# An initial-boundary problem for a hyperbolic equation with three lines of degenerating of the second kind

**Authors:**Urinov A.K.^{1}^{,2}, Usmonov D.A.^{1}-
**Affiliations:**- Fergana State University
- Institute of Mathematics named after V. I. Romanovsky of the Academy of Sciences of the Republic of Uzbekistan

**Issue:**Vol 26, No 4 (2022)**Pages:**672-693**Section:**Differential Equations and Mathematical Physics**URL:**https://journals.eco-vector.com/1991-8615/article/view/111846**DOI:**https://doi.org/10.14498/vsgtu1962**ID:**111846

Cite item

## Full Text

## Abstract

In present paper, a hyperbolic partial differential equation of the second kind degenerates on the sides and on the base of the rectangle has been considered. For the considered equation, an initial-boundary problem with non-local boundary conditions has been formulated. The uniqueness, existence, and stability of the solution to the stated problem were investigated. The uniqueness of the solution to the problem was proved by the method of energy integrals. The existence of obtained solution was investigated by the Fourier method based on the separation of variables. To do this first, a spectral problem for an ordinary differential equation was investigated, which arises from the formulated problem in virtue of the separation of variables. It was proved that this spectral problem can have only positive eigenvalues. Then, Green's function of the spectral problem has been constructed, and equivalently reduced to an integral Fredholm equation of the second kind with a symmetric kernel. Hence, on the basis of the theory of integral equations, it has been concluded that there is a countable number of eigenvalues and eigenfunctions of the spectral problem. Using the properties of constructed Green's function of the spectral problem, some lemmas, using to prove the existence of a solution to the problem on the uniform convergence of some bilinear series, were proved. Lemmas on the order of the Fourier coefficients of a given function have also been proved. The solution of the considered problem is derived 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 proved using the lemmas mentioned above. At the end of the paper, two estimates are obtained for the solution to the problem, one of which is in the space of square summable functions with weight, and the other is in the space of continuous functions. These inequalities imply the stability of the solution in the corresponding spaces.

## About the authors

### Akhmadjon K. Urinov

Fergana State University; Institute of Mathematics named after V. I. Romanovsky of the Academy of Sciences of the Republic of Uzbekistan
Email: urinovak@mail.ru

ORCID iD: 0000-0002-9586-1799

Scopus Author ID: 19639412400

http://www.mathnet.ru/person30024

Dr. Phys. & Math. Sci.; Professor; Dept. of Mathematical Analysis and Differential Equations; Leading Researcher

Uzbekistan, 150100, Fergana, Murabbiylar st., 19; 100174, Tashkent, Universitetskaya st., 46### Doniyor A. Usmonov

Fergana State University
**Author for correspondence.**

Email: usmonov-doniyor@inbox.ru

ORCID iD: 0000-0002-3574-075X

http://www.mathnet.ru/person191330

Researcher; Dept. of Mathematical Analysis and Differential Equations

Uzbekistan, 150100, Fergana, Murabbiylar st., 19## References

- Koshlyakov N. S., Smirnov M. M., Gliner E. B. Differential equations of mathematical physics. Amsterdam, North-Holland Publ., 1964, xvi+701 pp.
- Karol’ I. L. On the theory of equations of mixed type, Dokl. Akad. Nauk SSSR, 1953, vol. 88, no. 3, pp. 397–400 (In Russian).
- Tersenov S. A. On the Cauchy problem, with data given on a curve of degenerate type, for a hyperbolic equation, Differ. Uravn., 1966, vol. 2, no. 1, pp. 125–130 (In Russian).
- Tersenov S. A. On the theory of hyperbolic equations with data on a line of degeneration of type, Sibirsk. Mat. Zh., 1961, vol. 2, no. 6, pp. 913–935 (In Russian).
- Tersenov S. A. Vvedenie v teoriiu uravnenii, vyrozhdaiushchikhsia na granitse [Introduction to the Theory of Equations Degenerating at the Boundary]. Novosibirsk, Novosib. Gos. Univ., 1973, 144 pp. (In Russian)
- Smirnov M. M. Vyrozhdaiushchiesia giperbolicheskie uravneniia [Degenerate Hyperbolic Equations]. Minsk, Vyssh. shk., 1977, 157 pp. (In Russian)
- Khairullin R. S. Zadacha Trikomi dlia uravneniia vtorogo roda s sil’nym vyrozhdeniem [Tricome’s Problem for a Equation of the Second Kind with Strong Degeneration]. Kazan, Kazan Univ., 2015, 336 pp. (In Russian). EDN: UWLDMB.
- Mamadaliev N. K. On representation of a solution to a modified Cauchy problem, Sib. Math. J., 2000, vol. 41, no. 5, pp. 889–899. DOI: https://doi.org/10.1007/BF02674745.
- Urinov A. K., Okboev A. B. Modified Cauchy problem for one degenerate hyperbolic equation of the second kind, Ukr. Math. J., 2020, vol. 72, no. 1, pp. 114–135. EDN: XDIOGV. DOI: https://doi.org/10.1007/s11253-020-01766-1.
- Urinov A. K., Okboev A. B. On a Cauchy type problem for a second kind degenerating hyperbolic equation, Lobachevskii J. Math., 2022, vol. 43, no. 3, pp. 793–803. EDN: QPEVQB. DOI: https://doi.org/10.1134/S1995080222060324.
- Urinov A. K., Usmonov D. A. About modified Cauchy problem for a second kind degenerated hyperbolic equation, Bull. Inst. Math., 2021, vol. 4, no. 1, pp. 46–63 (In Russian).
- Ehrgashev T. G. Generalized solutions of the degenerate hyperbolic equation of the second kind with a spectral parameter, Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2017, no. 46, pp. 41–49 (In Russian). EDN: YPDTPD. DOI: https://doi.org/10.17223/19988621/46/6.
- Baikuziev K. B. Mixed problem for one hyperbolic equation degenerating on a contour, Izv. Akad. Nauk UzSSR, Ser. Fiz.-Mat. Nauk, 1962, vol. 6, no. 2, pp. 83–85 (In Russian).
- Karimov D. H., Baikuziev K. B. A mixed problem for a hyperbolic equation degenerating on the boundary of the domain, Nauchn. Tr. Tashkent. Gos. Univ., 1962, vol. 208, pp. 90–97 (In Russian).
- Karimov D. H., Baikuziev K. B. The second mixed problem for one hyperbolic equation degenerating on the boundary of the domain, Izv. Akad. Nauk UzSSR, Ser. Fiz.-Mat. Nauk, 1964, vol. 8, no. 6, pp. 27–30 (In Russian).
- Baikuziev K. B. On the solvability of mixed problems for a class of non-linear equations of hyperbolic type that degenerate on the boundary of a domain, Izv. Akad. Nauk UzSSR, Ser. Fiz.-Mat. Nauk, 1967, vol. 11, no. 2, pp. 3–6 (In Russian).
- Baikuziev K. B., Karimov D. H. On the solvability of a mixed problem for hyperbolic equations that degenerates on the entire boundary of the domain, Tr. Tashkent. Gos. Univ., 1969, vol. 2, no. 350, pp. 8–20 (In Russian).
- Datkabayev D. A mixed problem for a system of second-order equations that degenerate on the entire boundary of the domain, Probl. Fiz.-Mat. Nauk. Tashkent, 1976, vol. 164, pp. 32–38 (In Russian).
- Krasnov M. L. Mixed boundary problems for degenerate linear hyperbolic differential equations second order, Mat. Sb. (N.S.), 1959, vol. 49(91), no. 1, pp. 29–84 (In Russian).
- Oleinik O. A. The Cauchy problem and the boundary value problem for second-order hyperbolic equations degenerating in a domain and on its boundary, Sov. Math., Dokl., 1966, vol. 7, pp. 969–973.
- Bryukhanov V. A. A mixed problem for a hyperbolic equation degenerate on a part of the bounday of a region, Differ. Uravn., 1972, vol. 8, no. 1, pp. 3–6 (In Russian).
- Vragov V. N. A mixed problem for a certain class of second order hyperbolic-parabolic equations, Differ. Uravn., 1976, vol. 12, no. 1, pp. 24–31 (In Russian).
- Bubnov B. A. A mixed problem for certain parabolic-hyperbolic equations, Differ. Uravn., 1976, vol. 12, no. 3, pp. 494–501 (In Russian).
- Baranovskii F. T. Mixed problem for a second-order hyperbolic equation which is strongly degenerate on the initial plane, Sib. Math. J., 1979, vol. 20, no. 3, pp. 338–346. DOI: https://doi.org/10.1007/BF00969936.
- Baranovskii F. T. A mixed boundary value problem for a hyperbolic equation with degenerate principal part, Sb. Math., 1982, vol. 43, no. 4, pp. 499–513. DOI: https://doi.org/10.1070/SM1982v043n04ABEH002577.
- Sabitov K. B., Suleimanova A. Kh. The Dirichlet problem for a mixed-type equation of the second kind in a rectangular domain, Russian Math. (Iz. VUZ), 2007, vol. 51, no. 4, pp. 42–50. DOI: https://doi.org/10.3103/S1066369X07040068.
- Sabitov K. B., Suleimanova A. Kh. The Dirichlet problem for a mixed-type equation with characteristic degeneration in a rectangular domain, Russian Math. (Iz. VUZ), 2009, vol. 53, no. 11, pp. 37–45. DOI: https://doi.org/10.3103/S1066369X0911005X.
- Sabitov K. B., Egorova I. P. On the correctness of boundary value problems for the mixed type equation of the second kind, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2019, vol. 23, no. 3, pp. 430–451 (In Russian). EDN: KBIAPC. DOI: https://doi.org/10.14498/vsgtu1718.
- Khairullin R. S. Dirichlet problem for a mixed type equation of the second kind in exceptional cases, Differ. Equat., 2018, vol. 54, no. 4, pp. 562–562. EDN: UXXARG. DOI: https://doi.org/10.1134/S0012266118040134.
- Khairullin R. S. Problem with a periodicity condition for an equation of the mixed type with strong degeneration, Differ. Equat., 2019, vol. 55, no. 8, pp. 1105–1117. EDN: CYEAGU. DOI: https://doi.org/10.1134/S0012266119080111.
- Erdélyi A., Magnus W., Oberhettinger F., Tricomi F. G. Higher transcendental functions, vol. II, Bateman Manuscript Project. New York, Toronto, London, McGraw-Hill Book Co., 1953, xvii+396 pp.
- Naimark M. A. Lineinye differentsial’nye operatory [Linear Differential Operators]. Moscow, Fizmatlit, 2010, 528 pp. (In Russian). EDN: RYRSSP.
- Mikhlin S. G. Lektsii po lineinym integral’nym uravneniiam [Lectures on Linear Integral Equations]. Moscow, Fizmatlit, 1959, 232 pp. (In Russian)
- Watson G. N. A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library. Cambridge, Cambridge Univ., 1995, vi+804 pp.

## Supplementary files

There are no supplementary files to display.