Geodesic
An initial boundary value problem for a partial differential equation of higher even order with a Bessel operator
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 26 (2022) no. 2, pp. 273-292.

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In present paper, an initial-boundary value problem is formulated in a rectangle for a higher even order partial differential equation with the Bessel operator. Applying the method of separation of variables to the considered problem a spectral problem is obtained for an ordinary differential equation of higher even order. The self-adjointness of the last problem is proved, which implies the existence of the system of its eigenfunctions, as well as the orthonormality and completeness of this system. The uniform convergence of some bilinear series and the order of the Fourier coefficients, depending on the found eigenfunctions, is investigated. The solution of the considered problem is found as the sum of the Fourier series with respect to the system of eigenfunctions of the spectral problem. The absolute and uniform convergence of this series, as well as the series obtained by its differentiating, have been proved. The uniqueness of the solution of the problem is proved by the method of spectral analysis. An estimate is obtained for the solution of the problem which implies the continuous dependence of the solution on the given functions.
Keywords: even order partial differential equation, Bessel operator, initial-boundary value problem, spectral method, Green's function, integral equation, uniqueness and stability of the solution.
Mots-clés : existence 
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A. K. Urinov; M. S. Azizov. An initial boundary value problem for a partial differential equation of higher even order with a  Bessel operator. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 26 (2022) no. 2, pp. 273-292. http://geodesic.mathdoc.fr/item/VSGTU_2022_26_2_a3/

[1] Tikhonov A. N., Samarskiy A. A., Uravneniia matematicheskoi fiziki [Equations of Mathematical Physics], Nauka, Moscow, 1972, 736 pp. (In Russian)

[2] Nakhushev A. M., Uravneniia matematicheskoi biologii [Equations of Mathematical Biology], Vyssh. shk., Moscow, 1995, 301 pp. (In Russian)

[3] Salakhitdinov M. S., Amanov D., “Solvability and spectral properties of a selfadjoint problem for a fourth-order equation”, Uzbek. Mat. Zh., 2005, no. 3, 72–77 (In Russian)

[4] Amanov D., Yuldasheva A. V., “Solvability and spectral properties of a selfadjoint problem for a fourth-order equation”, Uzbek. Mat. Zh., 2007, no. 4, 3–8 (In Russian) | Zbl

[5] Amanov D., Murzambetova M. B., “Boundary value problems for a fourth order equation with a spectral parameter”, Uzbek. Mat. Zh., 2012, no. 3, 22–30 (In Russian)

[6] Amanov D., Murzambetova M. B., “A boundary value problem for a fourth order partial differential equation with the lowest term”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2013, no. 1, 3–10 (In Russian)

[7] Otarova Zh. A., “The solvability and spectral properties of selfadjoint problems for a fourth-order equation”, Uzbek. Mat. Zh., 2008, no. 2, 74–80 (In Russian) | Zbl

[8] Otarova Zh. A., “Solvability and spectral properties of a selfadjoint problem for a fourth-order equation”, Dokl. AN RUz., 2008, no. 1, 10–14 (In Russian)

[9] Otarova Zh. A., “Volterra boundary value problem for a fourth order equation”, Dokl. AN RUz., 2008, no. 6, 18–22 (In Russian)

[10] Sabitov K. B., “Cauchy problem for the beam vibration equation”, Differ. Equ., 53:5 (2017), 658–664 | DOI | DOI

[11] Sabitov K. B., “Fluctuations of a beam with clamped ends”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 19:2 (2015), 311–324 (In Russian) | DOI | Zbl

[12] Sabitov K. B., “A remark on the theory of initial-boundary value problems for the equation of rods and beams”, Differ. Equ., 53:1 (2017), 86–98 | DOI | DOI

[13] Sabitov K. B., Fadeeva O. V., “Initial-boundary value problem for the equation of forced vibrations of a cantilever beam”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 25:1 (2021), 51–66 (In Russian) | DOI | Zbl

[14] Azizov M. S., “A boundary problem for the fourth order equation with a singular coefficient in a rectangular region”, Lobachevskii J. Math., 41:6 (2020), 1043–1050 | DOI

[15] Azizov M. S., “A mixed problem for a fourth-order nonhomogeneous equation with singular coefficients in a rectangular”, Bul. Inst. Math., 2020, no. 4, 50–59 (In Russian)

[16] Amanov D., Yuldasheva A. V., “Solvability and spectral properties of boundary value problems for equations of even order”, Malays. J. Math. Sci., 3:2 (2009), 227–248 https://mjms.upm.edu.my/lihatmakalah.php?kod=2009/July/3/2/227-248

[17] Amanov D., “About correctness of boundary value problems for equation of even order”, Uzbek Math. J., 2011, no. 4, 20–35

[18] Yuldasheva A. V., “On one proble for higher-order equation”, Bulletin KRASEC. Phys. Math. Sci., 9:2 (2014), 18–22 | DOI | DOI

[19] Yuldasheva A. V., “On a problem for a quasi-linear equation of even order”, J. Math. Sci., 241:4 (2019), 423–429 | DOI | MR | Zbl

[20] Amanov D., Ashyralyev A., “Well-posedness of boundary value problems for partial diffferential equations of even order”, AIP Conference Proceedings, 1470:1 (2012), 3 | DOI

[21] Ashurov R. R., Muhiddinova O. T., “Initial-boundary value problem for hyperbolic equations with an arbitrary order elliptic operator”, Vestnik KRAUNC. Fiz.-Mat. Nauki, 30:1 (2020), 8–19 (In Russian) | DOI

[22] Ashurov R. R., Muhiddinova O. T., “Initial-boundary value problem for a time-fractional subdiffusion equation with an arbitrary elliptic differential operator”, Lobachevskii J. Math., 42:2 (2021), 517–525 | DOI

[23] Karimov Sh. T., “Method of solving the Cauchy problem for one-dimensional polywave equation with singular Bessel operator”, Russian Math. (Iz. VUZ), 61:8 (2017), 22–35 | DOI

[24] Karimov Sh. T., “On some generalizations of properties of the Lowndes operator and their applications to partial differential equations of high order”, Filomat, 32:3 (2018), 873–883 | DOI

[25] Karimov Sh. T., “The Cauchy problem for the degenerated partial differential equation of the high even order”, Sib. Elektron. Mat. Izv., 2018, no. 15, 853–862 | DOI

[26] Karimov Sh. T., Urinov A. K., “Solution of the Cauchy problem for the four-dimensional hyperbolic equation with Bessel operator”, Vladikavkaz. Mat. Zh., 20:3 (2018), 57–68 (In Russian) | DOI

[27] Urinov A. K., Karimov Sh. T., “On the Cauchy problem for the iterated generalized two-axially symmetric equation of hyperbolic type”, Lobachevskii J. Math., 41:1 (2020), 102–110 | DOI

[28] Mikhlin S. G., Linear Integral Equations, Dover Publ., Mineola, NY, 2020, xv+223 pp. | Zbl

[29] Naimark M. A., Lineinye differentsial'nye operatory [Linear Differential Operators], Fizmatlit, Moscow, 1969, 528 pp. (In Russian) | Zbl