Volume 6, no. 2Pages 5 - 24

The Initial-Finite Problems for Nonclassical Models of Mathematical Physics

S.A. Zagrebina
The models of Mathematical Physics, whose representation in the form of equations or systems of partial differential equations do not fit one of the classical types such as elliptic, parabolic or hyperbolic, are called nonclassical. The article provides an overview of the author's results in the field of nonclassical models of Mathematical Physics for which the initial-finite problems, generalizing the Cauchy and Showalter, Sidorov conditions, are considered. Basic method for the research is the Sviridyuk relative spectrum theory. Abstract results are illustrated by the specific initial-finite problems for the equations and systems of equations in partial derivatives occurring in applications, namely, the theory of filtration, fluid dynamics and mesoscopic theory, considered on the sets of different geometrical structure.
Full text
Keywords
Plotnikov model, nonclassical models of Mathematical Physics, the Navier - Stokes system, the Barenblatt - Zheltov - Kochina equation, the (multipoint) initial-finite problems, the relative spectrum.
References
1. Plotnikov P.I., Starovoitov V.N. The Stefan Problem with Surface Tension as a Limit of the Phase Field Model. Differential Equations, 1993, vol. 29, no. 3, pp. 395-405.
2. Plotnikov P.I., Klepacheva A.V. The Phase Field Equations and Gradient Flows of Marginal Functions. Siberian Mathematical Journal, 2001, vol. 42, no. 3, pp.551-567.
3. Ladyzhenskaya O.A. Mathematical Problems in the Dynamics of a Viscous Incompressible Fluid. Moscow, Nauka, 1970. (in Russian)
4. Temam R. Navier-Stokes Equations. Theory and Numerical Analysis. Amsterdam, N.-Y., Oxford, North Holland Publ. Co., 1979.
5. 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.
6. Rutkas A.G. The Cauchy Problem for the Equations Ax'(t)+Bx(t)=f(t). Differential Equations, 1975, vol. 11, no. 11, pp. 1996-2010. (in Russian).
7. Ting T.W. Certain Non-Steady Flows of Second-Order Fluids. Arch. Rat. Mech. Anal., 1963, vol. 14, no. 1, pp. 28-57.
8. Chen P.J., Gurtin M.E. On a Theory of Heat Conduction Involving Two Temperatures. Z. Angew. Math. Phys., 1968, vol. 19, pp. 614-627.
9. Hallaire M. On a Theory of Moisture-Transfer. Inst. Rech. Agronom., 1964, no. 3, pp. 60-72.
10. Oskolkov A.P. Nonlocal Problems for Some Class Nonlinear Operator Equations Arising in the Theory Sobolev Type Equations. Zap. Nauchn. Sem. LOMI. Problems in the theory of representations of algebras and groups. Part 2, Nauka, St. Petersburg, 1991, vol. 198, pp. 31-48. (in Russian).
11. Sviridyuk G.A., Ankudinov A.V. The Phase Space of the Cauchy - Dirichlet Problem for a Nonclassical Equation. Differential Equations, 2003, vol. 39, no. 11, pp. 1639-1644.
12. Poincare H. Sur l'equilibre d'une mass fluide animee d'un mouvement de rotation. Acta Math., 1885, vol. 7, pp. 259-380.
13. Sobolev S.L. On a New Problem of Mathematical Physics. Izv. Akad. Nauk SSSR Ser. Mat., 1954, vol. 18, issue 1, pp. 3-50. (in Russian).
14. Demidenko G.V., Uspenskii S.V. Partial Differential Equations and Systems not Solvable with Respect to the Highest - Order Deriative. New York, Basel, Hong Kong, Marcel Dekker, Inc., 2003.
15. Pankov A.A., Pankova T.E. Nonlinear Evolution Equations with Irreversible Operator Coefficient for the Derivative. Dokl. Akad. Nauk Ukraine, 1993, no. 9, pp. 18-20.(in Russian)
16. Pyatkov S.G. Operator Theory. Nonclassical Problems. Utrecht, Boston, Koln, Tokyo, VSP, 2002.
17. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht, Boston, Koln, Tokyo, VSP, 2003.
18. Zamyshlyaeva A.A. Linear Sobolev Type Equations of High Order. Chelyabinsk, Publ. Center of the South Ural State University, 2012. (in Russian)
19. Manakova N.A. Optimal Control Problem for the Sobolev Type Equations. Chelyabinsk, Publ. Center of the South Ural State University, 2012. (in Russian)
20. Sagadeeva M.A. Dichotomy of Solutions of Linear Sobolev Type Equations. Chelyabinsk, Publ. Center of the South Ural State University, 2012. (in Russian)
21. Keller A.V. The Algorithm for Solution of the Showalter - Sidorov Problem for Leontief Type Models. Bulletin of the South Ural State University. Series "Mathematical Modelling, Programming & Computer Software", 2011, no. 4 (241), issue 7, pp. 40-46. (in Russian)
22. Sviridyuk G.A., Brychev S.V. Numerical Solution of Systems of Equations of Leontief Type. Russian Mathematics (Izvestiya VUZ. Matematika), 2003, vol. 47, no. 8, pp. 44-50.
23. Sviridyuk G.A., Burlachko I.V. An Algorithm for Solving the Cauchy Problem for Degenerate Linear Systems of Ordinary Differential Equations. Computational Mathematics and Mathematical Physics, 2003, vol. 43, no. 11, pp. 1613-1619.
24. Shestakov A.L., Sviridyuk G.A. Optimal Measurement of Dynamically Distorted Signals. Bulletin of the South Ural State University. Series "Mathematical Modelling, Programming & Computer Software", 2011, no. 17 (234), issue 8, pp. 70-75.
25. Shestakov A.L., Keller A.V., Nazarova E.I. Numerical Solution of the Optimal Measurement Problem. Automation and Remote Control, 2012, vol. 73, no. 1, pp. 97-104.
26. Sviridyuk G.A., Zagrebina S.A. Verigin's Problem for Linear Equations of the Sobolev Type with Relatively p-Sectorial Operators. Differential Equations, 2002, vol. 38, no. 12, pp. 1745-1752.
27. Zagrebina S.A. On the Showalter - Sidorov Problem. tRussian Mathematics (Izvestiya VUZ. Matematika), 2007, vol. 51, no. 3, pp. 19-24.
28. Zagrebina S.A., Solovyeva N.P. The Initial-Finish Problem for the Evolution of Sobolev-Type Equations on a Graph. Bulletin of the South Ural State University. Series "Mathematical Modelling, Programming & Computer Software", 2008, no. 15 (115), issue 1, pp. 23-26. (in Russian)
29. Manakova N.A., Dylkov A.G. On One Optimal Control Problem with a Penalty Functional in General Form. Vestnik Samarskogo Gosudarstvennogo Tekhnicheskogo Universiteta. Seriya "Fiziko-Matematicheskie Nauki", 2011, no. 4 (25), pp. 18-24.(in Russian)
30. Zamyshlyaeva A.A. The Initial-Finish Value Problem for Nonhomogenious Boussinesque - Love Equation. Bulletin of the South Ural State University. Series "Mathematical Modelling, Programming & Computer Software", 2011, no. 37 (254), issue 10, pp. 22-29. (in Russian)
31. Sidorov N.N. A Class of Degenerate Differential Equations with Convergence. Mathematical Notes, 1984, vol. 35, no. 4, pp. 300-305.
32. Sviridyuk G.A., Zagrebina S.A. The Showalter - Sidorov Problem as Phenomena of the Sobolev-Type Equations. News of Irkutsk State University. Series 'Mathematics', 2010, vol. 3, no. 1, pp. 51-72. (in Russian)
33. Zagrebina S.A. The Initial-Finish Problem for Sobolev Type Equations with Strongly (L,p)-Radial Operator. Math. Notes of YSU, 2012, vol. 19, no. 2, pp. 39-48. (in Russian)
34. Zagrebina S.A., Sagadeeva M.A. The Generalized Showalter - Sidorov Problem for the Sobolev type Equations with strongly (L,p)-radial operator. Vestnik Magnitogorskogo gosudarstvennogo universiteta. Seria "Matematika", [Bulletin of Magnitogorsk State University. Series "Mathematics"], 2006, issue 9, pp. 17-27. (in Russian)
35. Zagrebina S.A. The Showalter - Sidorov - Verigin Problem for the Linear Sobolev Type Equations. Neklassicheskie uravneniya matematicheskoy fiziki [Nonclassical Mathematical Physics Equations], Novosibirsk, 2007, pp. 150-157. (in Russian)
36. Zagrebina S.A. The Initial-Finish Problem for the Navier - Stokes Linear System. Bulletin of the South Ural State University. Series "Mathematical Modelling, Programming & Computer Software", 2010, no. 4 (221), issue 7, pp. 35-39. (in Russian)
37. Zagrebina S.A. The Multipoint Initial-Finish Problem for Hoff Linear Model. Bulletin of South Ural State University. Series "Mathematical Modelling, Programming & Computer Software", 2012, no. 5 (264), issue 11, pp. 4-12. (in Russian)
38. Zagrebina S.A., Konkina A.S. On a New Problem for the Barenblatt - Zheltov - Kochina Equation. Vestnik Magnitogorskogo gosudarstvennogo universiteta. Seria "Matematika", [Bulletin of Magnitogorsk State University. Series 'Mathematics'], 2012, issue 14, pp. 67-77. (in Russian)
39. Fedorov V.E. Degenerate Strongly Continuous Semigroups of Operators. St. Petersburg Mathematical Journal, 2001, vol. 12, issue 3, pp. 471-489.
40. Fedorov V.E About Some Relations in the Theory of Degenerate Operator Semigroups. Bulletin of te South Ural State University. Series "Mathematical Modelling, Programming & Computer Software", 2008, no. 15 (115), issue 7, pp. 89-99. (in Russian)
41. Zagrebina S.A. On the Existence and Stability of Solutions Navier - Stokes Equations. Vestnik Magnitogorskogo gosudarstvennogo universiteta. Seria "Matematika", [Bulletin of Magnitogorsk State University. Series "Mathematics"], 2005, issue 8, pp. 74-86. (in Russian)
42. Sviridyuk G.A. On a Model of the Dynamics of a Weakly Compressible Viscoelastic Fluid. Russian Mathematics (Izvestiya VUZ. Matematika), 1994, vol. 38, no. 1, pp. 59-68.
43. Sviridyuk G.A., Kuznetsov G.A. Relatively Strongly p-Sectorial Linear Operators. Doklady Mathematics, 1999, vol. 59, issue 2, pp. 298-300.
44. Sviridyuk G.A., Shemetova V.V. The Barenblatt - Zheltov - Kochina Equations on a Graph. Vestnik Magnitogorskogo gosudarstvennogo universiteta. Seria "Matematika", [Bulletin of Magnitogorsk State University. Series "Mathematics"], 2003, issue 4, pp. 129-139. (in Russian)