Nonlocal boundary-value problem with Bitsadze--Samarskii condition for equation of parabolic-hyperbolic type of the second kind

M. S. Salakhitdinov ; N. B. Islamov

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 6 (2015), pp. 43-52.

Résumé

We prove the unique solability of nonlocal boundary-value problem for a degenerate parabolic-hyperbolic equation of the second kind in the case when on the first part of a characteristic a nonlocal boundary condition is specified, while on parallel characteristic the Bitsadze–Samarskii condition is specified.

Détail

Keywords: degenerate parabolic-hyperbolic equation, nonlocal boundary-value problem, equation of the second kind, Bitsadze–Samarskii condition, uniqueness and existence of a solution, extremum principle, Fredholm integral equation.

Bibliographie

[1] Bitsadze A. V., Izbrannye trudy, Izd-vo KBNTsRAN, Rossiya, 2012

[2] Salakhitdinov M. S., Izbrannye nauchnye trudy, Izd-vo Mumtoz Suz, Tashkent, 2013

[3] Dzhuraev T. D., Kraevye zadachi dlya uravnenii smeshannogo i smeshanno-sostavnogo tipa, Izd-vo FAN, Tashkent, 1979 | MR

[4] Moiseev E. I., Uravneniya smeshannogo tipa so spektralnym parametrom, Izd-vo Mosk. un-ta, 1988 | MR

[5] Nakhushev A. M., Ob odnom klasse lineinykh kraevykh zadach dlya giperbolicheskogo i smeshannogo tipov uravnenii vtorogo poryadka, Izd-vo KBNTsRAN, Rossiya, 1992

[6] Kalmenov T. Sh., “O spektre zadachi Trikomi dlya uravneniya Lavrenteva–Bitsadze”, Differents. uravneniya, 13:8 (1977), 1718–1725 | MR

[7] Sabitov K. B., “Nelokalnaya zadacha dlya uravneniya parabolo-giperbolicheskogo tipa v pryamougolnoi oblasti”, Matem. zametki, 89:4 (2011), 596–602 | DOI | MR | Zbl

[8] Sabitov K. B., “Nachalno-granichnaya zadacha dlya parabolo-giperbolicheskogo uravneniya so stepenným vyrozhdeniem na perekhodnoi linii”, Differents. uravneniya, 47:10 (2011), 1474–1481 | MR | Zbl

[9] Salakhitdinov M. S., Islomov B. I., “Nelokalnaya kraevaya zadacha s konormalnoi proizvodnoi dlya uravneniya smeshannogo tipa s dvumya vnutrennimi liniyami i razlichnymi poryadkami vyrozhdeniya”, Izv. vuzov. Matem., 2011, no. 1, 49–58 | MR | Zbl

[10] Karol I. L., “Ob odnoi kraevoi zadache dlya uravneniya smeshannogo elliptiko-giperbolicheskogo tipa”, DAN SSSR, 88:2 (1953), 197–200 | MR | Zbl

[11] Karol I. L., “K teorii kraevykh zadach dlya uravneniya smeshannogo elliptiko-giperbolicheskogo tipa”, Matem. sb., 38(80):3 (1956), 261–282 | MR | Zbl

[12] Salakhitdinov M. C., Isamukhamedov S. S., “O nekotorykh smeshannykh zadachakh dlya uravneniya $\operatorname{sgn}y|y|^{-m}u_{xx}+u_{yy}=0$, $01$”, Izv. AN UzSSR. Ser. fiz.-matem. n., 1970, no. 5, 15–19 | MR | Zbl

[13] Smirnov M. M., “Kraevaya zadacha dlya uravneniya smeshannogo tipa $2$-go roda so smescheniem”, Differents. uravneniya, 13:5 (1977), 931–943 | MR | Zbl

[14] Ivashkina G. A., “O zadachakh tipa Bitsadze–Samarskogo dlya uravneniya $u_{xx}+\operatorname{sgn}y|y|^mu_{yy}=0$, $01$”, Differents. uravneniya, 17:6 (1981), 1078–1089 | MR | Zbl

[15] Sabitov K. B., Bibakova S. L., “Postroenie sobstvennykh funktsii zadachi Trikomi–Neimana dlya uravneniya smeshannogo tipa s kharakteristicheskim vyrozhdeniem i ikh primenenie”, Matem. zametki, 74:1 (2003), 76–87 | DOI | MR | Zbl

[16] Dzhuraev N., “Zadacha Trikomi dlya smeshannogo parabolo-giperbolicheskogo uravneniya s dvumya liniyami vyrozhdeniya vtorogo roda”, Izv. AN UzSSR. Ser. fiz.-matem. n., 1989, no. 2, 19–23 | MR | Zbl

[17] Salakhitdinov M. C., Ergashev T. G., “O dvukh kraevykh zadachakh so smescheniem dlya vyrozhdayuschegosya giperbolicheskogo uravneniya vtorogo roda”, DAN USSR, 1991, no. 7, 3–5 | MR

[18] Islamov B., Zhumoev B., “Analog zadachi Trikomi dlya vyrozhdayuschegosya parabolo-giperbolicheskogo uravneniya vtorogo roda”, Mezhdunar. Rossiisko-Bolgarskii simpozium “Uravneniya smeshannogo tipa i rodstvennye probl. analiza i informatiki”, Nalchik, 2010, 106–108

[19] Zhumoev B., “Nelokalnaya kraevaya zadacha dlya uravneniya parabolo-giperbolicheskogo tipa vtorogo roda, vyrozhdayuschegosya vnutri oblasti”, Uzb. matem. zhurn., 2012, no. 1, 38–46

[20] Salakhitdinov M. C., Mirsaburov M., Nelokalnye zadachi dlya uravneniya smeshannogo tipa s singulyarnymi koeffitsientami, Izd-vo Universitet, Tashkent, 2005

[21] Islamov N. B., “Nelokalnaya kraevaya zadacha dlya uravneniya parabolo-giperbolicheskogo tipa vtorogo roda”, Mezhdunar. konf. molodykh uchenykh “Matem. modelir. fraktalnykh protsessov, rodstvennye probl. analiza i informatiki”, Nalchik, 2011, 122–125

[22] Islamov N. B., “Ob odnoi nelokalnoi kraevoi zadache dlya uravneniya elliptiko-giperbolicheskogo tipa vtorogo roda”, Uzb. matem. zhurn., 2012, no. 4, 38–50

[23] Smirnov M. M., Uravneniya smeshannogo tipa, Vyssh. shkola, M., 1985 | MR

[24] Tersenov S. A., Vvedenie v teoriyu uravneniya, vyrozhdayuschegosya na granitse, NGU, Novosibirsk, 1973

[25] Islamov B., Akbarova S., “Kraevye zadachi dlya uravneniya parabolo-giperbolicheskogo tipa s dvumya liniyami i razlichnymi poryadkami vyrozhdeniya”, Tez. dokl. vsesoyuzn. konf. “Differents. uravneniya i optimalnoe upravlenie”, Ashkhabad, 1990, 65–66

[26] Mikhlin S. G., Lektsii po lineinym integralnym uravneniyam, Fizmatgiz, M., 1959 | MR

[27] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii, v. 1, Nauka, M., 1965

[28] Salakhitdinov M. S., Islamov B., Uravneniya smeshannogo tipa s dvumya linyami vyrozhdeniya, Izd. Mumtoz suz, Tashkent, 2009