A nonlocal problem with integral conditions for hyperbolic equation
Abstract
In this article, we consider two initial-boundary value problems with nonlocal conditions. The main goal is to show the method which allows to prove solvability of a nonlocal problem with integral conditions of the first kind. This method is based on equivalence of a nonlocal problem with integral conditions of the first kind and nonlocal problem with integral conditions of the second kind in special form. Existence and uniqueness of generalized solutions to both problems are proved.
Supporting agencies: The author wish to thank the referee for his/her careful reading of the manuscript and valuable suggestions.
References
1. Cannon J.R. The solution of the heat equation subject to the specification of energy // Quart. Appl. Math. — 1963. — V. 21. — P. 155-160.
2. Kamynin L.I. On a boundary problem in the theory of heat conduction with a nonclassical boundary condition // Zh. Vychisl. Math. Math. Fiz. – 1964. — V. 4, No. 6 – P. 1006-1024.
3. Ionkin N.I. Solution of one boundary value problem of heat conduction theory with a nonclassical boundary condition // Differ. Uravn. — 1977. — V. 13, No. 2. — P. 294-304.
4. Samarskii A.A. On certain problems of the modern theory of differential equations // Differ. Uravn. — 1980. — V. 16, No. 11. — P. 1221-1228.
5. Muravei L.A., Filinovskii A.V. On a problem with nonlocal boundary condition for a parabolic equation // Mathematics of the USSR-Sbornik. — 1993. — V. 74, No. 1. — P. 219-249.
6. Gordeziani D.G., Avalishvili G.A. Solutions of nonlocal problems for one-dimensional oscillations of the medium // Mat. Model. — 2000. — V. 12, No. 1. — P. 94-103.
7. Kamynin V.L. Unique solvability of the inverse problem of determination of the leading coefficient in a parabolic equation // Differ. Equ. — 2011. — V. 47, No. 1. — P. 91-101.
8. Cannon J.R., Lin Y. An inverse problem of finding a parameter in a semi-linear heat equation // J. Math. Anal. Appl. — 1990. — V. 145. — P. 470-484.
9. Bouziani A. Solution forte d’un problem mixte avec condition non locales pour une classe d’equations hyperboliques // Bull. de la Classe des Sciences, Academie Royale de Belgique. — 1997. — V. 8. — P. 53-70.
10. Dmitriev V.B. Nonlocal problem with integral conditions for the wave equation // Vestnik of Samara State University. — 2006. — V. 42, No. 2. — P. 15-27.
11. Ladyzhenskaya O.A. Boundary value problems of mathematical physics. — Moscow: Nauka, 1973.
12. Steklov V.A. A problem on cooling of nonhomogeneous rigid bar // Reports of Kharkov Math. Society. — 1896. — V. 5, No. 3-4. — P. 136-181.
13. Pulkina L.S. A nonlocal problem with integral conditions for a hyperbolic equation // Differ. Equ. — 2004. — V. 40, No. 7. — P. 947-953.
14. Pulkina L.S. Initial-boundary value problem with a nonlocal boundary condition for a multidimensional hyperbolic equation // Differ. Equ. — 2008. — V. 44, No. 8. — P. 1119-1125.
15. Kozhanov A.I., Pulkina L.S. On the solvability of boundary value problems with a nonlocal boundary condition of integral form for multidimensional hyperbolic equations // Differ. Equ. — 2006. — V. 42, No. 9. — P. 1233-1246.
16. Kozhanov A.I. On solvability of certain spatially nonlocal boundary problems for linear parabolic equations // Vestnik of Samara State University. — 2008. — V. 62, No. 3. — P. 165-174 .
17. Pulkina L.S., Duzheva A.V. Nonlocal problem with time-dependent Steklov’s boundary conditions for hyperbolic equation // Vestnik of Samara State University. — 2010. — V. 78, No. 4. — P. 56-64.
Review
For citations:
Pulkina L.S. A nonlocal problem with integral conditions for hyperbolic equation. Nanosystems: Physics, Chemistry, Mathematics. 2011;2(4):61-70.
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