T.K. Yuldashev
Reshetnev Siberian State Aerospace University, Krasnoyarsk, 660037 Russia
E-mail: tursun.k.yuldashev@gmail.com
Received November 7, 2016
Abstract
This paper considers the questions of solvability and constructing the solution of a nonlocal boundary value problem for the fourth-order Boussinesq type nonhomogeneous partial integro-differential equation with degenerate kernel. The Fourier method based on separation of variables has been used. The system of algebraic equations has been obtained. The criterion of unique solvability of the considered problem has been revealed. The theorem of solvability of the problem has been proved under this criterion.
Keywords: integro-differential equation, boundary value problem, degenerate kernel, integral conditions, solvability
References
1. Turbin M.V. Investigation of initial boundary value problem for the Herschel-Bulkley mathematical fluid model. Vestn. Voronezh. Gos. Univ., Ser. Fiz. Mat., 2013, no. 2, pp. 246–257. (In Russian)
2. Akhtyamov A.M., Ayupova A.P. On solving the problem of diagnosing defects in a small cavity in the rod. Zh. Srednevolzh. Mat. O-va., 2010, vol. 12, no. 3, pp. 37–42. (In Russian)
3. Shabrov S.A. About the estimates of the influence function of a mathematical model of the fourth order. Vestn. Voronezh. Gos. Univ., Ser. Fiz. Mat., 2015, no. 2, pp. 168–179. (In Russian)
4. Whitham G. Linear and Nonlinear Waves. New York, John Wiley & Sons, 629 p.
5. Benney D.J., Luke J.C. Interactions of permanent waves of finite amplitude. J. Math. Phys., 1964, vol. 43, pp. 309–313.
6. Il'in V.A. The solvability of mixed problems for hyperbolic and parabolic equations. Russ. Math. Surv., 1960, vol. 15, no. 1, pp. 85–142.
7. Moiseev E.I. On the solution of a nonlocal boundary value problem by the spectral method. Differ. Equations, 1999, vol. 35, no. 8, pp. 1105–1112.
8. Sabitov K.B. Nonlocal problem for a parabolic-hyperbolic equation in a rectangular domain. Math. Notes, 2011, vol. 89, no. 4, pp. 562–567.
9. Chernyatin V.A. Substantiation of the Fourier Method in Mixed Problems for Partial Differential Equations. Moscow, Izd. Mosk. Univ., 1991. 112 p. (In Russian)
10. Yuldashev T.K. On a boundary value problem for a three dimensional analog of the Boussinesq type differential equation. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2016, vol. 158, no. 3, pp. 424–433. (In Russian)
11. Yuldashev T.K. A certain Fredholm partial integro-differential equation of the third order. Russ. Math., 2015, vol. 59, no. 9, pp. 62–66. doi: 10.3103/S1066369X15090091.
12. Yuldashev T.K. Inverse problem for a nonlinear Benney-Luke type integro-differential equations with degenerate kernel. Russ. Math., 2016, vol. 60, no. 9, pp. 53–60. doi: 10.3103/S1066369X16090061.
For citation: Yuldashev T.K. On a nonlocal problem for the nonhomogeneous Boussinesq type integro-differential equation with degenerate kernel. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2017, vol. 159, no. 1, pp. 88–99. (In Russian)
The content is available under the license Creative Commons Attribution 4.0 License.