Vestnik КRAUNC. Fiz.-Mat. Nauki. 2023. vol. 42. no. 1. P. 123-139. ISSN 2079-6641
Non-Local Initial-Boundary Value Problem for a Degenerate Fourth-Order Equation with a Fractional Gerasimov-Caputo Derivative
A. K. Urinov^*, D. A. Usmonov^*
Fergana state university, Uzbekistan, 150100, Fergana, 19, Murabbiylar st.
Abstract. Recently, initial-boundary problems in a rectangular domain for differential equations in partial derivatives of both even and odd order have been intensively studied. In this case, non-degenerate equations or equations that degenerate on one side of the quadrilateral are taken as the object of study. But initialboundary problems (both local and non-local) for equations with two or three lines of degeneracy remain unexplored. In this paper, in a rectangular domain, a fourth-order equation degene-rating on three sides of the rectangular and contains the Gerasimov-Caputo fractional diffe-rentiation operator has been considered. For this equation, an initial-boundary problem is formulated and investigated, with non-local conditions connecting the values of the desired function and its derivatives up to the third order (inclusive), taken on the sides of the rectangle. From the beginning, the uniqueness of the solution of the formulated problem was proved by the method of energy integrals. Then, the spectral problem that arises when applying the Fourier method based on the separation of variables to the considered initial-boundary problem has been investigated. The Green’s function of the spectral problem was constructed, with the help of which it is equivalently reduced to an integral Fredholm equation of the second kind with a symmetric kernel, which implies the existence of a countable number of eigenvalues and eigenfunctions of the spectral problem. A theorem is proved for expanding a given function into a uniformly convergent series in terms of a system of eigenfunctions. Using the found integral equation and Mercer’s theorem, we prove the uniform convergence of some bilinear series depending on the found eigenfunctions. The order of the Fourier coeffi-cients have been established. The solution of the considered is 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 the series obtained from it by term-by-term differentiation is studied. An estimate for solution to problem is obtained, from which follows its continuous dependence on the given functions.
Key words: degenerate fourth order equation, initial boundary value problem, method of separation of variables, spectral problem, Green’s function, integral equation, existence, uniqueness and stability of the solution.
Received: 07.11.2023; Revised: 20.02.2023; Accepted: 24.03.2023; First online: 16.04.2023
For citation. Urinov A. K.∗, Usmonov D. A. Non-local initial-boundary value problem for a degenerate fourth-order equation with a fractional Gerasimov-Caputo derivative. Vestnik KRAUNC. Fiz.-mat. nauki. 2023, 42: 1, 123-139. EDN: INZPHJ. https://doi.org/10.26117/2079-6641-2023-42-1-123-139.
Funding. Not applicable.
Competing interests. The authors declare that there are no conflicts of interest regarding authorship and publication.
Contribution and Responsibility. All authors contributed to this article. Authors are solely responsible for providing
the final version of the article in print. The final version of the manuscript was approved by all authors.
^*Correspondence: E-mail: email@example.com; firstname.lastname@example.org
The content is published under the terms of the Creative Commons Attribution 4.0 International License
© Urinov A. K.∗, Usmonov D. A., 2023
© Institute of Cosmophysical Research and Radio Wave Propagation, 2023 (original layout, design, compilation)
- Dzhrbashyan M. M., Nersesyan A. B. Drobnyye proizvodnyye i zadacha Koshi dlya differentsial’nykh uravneniy drobnogo poryadka, Izv. AN Arm SSR, 1968, 3, 1, 3-29 (In Russian).
- Dzhrbashyan M. M. Krayevaya zadacha dlya differentsial’nogo operatora drobnogo poryadka tipa Shturma-Liuvillya, Izv. AN ArmSSR, 1970, 5, 2, 71-96 (In Russian).
- Nakhushev A. M. The Sturm-Liouville problem for a second-order ordinary differential equation with fractional derivatives in lower terms, Dokl. AN SSSR, 1977, 234, 2 , 308-311 (In Russian).
- Aleroev T. S. On the problem of the zeros of the Mittag-Leffler function and the spectrum of a fractional order differential operator, Differents. uravneniya, 2000, 36, 9, 1278–1279 (In Russian).
- Pskhu A. V. Uravneniya v chastnykh proizvodnykh drobnogo poryadka [Partial differential equations of fractional order], Moskva, Nauka, 2005, 199 (In Russian).
- Nakhushev A. M. Drobnoye ischisleniye i yego primeneniye [Fractional calculus and its applicatio], Moskva, Fizmatlit, 2003, 272 (In Russian).
- Samko S. G., Kilbas A. A., Marichev O. I. Integraly i proizvodnyye drobnogo poryadka i nekotoryye ikh prilozheniya [Fractional integrals and derivatives and some of their applications], Minsk, Nauka i tekhnika, 1987, 688 (In Russian).
- Berdyshev A. S., Cabada A., Kadirkulov B. J. The Samarskii–Ionkin type problem for the fourth order parabolic equation with fractional differential operator, Computers and Mathematics with Applications, 2011, 62, 3884-3893.
- Berdyshev A. S, Kadirkulov B. J. A Samarskii-Ionkin problem for two-dimensionalparabolic equation with the Caputo fractional differential operator, International Journal of Pure and Applied Mathematics, 113, 4, 2017, 53-64.
- Kerbal S., Kadirkulov B. J., Kirane M. Direct and inverse problems for a Samarskii-Ionkin type problem for a two dimensional fractional parabolic equation, Progr. Fract. Differ. Appl, 2018, 3, 147-160.
- Aziz S., Malik S. A. Identifcation of an unknown source term for a time fractional fourthorder parabolic equation, Electron. J. Differ. Equat., 2016, 293, 1–20.
- Berdyshev A. S., Kadirkulov B. J. On a Nonlocal Problem for a Fourth-Order Parabolic Equation with a Dzhrbashyan–Nersesyan Fractional Operator, Differentsial’nyye uravneniya, 2016, 52, 1, 123–127 (In Russian).
- Karasheva L. L. Problem in a half-strip for a fourth-order parabolic equation with a Riemann–Liouville operator by time variable, Izvestiya Kabardino-Balkarskogo nauchnogo tsentra RAN, 2019, 5, 91, 21-29 (In Russian).
- Yuldashev T. K., Kadirkulov B. J. Nonlocal problem for a mixed typefourth-order differential equation with Hilfer fractional operator, Ural mathematical journal, 2020, 6, 1, 153–167.
- Yuldashev T. K., Kadirkulov B.J. Inverse boundary value problem for a fractional differential equations of mixed type with integral redenition conditions, Lobachevskii Journal of Mathematics, 2021, 42, 3, 649–662.
- Ashurov R., Umarov S. Determination of the order of fractional derivative for subdiffusion equations, Fract. Calc. Appl. Anal., 23, 6, 2020, 1647–1662.
- Ashurov R., Fayziev Y. Inverse Problem for Finding the Order of the Fractional Derivative in the Wave Equation, Mathematical Notes, 2021, 110, 6, 842–852.
- Karimov D. KH., Kasimova M. Smeshannaya zadacha dlya lineynogo uravneniya chetvertogo poryadka, vyrozhdayushchegosya na granitse oblasti, Izv. AN UzSSR, ser. fiz.-mat. nauk, 1968, 2, 27-31 (In Russian).
- Baykuziyev K.B., Kasimova M. Smeshannaya zadacha dlya uravneniya chetvertogo poryadka, vyrozhdayushchegosya na granitse oblasti, Izv. AN UzSSR, ser. fiz. -mat. nauk, 1968, 5, 7-12 (In Russian).
- Kasimova M. Smeshannaya zadacha dlya lineynogo uravneniya chetverogo poryadka, vyrozhdayushchegosya na granitse oblasti, Izv. AN UzSSR, ser. fiz. -mat. nauk, 1968, 5, 35-39 (In Russian).
- Beytmen G., Erdeyi A. Vysshiye transtsendentnyye funktsii, Tom 1 [Higher transcendental functions. Vol. 1], Moskva, Nauka, 1965, 296 (In Russian).
- Naymark M. A. Lineynyye differentsial’nyye operatory [Linear differential operators], Moskva, Nauka, 1969, 528 (In Russian).
- Mikhlin S. G. Lektsii po lineynym integral’nym uravneniyam [Lectures on linear integral equations], Moscow, Fizmatlit, 1959, 232 (In Russian).
- Boudabsa L., Simon T. Some Properties of the Kilbas-Saigo Function, Mathematics, 2021, 9, 217.
Information about authors
Urinov Akhmadzhon Kushakovich – D. Sci. (Phys. & Math.), Professor, Professor of the Department of Mathematical Analysis and Differential Equations, Fergana State University, Fergana, Uzbekistan, https://orcid.org/0000-0002-9586-1799.
Usmonov Doniyor Abdumutolib ugli – Researcher, Department of Mathematical Analysis and Differential Equations, Fergana State University, Fergana, Uzbekistan,