On an analogue of the tricomi problem for a > loaded hyperbolic-parabolic equation

K. U. Khubiev

Vestnik KRAUNC. Fiziko-matematičeskie nauki, Tome 36 (2021) no. 3, pp. 29-39

Résumé

The unique solvability of an analogue of the Tricomi problem is investigated for a loaded hyperbolic-parabolic equation. The load is determined at boundary and interior fixed points of the domain in which the solutions are sought. Sufficient conditions are found for the existence and uniqueness of solutions.

Détail

Keywords: loaded equation, equation of mixed type, hyperbolic-parabolic equation, Tricomi problem.

Bibliographie

[1] Nakhushev A. M., Nagruzhennye uravneniya i ikh primenenie [Loaded equations and their applications], Nauka, Moscow, 2012, 232 pp. (In Russian)

[2] Dikinov Kh. Zh., KerefovA. A. , Nakhushev A. M., “A certain boundary value problem for a loaded heat equation”, Differ. Equ., 12:1 (1976), 191–-192

[3] Nakhushev A. M., Uravneniya matematicheskoy biologii [Equations of mathematical biology], Vysshaya shkola, Moscow, 1995, 301 pp. (In Russian)

[4] Gel'fand I. M., “Nekotorye voprosy analiza i differentsial'nykh uravneniy [Some questions of analysis and differential equations]”, Uspekhi Mat. Nauk, 14:3(87) (1959), 3–19 (In Russian)

[5] Struchina G. M., “Zadacha o sopryazhenii dvukh uravneniy”, Inzhenerno-fizicheskiy zhurnal, 4:11 (1961), 99–104 (In Russian)

[6] Uflyand Ya. S., “K voprosu o rasprostranenii kolebaniy v sostavnykh elektricheskikh liniyakh”, Inzhenerno-fizicheskiy zhurnal, 7:1 (1964), 89–92 (In Russian)

[7] Zolina L. A., U.S.S.R. Comput. Math. Math. Phys., 6:6 (1966), 63–78 | Zbl

[8] Kaziev V. M., Kaygermazov A. A., Kudaeva Ph. Kh., “Zadacha Koshi dlya nagruzhennogo vyrozhdayushchegosya giperbolicheskogo uravneniya [The Cauchy problem for a loaded degenerate hyperbolic equation]”, Vestnik BGU. Matematika, informatika, 2018, 95–99 (In Russian)

[9] Jenaliyev M. T., Ramazanov M. I., Nagruzhennye uravneniya kak vozmushcheniya differentsial'nykh uravneniy [Loaded equation — how perturbed differential equationsy], Gylym, Almaty, 2010, 334 pp. (In Russian)

[10] Nakhushev A. M., “Boundary value problems for loaded integro-differential equations of hyperbolic type and some of their applications to the prediction of ground moisture”, Differ. Uravn., 15:1 (1979), 96–105 (In Russian)

[11] Kaziev V M., “Ob odnom nagruzhennom obyknovennom differentsial'nom uravnenii [On a loaded ordinary differential equation]”, Collection of scientific papers, Non-local boundary value problems for loaded mixed-type equations and related problems of continuous analysis, KBSU, Nalchik, 1982, 130–133 (In Russian)

[12] Assanova A. T., Imanchiyev A. E., Kadirbayeva Zh. M., “Numerical solution of systems of loaded ordinary differential equations with multipoint conditions”, Computational Mathematics and Mathematical Physics, 58:4 (2018), 508–516 | DOI | Zbl

[13] Eleev V. A., “Some boundary value problems for mixed loaded equations of second and third orders”, Differ. Equ., 30:2 (1994), 210–217 | Zbl

[14] Khubiev K. U., “Boundary-value problems for a characteristically loaded hyperbolic-parabolic equation”, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 195 (2021), 127–138 (In Russian)

[15] Khubiev K. U., “On an analogue of the Tricomi problem for a mixed hyperbolic-parabolic equation with a derivative under load”, Dokl. Adyg. (Cherkess.) Mezdunar. Akad. Nauk, 11:2 (2009), 58–60 (In Russian)

[16] Khubiev K. U., “Analogue of Tricomi problem for characteristically loaded hyperbolic-parabolic equation with variable coefficients”, Ufa Math. J., 9:2 (2017), 92–101 | Zbl

[17] Sabitov K. B., “Initial-boundary problem for parabolic-hyperbolic equation with loaded summands”, Russian Math. (Iz. VUZ), 59:6 (2015), 23–33 | Zbl

[18] Attaev A. Kh., “The characteristic problem for the second-order hyperbolic equation loaded along one of its characteristics”, Vestnik KRAUNC. Fiz.-Mat. Nauki, 2018, no. 3(23), 14–18 (In Russian) | Zbl

[19] Sabitova Yu. K., “Dirichlet problem for Lavrent'ev-Bitsadze equation with loaded summands”, Russian Math. (Iz. VUZ), 62:9 (2018), 35–51 | Zbl

[20] Attaev A. Kh., “On a characteristic problem for a loaded hyperbolic equation”, Bulletin of the Karaganda University. Mathematics Series, 96:4 (2019), 15–21 | DOI

[21] Agarwal P., Baltaeva U., Alikulov Y., “Solvability of the boundary-value problem for a linear loaded integro-differential equation in an infinite three-dimensional domain”, Chaos, Solitons and Fractals, 140 (2020), 110108 | DOI

[22] Zikirov O. S., Kholikov D. K., “Solvability of some non–local problems for the loaded pseudoparabolic equation”, Sib. Èlektron. Mat. Izv., 17 (2020), 77–88 (In Russian) | Zbl

[23] Khubiev K. U., “Boundary-value problem for a loaded hyperbolic-parabolic equation with degeneracy of order in the hyperbolicity domain”, Journal of Mathematical Sciences, 250:5 (2020), 830–834 | DOI | Zbl

[24] Islomov B. I., Alikulov Y. K., “Boundary value problem for loaded equation of parabolic-hyperbolic type of the third order in an infinite three-dimensional domain”, International Journal of Applied Mathematics, 34:2 (2021), 377–389 | DOI

[25] Urinov A. K., Azizov M. S., “A boundary problem for the loaded partial differential equations of fourth order”, Lobachevskii Journal of Mathematics, 42:3 (2021), 621–631 | DOI | Zbl

[26] Yuldashev T. K., Abdullaev O. K., “Unique solvability of a boundary value problem for a loaded fractional parabolic-hyperbolic equation with nonlinear terms”, Lobachevskii Journal of Mathematics, 42:5 (2021), 1113–1123 | DOI | Zbl