Geodesic
Inverse problem for degenerate parabolic-hyperbolic equation with nonlocal boundary condition
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 1 (2015), pp. 46-59.

Voir la notice de l'article provenant de la source Math-Net.Ru

For a mixed type equation with a power-law degeneration we consider the inverse problem on finding an unknown right side. We establish a uniqueness criterion of solution to the problem with a nonlocal condition that connects the normal derivative of the sought-for solution, which belongs to different types of studied equations. The solution is constructed in the form of sums of a series in eigenfunctions of the corresponding one-dimensional spectral problem. We also prove stability of the solution to the non-local boundary condition.
Keywords: equation of mixed type, spectral method, uniqueness, stability.
Mots-clés : existence 
@article{IVM_2015_1_a3,
     author = {K. B. Sabitov and S. N. Sidorov},
     title = {Inverse problem for degenerate parabolic-hyperbolic equation with nonlocal boundary condition},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {46--59},
     publisher = {mathdoc},
     number = {1},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2015_1_a3/}
}
TY  - JOUR
AU  - K. B. Sabitov
AU  - S. N. Sidorov
TI  - Inverse problem for degenerate parabolic-hyperbolic equation with nonlocal boundary condition
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2015
SP  - 46
EP  - 59
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2015_1_a3/
LA  - ru
ID  - IVM_2015_1_a3
ER  - 
%0 Journal Article
%A K. B. Sabitov
%A S. N. Sidorov
%T Inverse problem for degenerate parabolic-hyperbolic equation with nonlocal boundary condition
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2015
%P 46-59
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2015_1_a3/
%G ru
%F IVM_2015_1_a3
K. B. Sabitov; S. N. Sidorov. Inverse problem for degenerate parabolic-hyperbolic equation with nonlocal boundary condition. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 1 (2015), pp. 46-59. http://geodesic.mathdoc.fr/item/IVM_2015_1_a3/

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

[2] Sabitov K. B., Rakhmanova L. Kh., “Nachalno-granichnaya zadacha dlya uravneniya smeshannogo parabolo-giperbolicheskogo tipa v pryamougolnoi oblasti”, Differents. uravneniya, 44:9 (2008), 1175–1181 | MR | Zbl

[3] 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

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

[5] Sabitov K. B., Sidorov S. N., “Ob odnoi nelokalnoi zadache dlya vyrozhdayuschegosya parabolo-giperbolicheskogo uravneniya”, Differents. uravneniya, 50:3 (2014), 356–365 | DOI | Zbl

[6] Kapustin N. Yu., “Zadacha Trikomi dlya parabolo-giperbolicheskogo uravneniya s vyrozhdayuscheisya giperbolicheskoi chastyu. I”, Differents. uravneniya, 23:1 (1987), 72–78 | MR | Zbl

[7] Kapustin N. Yu., “Zadacha Trikomi dlya parabolo-giperbolicheskogo uravneniya s vyrozhdayuscheisya giperbolicheskoi chastyu. II”, Differents. uravneniya, 24:8 (1988), 1379–1386 | MR | Zbl

[8] Sabitov K. B., Safin E. M., “Obratnaya zadacha dlya uravneniya parabolo-giperbolicheskogo tipa v pryamougolnoi oblasti”, Dokl. RAN, 429:4 (2009), 451–454 | Zbl

[9] Sabitov K. B., Safin E. M., “Obratnaya zadacha dlya uravneniya smeshannogo parabolo-giperbolicheskogo tipa v pryamougolnoi oblasti”, Izv. vuzov. Matem., 2010, no. 4, 55–62 | MR | Zbl

[10] Ivanov V. K., Vasin V. V., Tanan V. P., Teoriya lineinykh nekorrektnykh zadach i ee prilozheniya, Nauka, M., 1978 | MR

[11] Lavrentev M. M., Romanov V. G., Shishatskii S. T., Nekorrektnye zadachi matematicheskoi fiziki i analiza, Nauka, M., 1980 | MR

[12] Denisov A. M., Vvedenie v teoriyu obratnykh zadach, MGU, M., 1994

[13] Prilepko A. I., Orlovsky D. G., Vasin I. A., Methods for solving inverse problems in mathematical physics, Marcel Dekker, New York, 2000 | MR | Zbl

[14] Kabanikhin S. I., Obratnye i nekorrektnye zadachi, Sib. nauch. izd-vo, Novosibirsk, 2009

[15] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii, v. 2, 2-e izd., Nauka, M., 1974

[16] Sabitov K. B., Vagapova E. V., “Zadacha Dirikhle dlya uravneniya smeshannogo tipa s dvumya liniyami vyrozhdeniya v pryamougolnoi oblasti”, Differents. uravneniya, 49:1 (2013), 68–78 | MR | Zbl

[17] Ilin V. A., Poznyak V. G., Osnovy matematicheskogo analiza, Ch. 1, 7-e izd., Fizmatlit, M., 2005

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

[19] Lyuk Yu., Spetsialnye matematicheskie funktsii i ikh approksimatsii, Mir, M., 1980

[20] Prudnikov A. P., Brychkov Yu. A., Marichev O. I., Integraly i ryady. Spetsialnye funktsii, Nauka, M., 1983 | MR | Zbl