Direct and inverse problems for a singular system with slow and fast variables in chemical kinetics

L. I. Kononenko

Vladikavkazskij matematičeskij žurnal, Tome 17 (2015) no. 1, pp. 39-46

Résumé

Direct and inverse problems for singular systems with small parameter are stated, which describe catalytic reactions in chemical kinetics. The solution of the direct problem is based on the method of integral manifolds. The inverse problem reduces to finding the coefficients of the polynomial in the right-hand part of the slow equation according to the solution given on the slow surface of the system. The above arguments make it possible to obtain existence and uniqueness conditions for the coefficients in the right-hand part of the slow subsystem of the degenerate system.

Bibliographie

[1] Mitropolskii Yu. A., Lykova O. B., Integralnye mnogoobraziya v nelineinoi mekhanike, Nauka, M., 1963 | MR

[2] Vasileva A. V., Butuzov V. F., Singulyarno vozmuschennye uravneniya v kriticheskikh sluchayakh, Izd-vo MGU, M., 1978 | MR

[3] Goldshtein V. M., Sobolev V. A., Kachestvennyi analiz singulyarno vozmuschennykh sistem, Izd-vo In-ta matematiki SO AN SSSR, Novosibirsk, 1988

[4] Tikhonov A. N., “Sistemy differentsialnykh uravnenii, soderzhaschie malye parametry pri proizvodnykh”, Mat. sb., 31(73):3 (1952), 575–586 | MR | Zbl

[5] Tikhonov A. N., “O zavisimosti reshenii differntsialnykh uravnenii ot malogo parametra”, Mat. sb., 22(64):2 (1948), 193–204 | MR | Zbl

[6] Goldshtein V. M., Kononenko L. I., Lazman M. Z., Sobolev V. A., Yablonskii G. S., “Kachestvennyi analiz dinamicheskikh svoistv kataliticheskogo izotermicheskogo reaktora idealnogo smesheniya”, Matematicheskie problemy khimicheskoi kinetiki, Nauka, Novosibirsk, 1989, 176–204 | MR

[7] Kononenko L. I., “O gladkosti medlennoi poverkhnosti singulyarno vozmuschennykh sistem”, Sib. zhurn. industr. matematiki, 5:2(10) (2002), 109–125 | MR | Zbl

[8] Kononenko L. I., Volokitin E. P., “Parametrizatsiya i kachestvenyi analiz singulyarnoi sistemy v matematicheskoi modeli reaktsii kataliticheskogo okisleniya”, Sib. zhurn. industr. matematiki, 15:1(49) (2012), 44–52 | MR

[9] Voropaeva N. V., Sobolev V. A., Geometricheskaya dekompozitsiya singulyarno vozmuschennykh sistem, Fizmatlit, M., 2009

[10] Romanov V. G., “Obratnye zadachi dlya giperbolicheskikh sistem”, Vychislitelnye metody v matematicheskoi fizike, geofizike i optimalnom upravlenii, Nauka, Novosibirsk, 1978, 128–142

[11] Lavrentev M. M., Romanov V. G., Shishatskii S. P., Nekorrektnye zadachi matematicheskoi fiziki i analiza, Nauka, M., 1980, 286 pp. | MR

[12] Anikonov Yu. E., “Neskolko voprosov teorii obratnykh zadach dlya kineticheskikh uravnenii”, Obratnye zadachi matematicheskoi fiziki, Novosibirsk, 1985, 28–41 | MR | Zbl

[13] Alekseev A. S., “Nekotorye obratnye zadachi teorii rasprostraneniya voln 1, 2”, Izv. AN SSSR. Ser. geofizika, 1962, no. 11, 1514–1531

[14] Kabanikhin S. I., Obratnye i nekorrektnye zadachi, Sibirskoe nauchnoe izd-vo, Novosibirsk, 2009, 456 pp.

[15] Golubyatnikov V. P., “Obratnaya zadacha dlya uravneniya Gamiltona–Yakobi na zamknutom mnogoobrazii”, Sib. mat. zhurn., 38:2 (1997), 276–279 | MR

[16] Kozhanov A. I., “Nelineinye nagruzhennye uravneniya i obratnye zadachi”, Zhurn. vychisl. matematiki i mat. fiziki, 44:4 (2004), 694–716 | MR | Zbl

[17] Gainova I. A., Fadeev S. I., Elokhin V. I., Boronin A. I., “Reaktsiya okisleniya CO na polikristallicheskoi folge iridiya. Modelirovanie kinetiki poverkhnostnykh protsessov”, Mezhdunar. konf. po vychislitelnoi matematike, Trudy konferentsii. Chast I, Novosibirsk, 2004, 449–454

[18] Kononenko L. I., “Integralnye mnogoobraziya v matematicheskoi modeli reaktsii kataliticheskogo okisleniya”, Izv. RAEN. Ser. MMMIU, 1:4 (1997), 53–59 | MR | Zbl

[19] Kononenko L. I., “Medlennye poverkhnosti v zadachakh khimicheskoi kinetiki”, Mat. zametki YaGU, 19:2 (2012), 49–67 | Zbl

[20] Chumakov G. A., Chumakova N. A., “Relaxation oscillations in a kinetic model of catalytic hydrogen oxidation involving a chase on canards”, Chem. Eng. J., 91:2–3 (2003), 151–158 | DOI

[21] Kononenko L. I., “Relaksatsii v singulyarno vozmuschennykh sistemakh na ploskosti”, Vestn. NGU. Ser. Matematika, mekhanika, informatika, 9:4 (2009), 45–50 | Zbl

[22] Sobolev V. A., Schepakina E. A., “Integralnye poverkhnosti so smenoi ustoichivosti i traektorii-utki”, Izv. RAEN. Ser. MMMIU, 1:3 (1997), 176–187

[23] Bobkova A. S., Kolesov A. Yu., Rozov N. Kh., “Problema “vyzhivaniya utok” v trekhmernykh singulyarno vozmuschennykh sistemakh s dvumya medlennymi peremennymi”, Mat. zametki, 71:6 (2002), 818–831 | DOI | MR | Zbl

[24] Reshetnyak Yu. G., Kurs matematicheskogo analiza. Ch. 1. Kn. 2: Integralnoe ischislenie funktsii odnoi peremennoi. Differentsialnoe ischislenie funktsii mnogikh peremennykh, Izd-vo Instituta matematiki im. S. L. Soboleva SO RAN, Novosibirsk, 1999, 512 pp.

[25] Ermakova A., Metody makrokinetiki, primenyaemye pri matematicheskom modelirovanii khimicheskikh protsessov i reaktorov, Institut kataliza SO RAN im. G. K. Boreskova, Novosibirsk, 2001, 188 pp.

[26] Godunov S. K., Sovremennye aspekty lineinoi algebry, Nauchnaya kniga, Novosibirsk, 1997, 388 pp.

[27] Godunov S. K., Lektsii po sovremennym aspektam lineinoi algebry, Nauchnaya kniga, Novosibirsk, 2002, 216 pp.

[28] Romanov V. G., Slinyucheva L. I., “Obratnaya zadacha dlya lineinykh giperbolicheskikh sistem pervogo poryadka”, Matem. problemy geofiziki, 3, Izdatelstvo VTs SO AN SSSR, Novosibirsk, 1972, 187–215