
Computational Mechanics Publ.  
AMCA, Santa Fe  
Clarendon Press  
Pergamon Press  
Universidad Nacional de Rosario  
DunodGauthier Villars  
McGraw Hill  
Cambridge University Press  
American Mathematical Society  
They include the 62 % of the 210 titles which were published by a Publisher.
NOTE 4:
Some abbreviations and conventions has been used, e.g. : I) The bibliography is arranged alphabetically by the first author. A prefix is treated as part of the name so that e.g. Van Duyn is listed under V and not under D. Names beginning with, e.g., Mac, Mc and Di are listed under MAC, MC and DI respectively.
II) In general, the source of information (authors, titles, references, etc.) is indicated.
III) When a nonEnglish paper has an English translation, we quote, in general, the English version.
IV) Abbreviations of names of journals are those used in the current literature whenever they are available.
V) We have only considered as the year of the publication the one appearing in the source reference and not the year of the conference or journal reception. Moreover, when the publication year is indicated e.g. (1980/81) it was considered like (1980). This is criterion followed in preparing Table I.
VI) Important: Russian (in general, nonwestern languages) literature on this field (which has not been translated to the English language) will be introduced in forthcoming issues.
NOTE 5:
For the search of bibliography many scientific libraries were consulted during 19771999; Among them, I must mention the following: I) Hemeroteca del Instituto de Matemática “Beppo Levi”, Univ. Nac. de Rosario, Rosario (Argentina),
II) Biblioteca dell’Dipartimento di Matematica “Ulisse Dini”, Univ. di Firenze, Firenze (Italy),
III) Bibliothèque de Mathématiques Recherche, Univ. de Paris VI, Paris (France),
IV) Centre de documentation de 1’INRIA, Rocquencourt (France),
V) Bibliothèque de Physique Recherche, Univ. de Paris VI, Paris (France),
VI) Hemeroteca del CERIDE, Santa Fe (Argentina),
VII) Biblioteca do IMPA e do LNCC, Rio de Janeiro (Brazil),
VIII) Hemeroteca del Departamento de Matemática (UBA) y del IAM, Buenos Aires (Argentina),
IX) Bibliothèques de Mécanique Théorique, d’Analyse Numérique, de Chimie Recherche et de Sciences de la Terre, Univ. de Paris VI, Paris (France),
X) Library of ICTP, Trieste (Italy),
XI) Biblioteca del Grupo de Frontera Libre GIFLA (CONICETUA), Univ. Austral, Rosario (Argentina).
NOTE 6:
Undoubtedly, I am responsible for the mistakes and omissions that might have slipped in the text. For those I apologize. I would welcome information about them so that they could be corrected in a possible future edition.
I would greatly appreciate and would like to receive reprints, technical reports, proceeding, books, thesis, preprints or even references or photocopies of articles on MFBP which have not been included in this bibliography and also some hints for new research. These could be incorporated in a future edition.
It is my hope that this bibliography will reduce the duplication of effort and stimulate the crossfertilization of ideas among the different and numerous disciplines concerned with MFBP. Finally, I also hope that this long bibliography will be useful to researchers in moving and free boundary problems in partial differential equations particularly regarding the heatdiffusion equation.
NOTE 7: (Service to the International Scientific Community)
I have a copy of all the references mentioned in the present bibliography. Any person who for any reason has not been able to find any of the references could ask for a copy (in general, an interchange basis will be offer).
Acknowledgements:
This work has been partially sponsored by CONICET and Universidad Austral (Argentina). This financial support was granted to the Research Project “Free Boundary Problems for the HeatDiffusion Equation” (Problemas de Frontera Libre para la Ecuación del CalorDifusión).
I am grateful to the Institutions that have granted me fellowships, which were developed in: G.N.F.M. of the Italian CNR, ICTP and Univ. di Firenze (Italy); Univ. de Paris VI and INRIA (France); and LNCC (Brazil).
Thanks to my colleagues and friends Mary Dzioba, Graciela Garguichevich, Riccardo Ricci, Cristina Turner and Luis Villa for your help and interest in this work during the last two decades. Special thanks to Professors. A. Fasano and M. Primicerio for their constant help during the last fifteen years and for having published my previous bibliography in 1988.
Thanks to Cristian Conde, Evangelina Fernández, Ariel Lombardi and Mariela Olguín for their help in the writing and organization of the data base files during different periods in the last decade. Special thanks to Adolfo Giancarelli for his help in the printing of the bibliographical database.
Finally, I am grateful to the great number of people (researchers, librarians, etc.) and Institutions that have silently contributed to this work.
I dedicate this work to Norma and my sons María Silvina and Pablo Alberto.
Domingo Alberto Tarzia
Rosario, Octubre de 1999
REFERENCES
We include now the 5869 references in the following way:
1. Books or booklets exclusively devoted to the subject (with 24 titles).
2. Thesis on the subject (with 59 titles).
3. Books that devote several chapters or sections to the subject (with 98 titles)
4. Editors of conferences, meetings or seminars completely devoted to the subject (with 35 titles).
5. Papers published on conferences, meetings or seminars completely devoted to the subject (with 622 titles).
6. Editors of a symposium not specifically on the subject but which includes at least three papers on the subject (with 53 titles).
7. Papers published on a symposium not specifically on the subject but which includes at least three papers on the subject (with 348 titles).
8. Papers on the subject published in a symposium not considered before (with 312 titles).
9. Papers on the subject published on scientific journal or some collections (with 4039 titles).
10. Papers on the subject to appear on scientific journal (with 32 titles).
11. Technical reports on the subject (with 247 titles).
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Para descargar a su computadora haga click con el botón derecho sobre el vínculo y seleccione la opción”Guardar destino como” o “Guardar enlace como” dependiendo de su navegador.  Full Edition PDF (4532 KB)
Para descargar a su computadora haga click con el botón derecho sobre el vínculo y seleccione la opción”Guardar destino como” o “Guardar enlace como” dependiendo de su navegador.