# Nonclassical Mathematical Physics Models

G. A. Sviridyuk, S. A. ZagrebinaNonclassical called the models of mathematical physics, whose representation in the form of equations or systems of partial differential equations do not fit into one of the classical types - elliptic, parabolic or hyperbolic. In particular, the non-classical model are described by the equations of mixed type (eg, Tricomi equation), the degenerate equation (for example, the Keldysh equation) or the equations of Sobolev type (eg, Barenblatt - Zheltov - Kochina equation). The article provides an overview of some, in our opinion - the main A.I. Kozhanov achievements in the field of non-classical models of mathematical physics. His major achievements in the field of non-classical linear models belong to the theory of composite type equations, where he developed almost to perfection the method of a priori estimates and did the maximum possible generalization. Furthermore, the method of a priori estimates, along with the principle of comparing A.I. Kozhanov very effectively applied to the study of non-linear non-classical models such as the generalized Boussinesq filtration equation and classical nonlinear models, including models of the Josephson junction. Special place in activity of A.I. Kozhanov take the inverse problem, which, along with the decision and want to find another unknown factor. Here he received outstanding results in both linear and nonlinear cases.Full text

- Keywords
- composite type equations, Sobolev type equations, weakened Showalter - Sidorov problem, generalized filtration Boussinesq equation, inverse coefficient problems.
- References
- 1. Vragov V.N. Boundary Problems for Nonclassical Equations of Mathematical Physics [Kraevye zadachi dlja neklassicheskih uravnenij matematicheskoj fiziki]. Novosibirsk, Novosibirskij gos. univ., 1983. (in Russian)

2. Pyatkov S.G. Operator Theory. Nonclassical Problems. Utrecht, Boston, Köln, Tokyo, VSP, 2002.

3. Demidenko G.V., Uspenskii G.V. Partial Differential Equations and Systems not Solvable with Respect to the Highest-Order Derivative. New York, Basel, Hong Kong, Marcel Dekker, Inc., 2003.

4. Sveshnikov A.G., Al'shin A.B., Korpusov M.O., Pletner Ju.D. Linear and Nonlinear Sobolev Type Equation [Linejnye i nelinejnye uravnenija sobolevskogo tipa]. Moscow, Fizmatlit, 2007. (in Russian)

5. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht, Boston, Köln, Tokyo, VSP, 2003.

6. Sviridyuk G.A., Zagrebina S.A. The Showalter - Sidorov Problem as Phenomena of the Sobolev-Type Equations. News of Irkutsk State University. Ser. Mathematics, 2010, vol. 3, no. 1, pp. 51-72. (in Russian)

7. Kozhanov A.I. A Problem with Oblique Derivative for Some Pseudoparabolic Equations and Equations Close to Them. Siberian Mathematical J., 1996, vol. 37, no. 6, pp. 1171-1181.

8. Kozhanov A.I., Lar'kin N. A. On Solvability of Boundary-Value Problems for the Wave Equation with a Nonlinear Dissipation in Noncylindrical Domains. Siberian Mathematical J., 2001, vol. 42, no. 6, pp. 1062-1081.

9. Kozhanov A. I. On the Solvability of Boundary Value Problems for Quasilinear Ultraparabolic Equations in Some Mathematical Models of the Dynamics of Biological Systems. J. of Applied and Industrial Mathematics, 2010, vol. 4, no. 4, pp. 512-525.

10. Kozhanov A. I. Boundary Value Problems for Equations of Mathematical Physics of Odd Order [Kraevye zadachi dlja uravnenij matematicheskoj fiziki nechetnogo porjadka]. Novosibirsk, Novosibirskij gos. univ., 1990. (in Russian)

11. Barenblatt G. I., Zheltov Yu. P., Kochina I. N. Basic Concepts in the Theory of Seepage of Homogeneous Fluids in Fissurized Rocks. J. Applied Mathematics and Mechanics (PMM), 1960, vol. 24, no. 5, pp. 1286-1303.

12. Hallaire, M. On a Theory of Moisture-Transfer. Inst. Rech. Agronom., 1964, no. 3, pp. 60-72.

13. Chen P.J., Gurtin M.E. On a Theory of Heat Conduction Involving Two Temperatures. J. Angew. Math. Phys., 1968, vol. 19, pp. 614-627.

14. Kozhanov A.I. Boundary Value Problems for Some Classes of Higher-Order Equations that are Unsolved with Respect to the Highest Derivative. Siberian Mathematical J., 1994, vol. 35, no. 2, pp. 324-340.

15. Zagrebina S.A. The Initial-Finish Problem for the Navier - Stokes Linear System. Bulletin of South Ural State University. Ser. 'Mathematical Modelling, Programming & Computer Software', 2010, no. 4 (221), issue 7, pp. 35-39. (in Russian)

16. Sviridyuk G.A., Zamyshlyaeva A.A. The Phase Spaces of a Class of Linear Higher-Order Sobolev Type Equations. Differential Equations, 2006, vol. 42, no. 2, pp. 269-278.

17. Zamyshlyaeva A.A. The Initial-Finish Value Problem for Nonhomogenious Boussinesque - Löve Equation. Bulletin of South Ural State University. Ser. 'Mathematical Modelling, Programming & Computer Software', 2011, no. 37 (254), issue 10, pp. 22-29. (in Russian)

18. Polubarinova-Kochina P.Y. Theory of Ground Water Movement. Princeton, New Jersey, Princeton University Press. 1962.

19. Dzektser E.S. Generalization of the Groundwater Flow from Free Surface. Doklady Mathematics (Doklady Akademii Nauk), 1972, vol. 202, no. 5, pp. 1031-1033. (in Russian)

20. Furaev V.Z., Shadrin G.A. Derivation of the Equation for the Free Surface of the Filtered Liquid in a Layer of Finite Depth. Calculate. Mathematics and Math. Physics, Moscow, 1982, vol. 10, pp. 66-71. (in Russian)

21. Sviridyuk G.A. A Problem of Generalized Boussinesq Filtration Equation. Soviet Mathematics (Izvestiya VUZ. Matematika), 1989, vol. 33, no. 2, pp. 62-73.

22. Manakova N.A. The Optimal Control Problem for a Generalized Boussinesq Filtration. Vestnik Magnitogorsk. gos. univ. Ser. Mathematics, Magnitogorsk, 2005, issue 8, pp. 113 - 122. (in Russian)

23. Kozhanov A.I. Initial Boundary Value Problem for Generalized Boussinesque Type Equations with Nonlinear Source. Math. Notes, 1999, vol. 65, no. 1, pp. 59-63.

24. Kozhanov A.I. Some Classes of Nonstationary Equations with Growing Lower Terms. Siberian Mathematical J., 1998, vol. 46, no. 4, pp. 755-764.

25. Kozhanov A.I. Solvability of the Inverse Problem of Finding Thermal Conductivity. Siberian Mathematical J., 2005, vol. 46, no. 5, pp. 841-856.

26. Kozhanov A.I. Composite Type Equations and Inverse Problems. Utrecht, Boston, Köln, Tokyo, VSP, 1999.