Explicit and implicit parametrizations in the construction of branching solutions by iterative methods

N. A. Sidorov

Sbornik. Mathematics, Tome 186 (1995) no. 2, pp. 297-310

Résumé

We give iterative methods for calculating branching solutions of nonlinear equations using explicit and implicit parametrization in a neighbourhood of a branch point. We consider the regularization problem for these schemes in the presence of errors of calculation. We construct modifications of these methods developing results in (R. Zh. Mat., 1991, 12B 1058) on the application of group analysis (R. Zh. Mat., 1988, 11B 883k) in iteration schemes in branching theory.

Bibliographie

[1] Vainberg M. M., Trenogin V. A., Teoriya vetvleniya reshenii nelineinykh uravnenii, Nauka, M., 1969 | MR

[2] Keller H. B., “Numerical solution of bifurcation and Nonlinear problems”, Applications of bifurcation theory, ed. P. Rabinowitz, Academic Press, New York, 1977, 359–384 | MR

[3] Moose G., “The Numerical treatment of Nontrivial bifurcation points”, Numer. funct. anal. and optimiz., 2(6) (1980), 441–472 | DOI | MR

[4] Langford W. F., “Numerical solution of bifurcation problems for ordinary differential equations”, Numer. Math., 28 (1977), 171–190 | DOI | MR | Zbl

[5] Westreich D., Vasol Y. L., “Numerical bifurcation at simple eigenvalues”, SIAM J. Numer. Anal., 16 (1979), 538–546 | DOI | MR | Zbl

[6] Sidorov N. A., Obschie voprosy regulyarizatsii v zadachakh teorii vetvleniya, Izd-vo Irkut. un-ta, Irkutsk, 1982 | MR | Zbl

[7] Sidorov N. A., “Otyskanie reshenii nelineinykh uravnenii vblizi tochki vetvleniya metodom dvukhstupenchatykh iteratsii”, Trudy Irkut. un-ta. Ser. matem., 74:6 (1970), 23–29 | MR

[8] Tolstonogov D. A., Svoistva reshenii integrodifferentsialnykh uravnenii fiziki plazmy, Kand. diss., Irkutskii VTs SO RAN, 1991

[9] Ovsyannikov L. V., Lektsii po teorii gruppovykh svoistv differentsialnykh uravnenii, Izd-vo NGU, Novosibirsk, 1966

[10] Yudovich V. I., “Svobodnaya konvektsiya i vetvlenie”, PMM, 31:1 (1967), 101–111 | Zbl

[11] Loginov B. V., Teoriya vetvleniya reshenii nelineinykh uravnenii v usloviyakh gruppovoi invariantnosti, Izd-vo FAN, Tashkent, 1985 | MR

[12] Loginov B. V., Sidorov N. A., “Gruppovaya simmetriya uravneniya razvetvleniya Lyapunova–Shmidta i iteratsionnye metody v zadache o tochke bifurkatsii”, Matem. sb., 182(224) (1991), 681–691 | MR | Zbl

[13] Trenogin V. A., Sidorov N. A., Loginov B. V., “Potentiality, group symmetry and bifurcation in the theory of branching equation”, Diff. and Integral Equations, Ohio, USA, 3:1 (1990), 145–154 | MR | Zbl

[14] Elenin G. G., Troischev Yu. V., “Chislennyi metod vetvleniya v slozhnykh tochkakh bifurkatsii pri prodolzhenii po parametru”, Matem. modelirovanie, 3:7 (1991), 101–110 | MR

[15] Tikhonov A. N., Ivanov V. K., Lavrentev M. M., “Nekorrektno postavlennye zadachi”, Trudy simpoziuma, posvyaschennogo 60-letiyu akademika S. L. Soboleva, Nauka, M., 1970, 224–238

[16] Trenogin V. A. and Sidorov N. A., “Regularization of computation of branching solutions of nonlinear equations”, Lecture Notes in Mathematics, 594, 1977, 491–506 | MR

[17] Sidorov N. A., “Ob optimalnom vybore nachalnykh priblizhenii reshenii regulyarizovannykh uravnenii v teorii vetvleniya”, Matem. zametki, 20:2 (1976), 273–278 | MR | Zbl

[18] Trenogin V. A., Funktsionalnyi analiz, Nauka, M., 1980 | MR | Zbl

[19] Loginov B. V., Sidorov N. A., “Gruppovaya simmetriya uravneniya razvetvleniya i iteratsionnye metody”, Lavrentevskie chteniya po matematike, mekhanike i fizike, Tretya Mezhdunarodnaya konferentsiya, Novosibirsk, 1990, 60–61 | MR

[20] Sidorov N. A., “Asimptotiki i iteratsii v okrestnosti tochek bifurkatsii reshenii nelineinykh uravnenii”, Nekorrektno postavlennye zadachi v estestvennykh naukakh, Mezhdunarodnaya konferentsiya, Moskva, 1991, 243