Geodesic
On the solvability of spatial nonlocal boundary value problems for one-dimensional pseudoparabolic and pseudohyperbolic equations
Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, no. 3 (2015), pp. 29-43 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In the present work we study the solvability of spatial nonlocal boundary value problems for linear one-dimensional pseudoparabolic and pseudohyperbolic equations with constant coefficients, but with general nonlocal boundary conditions by A.A. Samarsky and integral conditions with variables coefficients. The proof of the theorems of existence and uniqueness of regular solutions is carried out by the method of Fourier. The study of solvability in the classes of regular solutions leads to the study of a system of integral equations of Volterra of the second kind. In particular cases nongeneracy conditions of the obtained systems of integral equations in explicit form are given.
Mots-clés : pseudoparabolic equation
Keywords: pseudohyperbolic equation, Sobolev space, initial-boundary value problem, Fourier's method, regular solution, integral equation of Volterra. 
@article{VSGU_2015_3_a2,
     author = {N. S. Popov},
     title = {On the solvability of spatial nonlocal boundary value problems for one-dimensional pseudoparabolic and pseudohyperbolic equations},
     journal = {Vestnik Samarskogo universiteta. Estestvennonau\v{c}na\^a seri\^a},
     pages = {29--43},
     year = {2015},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSGU_2015_3_a2/}
}
TY  - JOUR
AU  - N. S. Popov
TI  - On the solvability of spatial nonlocal boundary value problems for one-dimensional pseudoparabolic and pseudohyperbolic equations
JO  - Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ
PY  - 2015
SP  - 29
EP  - 43
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/VSGU_2015_3_a2/
LA  - ru
ID  - VSGU_2015_3_a2
ER  - 
%0 Journal Article
%A N. S. Popov
%T On the solvability of spatial nonlocal boundary value problems for one-dimensional pseudoparabolic and pseudohyperbolic equations
%J Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ
%D 2015
%P 29-43
%N 3
%U http://geodesic.mathdoc.fr/item/VSGU_2015_3_a2/
%G ru
%F VSGU_2015_3_a2
N. S. Popov. On the solvability of spatial nonlocal boundary value problems for one-dimensional pseudoparabolic and pseudohyperbolic equations. Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, no. 3 (2015), pp. 29-43. http://geodesic.mathdoc.fr/item/VSGU_2015_3_a2/

[1] Bitsadze A. V., Samarski A. A., “About some of the simplest generalizations of elliptic tasks”, DAN SSSR, 185:4 (1969), 739–740 (in Russian) | MR | Zbl

[2] Kozhanov A. I., “The problem with integral conditions for some classes of nonstationary equations”, Reports of the Academy of Sciences, 457:2 (2014), 152–156 (in Russian) | DOI | Zbl

[3] Lazetic N., “On a classical solutions of mixed boundary problems for one-dimensional parabolic equation of second orde”, Publ. de l'Institut Mathematique, Nouvelle Serie, 67(81) (2000), 53–75 | MR | Zbl

[4] Lazetic N., “On the classical solvability of the mixed problem for one-dimensional hyperbolic equations of the second order”, Differential equations, 42:8 (2006), 1072–1077 (in Russian) | MR | Zbl

[5] Pulkina L. S., “Nonlocal problem with integral conditions for hyperbolic equations”, Differential equations, 40:7 (2004), 887–892 (in Russian) | MR | Zbl

[6] Pulkina L. S., “Nonlocal problem with two integral conditions for hyperbolic equations in the plane”, Nonclassical equations of mathematical physics, IM SO RAN, Novosibirsk, 2007, 232–236 (in Russian)

[7] Kozhanov A. I., Pulkina L. S., “On the solvability of boundary value problems with nonlocal boundary condition of integral type for multidimensional hyperbolic equations”, Differential equations, 42:9 (2006), 1116–1172 (in Russian)

[8] Kozhanov A. I., Pulkina L. S., “On the solvability of certain boundary value problems with offset for linear hyperbolic equations”, Mathematical journal, 9:2(32) (2009), 78–92 (in Russian)

[9] Skubachevskii A. L., Elliptic Functional Differential Equations and Applications, Operator Theory: Advances and Applications, 91, Birkhäuser Verlag, Basel, 1997 | MR | Zbl

[10] Nakhushev A. M., Tasks with offset for partial differential equations, Nauka, M., 2006, 287 pp. (in Russian)

[11] Nakhushev A. M., Laden equations and their applications, Nauka, M., 2012 (in Russian)

[12] Kozhanov A. I., “About one loaded nonlinear parabolic equation and associated inverse problem”, Mathematical notes, 76:6 (2004), 840–853 (in Russian) | DOI | MR | Zbl

[13] Telesheva L. A., “On the solvability of a linear inverse problem for parabolic equations of high order”, Mathematical notes of YaSU, 20:2 (2013), 186–196 (in Russian) | Zbl

[14] Krasnov M. L., Integral equations. (Introduction to the theory), Fizmatlit, M., 1975 (in Russian)

[15] Kozhanov A. I., “About solvability of a regional problem with nonlocal boundary condition for linear parabolic equations”, Vestnik of Samara State Technical University, 2004, no. 30, 63–69 (in Russian) | MR

[16] Kozhanov A. I., Popov N. S., “On the solvability of some tasks with offset for pseudoparabolic equations”, Vestnik of NGU. Series: Mathematics, Mechanics, Informatics, 10:3 (2010), 63–75 (in Russian) | MR | Zbl

[17] Prudnikov A. P., Brychkov Yu. A., Marichev O. I., Integrals and series, Nauka, M., 1981 (in Russian)

[18] Yakubov S. Ya., Linear differential-operator equations and their applications, Elm, Baku, 1985 (in Russian)

[19] Kozhanov A. I., Composite Type Equations and Inverse Problems, VSP, Utrecht, 1999, 171 pp. | MR | Zbl

[20] Popov N. S., “On solvability of boundary value problems for multidimensional pseudo-parabolic equations with nonlocal boundary condition of an integral type”, Mathematical notes of YaSU, 19:1 (2012), 82–95 (in Russian) | MR | Zbl