Construction of adjoint operators in non-linear problems of mathematical physics

G. I. Marchuk

Sbornik. Mathematics, Tome 189 (1998) no. 10, pp. 1505-1516

Résumé

The principles of the construction of adjoint operators in non-linear problems are reviewed. The aim is to draw attention to new approaches in the method of adjoint operators which often enable one to obtain some information about physical processes and unknown parameters of complex systems that may be necessary for the development of the technology of experimental design required in the solution of applied problems.

Bibliographie

[1] Rellich F., “Störungstheeorie des Spektralzerlegung. I–V”, Math. Ann., 113 (1936), 600–619 ; 667–685 ; 116 (1939), 555–570 ; 117 (1940), 346–382 ; 118 (1942), 462–484 | DOI | MR | Zbl | DOI | MR | Zbl | MR | DOI | Zbl

[2] Rellich F., “Störungstheeorie des Spektralzerlegung. I–V”, Proc. Internat. Mathematicians, V. 1 (Cambridge, Maas., 1950), Providence, New York, 1952, 606–613 | MR | Zbl

[3] Rellich F., Perturbation Theory of Eigenvalue Problem, Cordon and Breach Sci. Pub., New York, 1969 | MR

[4] Weinberg A., Wigner E., The Physical Theory of Neutron Chain Reactors, Univ. of Chicago Press, Chicago, 1958 | MR

[5] Usachev L. N., “Uravnenie dlya tsennosti neitronov kineticheskogo reaktora v teorii vozmuschenii”, Reaktorostroenie i teoriya reaktorov, Izd-vo AN SSSR, M., 1955, 251 pp.

[6] Usachev L. N., “Teoriya vozmuschenii dlya koeffitsienta vosproizvodstva i drugikh otnoshenii chisel razlichnykh protsessov v reaktore”, Atom. energiya, 15:6 (1963), 472

[7] Usachev L. N., Bobkov Yu. G., Teoriya vozmuschenii i planirovanie eksperimenta v probleme yadernykh dannykh dlya reaktorov, Atomizdat, M., 1980

[8] Marchuk G. I., Orlov V. V., “K teorii sopryazhennykh funktsii”, Neitronnaya fizika, Gosatomizdat, M., 1961, 30–45

[9] Pupko V. Ya., Zrodnikov A. V., Likhachev Yu. I., Metod sopryazhennykh funktsii v inzhenerno-fizicheskikh issledovaniyakh, Energoatomizdat, M., 1984

[10] Kadomtsev B. B., “O funktsii vliyaniya v teorii perenosa luchistoi energii”, Dokl. AN SSSR, 113:3 (1957), 541–543 | Zbl

[11] Marchuk G. I., Chislennoe reshenie zadach dinamiki atmosfery i okeana, Gidrometeoizdat, L., 1974

[12] Marchuk G. I., “Osnovnye i sopryazhennye uravneniya dinamiki atmosfery i okeana”, Meteorologiya i gidrologiya, 1974, no. 2, 9–37

[13] Maslov V. P., Teoriya vozmuschenii i asimptoticheskie metody, Izd-vo MGU, M., 1965

[14] Vladimirov V. S., Volovich I. V., “Zakony sokhraneniya dlya nelineinykh uravnenii”, Aktualnye problemy vychislitelnoi matematiki i matematicheskogo modelirovaniya, Nauka, Novosibirsk, 1985, 147–162 | MR

[15] Vainberg M. M., Funktsionalnyi analiz, Prosveschenie, M., 1979 | MR | Zbl

[16] Marchuk G. I., Agoshkov V. I., “Sopryazhennye uravneniya v nelineinykh zadachakh i ikh prilozheniya”, Funktsionalnye i chislennye metody matematicheskoi fiziki, Naukova dumka, Kiev, 1988, 138–142 | MR

[17] Marchuk G. I., Agoshkov V. I., “Conjugate operators and algorithms of perturbation in non-linear problems. I; II”, Soviet J. Numer. Anal. and Math. Model., 1988, no. 1, 21–46 ; no. 2, 115–136 | MR | Zbl

[18] Marchuk G. I., Agoshkov V. I., Shutyaev V. P., Sopryazhennye uravneniya i metody vozmuschenii v nelineinykh zadachakh matematicheskoi fiziki, Nauka, M., 1993 | MR

[19] Bellman R., Perturbation Techniques in Mathematics, Physics and Engineering, Holft, New York, 1964 | MR

[20] Pontryagin L. S., Izbrannye trudy, T. 2, Nauka, M., 1988 | Zbl

[21] Krasovskii N. N., Upravlenie dinamicheskoi sistemoi. Zadacha o minimume garantirovannogo rezultata, Nauka, M., 1985 | MR

[22] Lions J.-L., Contrôllabilité exacte perturbations de systèms distributés, V. 1 ; “Contrôllabilité exacte”, Perturbations, V. 2, Masson, Paris, 1988 | Zbl | Zbl

[23] Glowinski R., Numerical methods for non-linear variational problems, Springer-Verlag, New York, 1984 | MR | Zbl

[24] Marchuk G. I., Metody vychislitelnoi matematiki, Nauka, M., 1980 | MR

[25] Avdoshin S. M., Belov V. V., Maslov V. P., Matematicheskie aspekty sinteza vychislitelnykh sred, MIEM, M., 1984

[26] Marchuk G. I., Sopryazhennye uravneniya i analiz slozhnykh sistem, Nauka, M., 1992 | MR

[27] Marchuk G. I., Agoshkov V. I., Shutyaev V. P., Adjoint equations and perturbation algorithms in non-linear problems, CRC Press Inc., New York, 1996 | MR