Abstract
for the one-dimensional wave equation given in a half-strip, we consider a boundary value problem with the Cauchy conditions and two nonlocal integral conditions. Each integral condition is a linear combination of a linear Fredholm integral operator along the lateral side of the half-strip applied to the solution and the values of the solution and of a given function on the corresponding base of the half-strip. Under the assumption of appropriate smoothness of the right-hand side of the equation and the initial data, we obtain a necessary and sufficient condition for the existence and uniqueness of a classical solution of this problem and propose a method for finding it in analytical form. A classical solution is understood as a function that is defined everywhere in the closure of the domain where the equation is considered and has all classical derivatives occurring in the equation and in the conditions of the problem.
Similar content being viewed by others
References
Korzyuk, V.I., Erofeenko, V.T., and Pulko, Yu.V., Classical solution of the initial-boundary value problem for the wave equation with an integral boundary condition with respect to time, Dokl. Nats. Akad. Nauk Belarusi, 2009, vol. 53, no. 5, pp. 36–41.
Korzyuk, V.I., Kozlovskaya, I.S., and Kovnatskaya, O.A., Classical solution of problem of control boundary conditions in case of the first mixed problem for one-dimensional wave equation, Computer Algebra Systems in Teaching and Research. Mathematical Physics and Modeling in Economics, Finance and Education, Siedlce: AKADEMA, Univ. of Podlasie, 2011, pp. 68–78.
Korzyuk, V.I. and Kozlovskaya, I.S., On compatibility conditions in boundary value problems for hyperbolic equations, Dokl. Nats. Akad. Nauk Belarusi, 2013, vol. 57, no. 5, pp. 37–42.
Il’in, V.A. and Moiseev, E.I., Optimization of boundary controls of string vibrations, Russian Math. Surveys, 2005, vol. 60, no. 6 (366), pp. 1093–1119.
Il’in, V.A. and Moiseev, E.I., Optimization of the control by elastic boundary forces at two ends of a string in an arbitrarily large time interval, Differ. Equations, 2008, vol. 44, no. 1, pp. 92–114.
Samarskii, A.A., Some problems of the theory of differential equations, Differ. Equations, 1980, vol. 16, no. 11, pp. 1925–1935.
Pul’kina, L.S., A mixed problem with an integral condition for a hyperbolic equation, Math. Notes, 2003, vol. 74, no. 3, pp. 411–421.
Pul’kina, L.S., A nonlocal problem with integral conditions for a hyperbolic equation, Differ. Equations, 2004, vol. 40, no. 7, pp. 947–953.
Kozhanov, A.I. and Pul’kina, L.S., Boundary value problems with integral conditions for multidimensional hyperbolic equations, Dokl. Math., 2005, vol. 72, no. 2, pp. 743–745.
Korzyuk, V.I., Erofeenko, V.T., and Sheika, J.V., Classical solution for initial boundary value problem for wave equation with integral boundary condition, Math. Model. Anal., 2012, vol. 17, no. 3, pp. 309–329.
Moiseev, E.I., Korzyuk, V.I., and Kozlovskaya, I.S., Classical solution of a problem with an integral condition for the one-dimensional wave equation, Differ. Equations, 2014, vol. 50, no. 10, pp. 1364–1377.
Korzyuk, V.I. and Vin’, N.V., Classical solution of a problem with an integral condition for the one-dimensional biwave equation, Vestsi Akad. Navuk Belarusi Ser. Fiz.-Mat. Navuk, 2016, no. 3, pp. 16–29.
Korzyuk, V.I. and Stolyarchuk, I.I., Classical solution of the mixed problem for the wave equation with the integral condition, Dokl. Nats. Akad. Nauk Belarusi, 2016, vol. 60, no. 6, pp. 22–27.
Korzyuk, V.I. and Kozlovskaya, I.S., Klassicheskie resheniya zadach dlya giperbolicheskikh uravnenii (Classical Solutions of Problems for Hyperbolic Equations), A Lecture Course for Students of Mathematical and Physical-Mathematical Specialities, in 10 parts, Minsk: Belarus. Gos. Univ., 2017, Pt. 2.
Korzyuk, V.I., Uravneniya matematicheskoi fiziki (Equations of Mathematical Physics), Minsk: Belarus. Gos. Univ., 2011.
Koshlyakov, N.S., Gliner, E.V., and Smirnov, M.M., Uravneniya v chastnykh proizvodnykh (Partial Differential Equations), Moscow: Vysshaya Shkola, 1970.
Author information
Authors and Affiliations
Corresponding authors
Rights and permissions
About this article
Cite this article
Korzyuk, V.I., Kozlovskaya, I.S. & Naumavets, S.N. Classical Solution of a Problem with Integral Conditions of the Second Kind for the One-Dimensional Wave Equation. Diff Equat 55, 353–362 (2019). https://doi.org/10.1134/S0012266119030091
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0012266119030091