Finding of accessory parameters for mixed inverse boundary value problem with polygonal known part of boundary

S. R. Nasyrov ; L. Yu. Nizamieva

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 11 (2011) no. 4, pp. 34-40.

Résumé

We consider a mixed inverse boundary value problem with respect to parameter $x$ for the case when the known part of the boundary $L_z^1$ is a polygonal line. Integral representation of solution to the problem depends on real parameters being the pre-images of the vertices of $L_z^1$ under conformal mapping. By analogy with Schwartz–Christoffel integrals, we name them accessory parameters. It is suggested a new method of determining the accessory parameters. Is is based on consideration of one-parametric family of solutions to the problem corresponding to the case when the known part of the boundary is the union of two rays and the stretching slit the endpoint of which goes along the given polygonal line $L_z^1$.

Bibliographie

[1] Monakhov V. N., Kraevye zadachi so svobodnymi granitsami dlya ellipticheskikh sistem uravnenii, Novosibirsk, 1977, 424 pp. | MR

[2] Nasyrov S. R., O metode poligonalnoi approksimatsii v smeshannykh obratnykh kraevykh zadachakh po parametru $x$, Dep. v VINITI 17.05.1982, No 2459-82, Kazan, 1982

[3] Nasyrov S. R., “Smeshannaya obratnaya kraevaya zadacha na rimanovykh poverkhnostyakh”, Izv. vuzov. Matematika, 1990, no. 10, 25–36 | MR | Zbl

[4] Aksentev L. A., Ilinskii N. B., Salimov R. B., “O knige V. N. Monakhova "Kraevye zadachi so svobodnymi granitsami dlya ellipticheskikh sistem uravnenii”, Tr. seminara po kraevym zadacham, 15, Kazan, 1978, 202–212 | MR

[5] Salimov R. B., Strezhneva E. V., K resheniyu obratnoi smeshannoi kraevoi zadachi, Dep. v VINITI 29.06.89, No 4312–V89, Kazan, 1989

[6] Salimov R. B., Strezhneva E. V., Reshenie obratnoi smeshannoi kraevoi zadachi dlya dvusvyaznoi oblasti v vidoizmenennoi postanovke, Dep. v VINITI 29.12.90, No 0487–V90, Kazan, 1990

[7] Strezhneva E. V., Reshenie obratnoi kraevoi zadachi dlya dvusvyaznoi oblasti v odnom sluchae, Dep. v VINITI 29.12.90, No 2736–V91, Kazan, 1990

[8] Tlyusten S. R., “Smeshannaya kraevaya zadacha so svobodnoi granitsei v neodnolistnykh oblastyakh”, Dinamika sploshnoi sredy, 76, Novosibirsk, 1986, 148–156 | MR | Zbl

[9] Tlyusten S. R., “Neodnolistnye otobrazheniya so svobodnoi granitsei”, Dinamika sploshnoi sredy, 86, Novosibirsk, 1988, 141–148 | MR | Zbl

[10] Tlyusten S. R., “Apriornye otsenki reshenii smeshannoi kraevoi zadachi so svobodnoi granitsei dlya analiticheskikh funktsii”, Dinamika sploshnoi sredy, 92, Novosibirsk, 1989, 108–121 | MR | Zbl

[11] Tlyusten S. R., “Geometricheskie svoistva reshenii smeshannoi obratnoi kraevoi zadachi so svobodnoi granitsei”, Dinamika sploshnoi sredy, 97, Novosibirsk, 1990, 114–123 | MR | Zbl

[12] Monakhov V. N., “Ob odnom variatsionnom metode resheniya zadach po gidrodinamike so svobodnoi granitsei”, Sib. matem. zhurn., 41:5 (2000), 1106–1121 | MR | Zbl

[13] Nasyrov S. R., Faizov I. Z., “Lokalnaya edinstvennost resheniya smeshannoi obratnoi kraevoi zadachi na poligonalnykh rimanovykh poverkhnostyakh s prostymi tochkami vetvleniya”, Uchen. zapiski Kazansk. gos. un-ta, 148, no. 2, 2006, 97–108 | Zbl

[14] Galiullina G. R., Nasyrov S. R., “Uravnenie Gakhova dlya vneshnei smeshannoi obratnoi kraevoi zadachi po parametru $x$”, Izv. vuzov. Matematika, 2002, no. 10, 48–55 | MR | Zbl

[15] Nasyrov S. R., Nizamieva L. Yu., “Uravnenie Gakhova dlya vneshnei smeshannoi obratnoi kraevoi zadachi po parametru $x$ na poligonalnoi rimanovoi poverkhnosti s prostoi tochkoi vetvleniya na beskonechnosti”, Uchen. zapiski Kazansk. gos. un-ta. Ser. fiz.-mat., 150, no. 1, 2008, 91–101 | Zbl

[16] Nasyrov S. R., Nizamieva L. Yu., “Uravnenie Gakhova dlya vneshnei smeshannoi obratnoi kraevoi zadachi na rimanovoi poverkhnosti s tochkoi vetvleniya na beskonechnosti proizvolnogo poryadka”, Vestn. Samarsk. gos. un-ta. Ser. estestvennonauchn., 2009, no. 4, 28–42

[17] Lavrentev G. V., “Chislennye raschety zadach gidrodinamiki so svobodnymi granitsami na osnove analiticheskogo predstavleniya reshenii”, Dinamika sploshnoi sredy, 6, Novosibirsk, 1970, 208–212

[18] Nizamieva L. Yu., Ispolzovanie kraevykh zadach pri nakhozhdenii aktsessornykh parametrov v integrale Kristoffelya–Shvartsa, Dep. v VINITI 06.07.10, No 421–V2010, Kazan, 2010

[19] Aleksandrov I. A., Parametricheskie prodolzheniya v teorii odnolistnykh funktsii, M., 1976, 344 pp. | MR