Geodesic
Construction of the Riemann--Hadamard function for the three-dimensional Bianchi equation
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 3 (2021), pp. 76-82.

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For a third-order equation with a dominant partial derivative (the Bianchi equation), the statement of the Darboux problem and the definition of the Riemann–Hadamard function are given. Based on the possibility of representing the Riemann function explicitly for a class of third-order Bianchi equations equivalent in function, sufficient conditions are proposed for the coefficients of the Bianchi equation that provide construction of the Riemann–Hadamard function in terms of hypergeometric functions.
Mots-clés : Bianchi equation, Laplace invariants.
Keywords: Darboux problem, Riemann–Hadamard function, Riemann function 
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A. N. Mironov. Construction of the Riemann--Hadamard function for the three-dimensional Bianchi equation. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 3 (2021), pp. 76-82. http://geodesic.mathdoc.fr/item/IVM_2021_3_a6/

[1] Bitsadze A. V., Nekotorye klassy uravnenii v chastnykh proizvodnykh, Nauka, M., 1981

[2] Moiseev E. I., “Ob odnom integralnom predstavlenii resheniya zadachi Darbu”, Matem. zametki, 32:2 (1982), 175–186 | MR | Zbl

[3] Moiseev E. I., “O priblizhenii klassicheskogo resheniya zadachi Darbu gladkimi resheniyami”, Differents. uravneniya, 20:1 (1984), 73–87 | MR | Zbl

[4] Moiseev E. I., Uravneniya smeshannogo tipa so spektralnym parametrom, Izd-vo MGU, M., 1988

[5] Sabitov K. B., “Postroenie v yavnom vide reshenii zadach Darbu dlya telegrafnogo uravneniya i ikh primenenie pri obraschenii integralnykh uravnenii. I”, Differents. uravneniya, 26:6 (1990), 1023–1032 | MR

[6] Sabitov K. B., Sharafutdinova G. G., “Zadachi Koshi–Gursa dlya vyrozhdayuschegosya giperbolicheskogo uravneniya”, Izv. vuzov. Matem., 2003, no. 5, 21–29 | Zbl

[7] Dzhokhadze O. M., Kharibegashvili S. S., “Nekotorye svoistva funktsii Rimana i Grina–Adamara dlya lineinykh giperbolicheskikh uravnenii vtorogo poryadka i ikh prilozheniya”, Differents. uravneniya, 47:4 (2011), 477–492 | MR

[8] Zhegalov V. I., “Trekhmernyi analog zadachi Gursa”, Neklassicheskie uravneniya i uravneniya smeshannogo tipa, IM SO AN SSSR, Novosibirsk, 1990, 94–98 | MR | Zbl

[9] Volkodavov V. F., Nikolaev N. Ya., Bystrova O. K., Zakharov V. N., Funktsii Rimana dlya nekotorykh differentsialnykh uravnenii v $n$-mernom evklidovom prostranstve i ikh primeneniya, Samarsk. un-t, Samara, 1995

[10] Korzyuk V. I., “Granichnaya zadacha dlya uravneniya Manzherona tretego poryadka”, Differents. uravneniya, 33:12 (1997), 1683–1690 | MR | Zbl

[11] Sevastyanov V. A., “Metod Rimana dlya trekhmernogo giperbolicheskogo uravneniya tretego poryadka”, Izv. vuzov. Matem., 1997, no. 5, 69–73 | Zbl

[12] Dzhokhadze O. M., “Zadacha tipa Darbu v trekhgrannom ugle dlya uravneniya tretego poryadka giperbolicheskogo tipa”, Izv. vuzov. Matem., 1999, no. 3, 22–30 | MR | Zbl

[13] Zhegalov V. I., Mironov A. N., Differentsialnye uravneniya so starshimi chastnymi proizvodnymi, Izd-vo Kazansk. matem. ob-va, Kazan, 2001

[14] Zhegalov V. I., Mironov A. N., Utkina E. A., Uravneniya s dominiruyuschei chastnoi proizvodnoi, Izd-vo Kazansk. matem. ob-va, Kazan, 2014

[15] Mironov A. N., “Nekotorye klassy uravnenii Bianki tretego poryadka”, Matem. zametki, 94:3 (2013), 389–400 | Zbl

[16] Mironov A. N., “Zadacha Darbu dlya uravneniya Bianki tretego poryadka”, Matem. zametki, 102:1 (2017), 64–71 | MR | Zbl

[17] Zhegalov V. I., “O trekhmernoi funktsii Rimana”, Sib. matem. zhurn., 38:5 (1997), 1074–1079 | MR | Zbl

[18] Zhegalov V. I., Kotukhov M. P., “Ob integralnykh uravneniyakh dlya funktsii Rimana”, Izv. vuzov. Matem., 1998, no. 1, 26–30 | Zbl

[19] Mironov A. N., “O postroenii funktsii Rimana dlya odnogo uravneniya v $n$-mernom prostranstve”, Izv. vuzov. Matem., 1999, no. 7, 78–80 | Zbl

[20] Mironov A. N., “O postroenii funktsii Rimana dlya odnogo uravneniya chetvertogo poryadka”, Differents. uravneniya, 37:12 (2001), 1698–1701 | MR | Zbl

[21] Zhegalov V. I., “K sluchayam razreshimosti giperbolicheskikh uravnenii v terminakh spetsialnykh funktsii”, Neklassicheskie uravneniya matem. fiziki, IM SO AN SSSR, Novosibirsk, 2002, 73–79

[22] Zhegalov V. I., “O sluchayakh razreshimosti giperbolicheskikh uravnenii v kvadraturakh”, Izv. vuzov. Matem., 2004, no. 7, 47–52 | MR | Zbl

[23] Koscheeva O. A., “O postroenii funktsii Rimana dlya uravneniya Bianki v $n$-mernom prostranstve”, Izv. vuzov. Matem., 2008, no. 9, 40–46

[24] Dzhokhadze O. M., “Ob invariantakh Laplasa dlya nekotorykh klassov lineinykh differentsialnykh uravnenii v chastnykh proizvodnykh”, Differents. uravneniya, 40:1 (2004), 58–68 | MR | Zbl

[25] Ovsyannikov L. V., Gruppovoi analiz differentsialnykh uravnenii, Nauka, M., 1978

[26] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii, v. 1, Nauka, M., 1973