Geodesic
The Dirichlet problem for a~loaded mixed-type equation in a~rectangular domain
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2013), pp. 62-76.

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We establish necessary and sufficient conditions for the unique solvability of the first boundary problem for a loaded equation with the Lavrent'ev–Bitsadze operator in a rectangular domain. We obtain a solution to the stated problem as the sum of the eigenfunction series for the corresponding one-dimensional problem with respect to eigenvalues. We prove the stability of the solution with respect to boundary functions.
Keywords: loaded mixed-type equation, Dirichlet problem, spectral method, uniqueness, stability.
Mots-clés : existence 
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K. B. Sabitov; E. P. Melisheva. The Dirichlet problem for a~loaded mixed-type equation in a~rectangular domain. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2013), pp. 62-76. http://geodesic.mathdoc.fr/item/IVM_2013_7_a5/

[1] Sabitov K. B., “Nachalno-granichnaya zadacha dlya nagruzhennogo uravneniya parabolo-giperbolicheskogo tipa”, Dokl. AMAN (Nalchik), 11:1 (2009), 66–73 | MR

[2] Sabitov K. B., “Zadacha Trikomi dlya uravneniya smeshannogo parabolo-giperbolicheskogo tipa v pryamougolnoi oblasti”, Matem. zametki, 86:2 (2009), 273–279 | DOI | MR | Zbl

[3] Cannon J. R., “A Dirichlet problem for an equation of mixed type with a discontinuous coefficient”, Ann. Math. Pura Appl. IV Ser., 62 (1963), 371–377 | DOI | MR | Zbl

[4] Shabat B. V., “Primery resheniya zadachi Dirikhle dlya uravneniya smeshannogo tipa”, DAN SSSR, 112:3 (1957), 386–389 | Zbl

[5] Nakhushev A. M., “Kriterii edinstvennosti zadachi Dirikhle dlya uravnenii smeshannogo tipa v tsilindricheskoi oblasti”, Differents. uravneniya, 6:1 (1970), 190–191 | Zbl

[6] Khachev M. M., “Zadacha Dirikhle dlya obobschennogo uravneniya Lavrenteva–Bitsadze v pryamougolnoi oblasti”, Differents. uravneniya, 14:1 (1978), 136–139 | MR | Zbl

[7] Zhegalov V. I., “Nelokalnaya zadacha Dirikhle dlya uravneniya smeshannogo tipa”, Neklassicheskie uravneniya matem. fiziki, IM SO AN SSSR, Novosibirsk, 1985, 168–172

[8] Soldatov A. P., “Zadacha Dirikhle dlya uravneniya Lavrenteva–Bitsadze”, Differents. uravneniya, 30:11 (1994), 201–209 | MR

[9] Sabitov K. B., “Zadacha Dirikhle dlya uravneniya smeshannogo tipa v pryamougolnoi oblasti”, Dokl. RAN, 413:1 (2007), 23–26 | MR | Zbl

[10] Sabitova Yu. K., Bakhristova A. A., “Zadacha Dirikhle dlya uravneniya smeshannogo tipa s operatorom Lavrenteva–Bitsadze”, Tr. Sterlitamaksk. filiala AN RB. Ser. Fiz.-matem. i tekhn. nauki, 6, Gilem, Ufa, 2009, 103–110

[11] Nakhushev A. M., “O zadache Darbu dlya odnogo vyrozhdayuschegosya nagruzhennogo integrodifferentsialnogo uraneniya vtorogo poryadka”, Differents. uravneniya, 12:1 (1976), 103–108 | MR | Zbl

[12] Kaziev V. M., “Zadacha Trikomi dlya nagruzhennogo uravneniya Lavrenteva-Bitsadze”, Differents. uravneniya, 15:1 (1979), 173–175 | MR | Zbl

[13] Nakhushev A. M., “Nagruzhennye uravneniya i ikh prilozheniya”, Differents. uravneniya, 19:1 (1983), 86–94 | MR | Zbl

[14] Dzhenaliev M. T., K teorii lineinykh kraevykh zadach dlya nagruzhennykh differentsialnykh uravnenii, Almaty, 1995

[15] Pulkina L. S., “Nelokalnaya zadacha dlya nagruzhennogo giperbolicheskogo uravneniya”, Tr. MIAN, 236, 2002, 298–303 | MR | Zbl

[16] Kozhanov A. I., “Nelineinye nagruzhennye uravneniya i obratnye zadachi”, Vychisl. matem. i matem. fiz., 44:4 (2004), 694–716 | MR | Zbl

[17] Khubiev K. U., Lokalnye i nelokalnye kraevye zadachi dlya nagruzhennykh uravnenii smeshannogo giperbolo-parabolicheskogo tipa, Avtoreferat diss. $\dots$ kand. fiz.-matem. nauk, Belgorod, 2009

[18] Melisheva E. P., “Zadacha Dirikhle dlya nagruzhennogo uravneniya Lavrenteva–Bitsadze”, Vest. SamGU. Estestvennonauchnaya seriya, 80:6 (2010), 39–47

[19] Arnold V. I., “Malye znamenateli i problemy ustoichivosti dvizheniya v klassicheskoi i nebesnoi mekhanike”, UMN, 18:6 (1963), 91–192 | MR | Zbl

[20] Khinchin A. Ya., Tsepnye drobi, Nauka, M., 1978 | MR

[21] Martemyanova N. V., Nelokalnaya obratnaya zadacha dlya uravnenii smeshannogo elliptiko-giperbolicheskogo tipa, Avtoreferat diss. $\dots$ kand. fiz.-matem. nauk, Kazan, 2012