Previous |  Up |  Next


partial differential equations
New types of variational principles, each of them equivalent to the linear mixed problem for parabolic equation with initial and combined boundary conditions having been suggested by physicists, are discussed. Though the approach used here is purely mathematical so that it makes possible application to all mixed problems of mathematical physics with parabolic equations, only the example of heat conductions is used to show the physical interpretation. The principles under consideration are of two kinds. The first kind presents a variational characterization of the original problem, expressed in terms of a scalar function (temperature). The principles of the second kind characterize the same problem, formulated in terms of other variables, e.g. of a vector function (heat flux or entropy displacements).
[1] G. Adler: Sulla caratterizzabilita dell'equazione del calore dal punto di vista del calcolo delle variazioni. Matematikai kutató intézenétek közlemenyei II. (1957), 153-157. MR 0102656
[2] P. Rosen: On variational principles for irreversible processes. Proceedings of the Iowa Thermodynamic symposium (1953), 34-42. MR 0057190
[3] M. A. Biot: Thermoelasticity and irreversible thermodynamics. J. Appl. Phys. 27 (1956), 240-253. DOI 10.1063/1.1722351 | MR 0077441 | Zbl 0071.41204
[4] M. A. Biot: Variational principles in irreversible thermodynamics with application to viscoelasticity. Physical Review 97, (1955), 1463-1469. MR 0070514 | Zbl 0065.42003
[5] M. A. Biot: Linear thermodynamics and the mechanics of solids. Proceedings of the Third U. S. National Congress of Applied Mechanics (1958), 1-18. MR 0102941
[6] R. A. Schapery: Irreversible thermodynamics and variational principles with applications to viscoelasticity. Aeronaut. Research Labs., Wright-Patterson Air Force Base, Ohio (1962).
[7] M. E. Gurtin: Variational principles for linear initial value problems. Quart. Appl. Math., 22, (1964), 252-256. Zbl 0173.37602
[8] G. Herrmann: On variational principles in thermoelasticity and heat conduction. Quart. Appl. Math. 27 (1963), 2, 151-155. MR 0161512
[9] M. Ben-Amoz: On a variational theorem in coupled thermoelasticity. Trans. ASME (1965), E 32, 4, 943-945. DOI 10.1115/1.3627345
[10] M. E. Gurtin: Variational principles for linear elastodynamics. Archive for Ratl. Mech. Anal. 16 (1964), 1, 34-50. MR 0214322 | Zbl 0124.40001
[11] R. Courant D. Hilbert: Методы математической физики I. (Methoden der Mathematischen Physik I). Гостехиздат 1951, 231-232.
[12] I. Hlaváček: Derivation of non-classical variational principles in the theory of elasticity. Aplikace matematiky 12 (1967), 1, 15 - 29. MR 0214324
[13] В. А. Диткин А. 71. Прудников: Операционное исчисление. Издат. Высшая школа, Москва 1966. Zbl 1230.03072
[14] I. Hlaváček: Variational principles in the linear theory of elasticity for general boundary conditions. Aplikace matematiky 12 (1967), 6, 425-448. MR 0231575
[15] M. И. Вишик: Задача Копти для уравнений с операторноми коэффициентами, смешанная краевая задача для систем дифференциальных уравнений и приближенный метод их решения. Матем. сборник, 39 (81), (1956), 1, 51-148. MR 0080248 | Zbl 0995.90522
[16] О. А. Ладыженская: О решении нестационарных операторных уравнений. Матем. сборник, 39 (81), (1956), 4, 491-524. MR 0086987 | Zbl 0995.90522
Partner of
EuDML logo