Section
|
Mathematics
|
Title
|
On the solvability of nonlocal initial-boundary value problems for a partial differential equation of high even order
|
Author(-s)
|
Urinov A.K.ab,
Azizov M.S.a
|
Affiliations
|
Ferghana State Universitya,
Institute of Mathematics, National Academy of Sciences of Uzbekistanb
|
Abstract
|
In the present paper, two non-local initial-boundary value problems have been formulated for a partial differential equation of high even order with a Bessel operator in a rectangular domain. The correctness of one of the considered problems has been investigated. To do this, applying the method of separation of variables to the problem under consideration, the spectral problem was obtained for an ordinary differential equation of high even order. The self-adjointness of the last problem was proved, which implies the existence of the system of its eigenfunctions, as well as orthonormality and completeness of this system. Further, the Green's function of the spectral problem was constructed, with the help of which it was equivalently reduced to the Fredholm integral equation of the second kind with symmetrical kernel. Using this integral equation and Mercer's theorem, the uniform convergence of some bilinear series depending on found eigenfunctions has been studied. The order of the Fourier coefficients was established. The solution of the considered problem has been written as the sum of a Fourier series with respect to the system of eigenfunctions of the spectral problem. The uniform convergence of this series and also the series obtained from it by term-by-term differentiation was proved. Using the method of spectral analysis, the uniqueness of the solution of the problem was proved. An estimate for the solution of the problem was obtained, from which its continuous dependence on the given functions follows.
|
Keywords
|
differential equation of even order, nonlocal problem, Green's function, integral equation
|
UDC
|
517.956
|
MSC
|
35G15
|
DOI
|
10.35634/vm220206
|
Received
|
1 March 2022
|
Language
|
Russian
|
Citation
|
Urinov A.K., Azizov M.S. On the solvability of nonlocal initial-boundary value problems for a partial differential equation of high even order, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2022, vol. 32, issue 2, pp. 240-255.
|
References
|
- Tikhonov A.N., Samarskii A.A. Uravneniya matematicheskoi fiziki (Equations of mathematical physics), Moscow: Nauka, 1977.
- Nakhushev A.M. Uravneniya matematicheskoi biologii (Equations of mathematical biology), Moscow: Vysshaya Shkola, 1995.
- Korenev B.G. Voprosy rascheta balok i plit na uprugom osnovanii (Issues of calculation of beams and slabs on an elastic foundation), Moscow: Stroiizdat, 1954.
- Makhover E.V. Bending of a plate of variable thickness with a sharp edge, Uch. zap. LGP im. Gertsena, 1957, vol. 17, no. 2, pp. 28-39 (in Russian).
- Makhover E.V. On the natural frequency spectrum of a plate with a sharp edge, Uch. zap. LGP im. Gertsena, 1958, vol. 197, pp. 113-118 (in Russian).
- Sabitov K.B. Cauchy problem for the beam vibration equation, Differential Equations, 2017, vol. 53, no. 5, pp. 658-664. https://doi.org/10.1134/S0012266117050093
- Karimov Sh.T. The Cauchy problem for the degenerated partial differential equation of the high even order, Siberian Electronic Mathematical Reports, 2018, vol. 15, pp. 853-862. https://doi.org/10.17377/semi.2018.15.073
- Karimov Sh.T. On some generalizations of properties of the Lowndes operator and their applications to partial differential equations of high order, Filomat, 2018, vol. 32, issue 3, pp. 873-883. https://doi.org/10.2298/FIL1803873K
- Karasheva L.L. Cauchy problem for high even order parabolic equation with time fractional derivative, Siberian Electronic Mathematical Reports, 2018, vol. 15, pp. 696-706. https://doi.org/10.17377/semi.2018.15.055
- Salakhitdinov M.S., Amanov D. Solvability and spectral properties of a self-adjoint problem for a fourth-order equation, Uzbekskii Matematicheskii Zhurnal, 2005, no. 3, pp. 72-77 (in Russian).
- Otarova Zh.A. Volterra boundary value problem for a fourth order equation, Doklady Akademii Nauk Respubliki Uzbekistan, 2008, no. 6, pp. 18-22 (in Russian).
- Urinov A.K., Azizov M.S. A boundary problem for the loaded partial differential equations of fourth order, Lobachevskii Journal of Mathematics, 2021, vol. 42, no. 3, pp. 621-631. https://doi.org/10.1134/S1995080221030197
- Urinov A.K., Azizov M.S. Boundary value problems for a fourth order partial differential equation with an unknown right-hand part, Lobachevskii Journal of Mathematics, 2021, vol. 42, no. 3, pp. 632-640. https://doi.org/10.1134/S1995080221030203
- Amanov D., Murzambetova M.B. A boundary value problem for a fourth order partial differential equation with the lowest term, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2013, issue 1, pp. 3-10 (in Russian). https://doi.org/10.20537/vm130101
- Sabitov K.B. A remark on the theory of initial-boundary value problems for the equation of rods and beams, Differential Equations, 2017, vol. 53, no. 1, pp. 86-98. https://doi.org/10.1134/S0012266117010086
- Megraliev Ya.T., Alizade F.Kh. Inverse boundary value problem for a Boussinesq type equation of fourth order with nonlocal time integral conditions of the second kind, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2016, vol. 26, issue 4, pp. 503-514 (in Russian). https://doi.org/10.20537/vm160405
- Amanov D., Yuldasheva A.V. Solvability and spectral properties of boundary value problems for equations of even order, Malaysian Journal of Matematical Sciences, 2009, vol. 3, no. 2, pp. 227-248. https://mjms.upm.edu.my/lihatmakalah.php?kod=2009/July/3/2/227-248
- Amanov D., Ashyralyev A. Well-posedness of boundary value problems for partial differential equations of even order, AIP Conference Proceedings, 2012, vol. 1470, issue 1, pp. 3-7. https://doi.org/10.1063/1.4747625
- Irgashev B.Yu. On one boundary-value problem for an equation of higher even order, Russian Mathematics, 2017, vol. 61, issue 9, pp. 10-26. https://doi.org/10.3103/S1066369X1709002X
- Urinov A.K., Azizov M.S. Initial-boundary value problem for higher even order partial differential equation with Bessel operator, Differentsial'nye uravneniya, matematicheskoe modelirovanie i vychislitel'nye algoritmy: tez. dokl. Mezhdunar. konf., Belgorod: Belgorod State University, 2021, pp. 241-242 (in Russian).
- Azizov M.S. About a initial-boundary value problem for a higher even order partial differential equation with Bessel's operator, “Modern problems of applied mathematics and information technologies al-Khwarizmi 2021”: Abstracts of Int. Conf. dedicated to the 100th anniversary of the academician Vasil Kabulovich Kabulov, Fergana: Fergana State University, 2021, p. 110.
- Sabitov K.B. The Dirichlet problem for higher-order partial differential equations, Mathematical Notes, 2015, vol. 97, issues 1-2, pp. 255-267. https://doi.org/10.1134/S0001434615010277
- Irgashev B.Yu. Spectral problem for an equation of high even order, Russian Mathematics, 2016, vol. 60, issue 7, pp. 37-46. https://doi.org/10.3103/S1066369X16070069
- Irgashev B.Yu. On a boundary value problem for a high order mixed type equation, Siberian Electronic Mathematical Reports, 2020, vol. 17, pp. 899-912. https://doi.org/10.33048/semi.2020.17.066
- Cherfaoui S., Kessab A., Kheloufi A. On $2m$-th order parabolic equations with mixed boundary conditions in non-rectangular domains, Siberian Electronic Mathematical Reports, 2017, vol. 14, pp. 73-91. https://doi.org/10.17377/semi.2017.14.009
- Kozhanov A.I., Pinigina N.R. Boundary value problems for certain classes of high order composite type equations, Siberian Electronic Mathematical Reports, 2015, vol. 12, pp. 842-853. https://doi.org/10.17377/semi.2015.12.070
- Yuldasheva A.V. On a problem for a quasi-linear equation of even order, Journal of Mathematical Sciences, 2019, vol. 241, issue 4, pp. 423-429. https://doi.org/10.1007/s10958-019-04434-3
- Markov V.G. Solvability of a boundary-value problem for even-order mixed-type equations, AIP Conference Proceedings, 2017, vol. 1907, issue 1, 030004. https://doi.org/10.1063/1.5012626
- Popov S.V., Markov V.G. Boundary value problems for parabolic equations of high order with a changing time direction, Journal of Physics: Conference Series, 2017, vol. 894, 012075. https://doi.org/10.1088/1742-6596/894/1/012075
- Dzhamalov S.Z., Pyatkov S.G. On some boundary value problems for multidimensional higher order equations of mixed type, Siberian Mathematical Journal, 2020, vol. 61, no. 4, pp. 610-625. https://doi.org/10.1134/S0037446620040059
- Kipriyanov I.A. Singulyarnye ellipticheskie kraevye zadachi (Singular elliptic boundary value problems), Moscow: Nauka, 1997.
- Sitnik S.M., Shishkina E.L. Metod operatorov preobrazovaniya dlya differentsial'nykh uravnenii s operatorom Besselya (Method of transmutation operators for differential equations with Bessel operator), Moscow: Fizmatlit, 2019.
- Glushak A.V., Kononenko V.I., Shmulevich S.D. On a singular abstract Cauchy problem, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 1986, no. 6, pp. 55-56. http://mi.mathnet.ru/eng/ivm7572
- Glushak A.V. The Bessel operator function, Doklady Mathematics, 1997, vol. 55, no. 1, pp. 103-105. https://zbmath.org/?q=an:0965.34052
- Glushak A.V. The operator Bessel function, related semigroups, and a modified Hilbert transform, Differential Equations, 1999, vol. 35, issue 1, pp. 130-132.
- Glushak A.V. Criterion for the solvability of the weighted Cauchy problem for an abstract Euler-Poisson-Darboux equation, Differential Equations, 2018, vol. 54, no. 5, pp. 622-632. https://doi.org/10.1134/S0012266118050063
- Lyakhov L.N., Polovinkin I.P., Shishkina E.L. Formulas for the solution of the Cauchy problem for a singular wave equation with Bessel time operator, Doklady Mathematics, 2014, vol. 90, no. 3, pp. 737-742. https://doi.org/10.1134/S106456241407028X
- Naimark M.A. Lineinye differentsial'nye operatory (Linear differential operators), Moscow: Nauka, 1969.
- Mikhlin S.G. Lektsii po lineinym integral'nym uravneniyam (Lectures on linear integral equations), Moscow: Fizmatlit, 1959.
- Sobolev V.I. Lektsii po dopolnitel'nym glavam matematicheskogo analiza (Lectures on additional chapters of mathematical analysis), Moscow: Nauka, 1968.
|
Full text
|
|