Geodesic
Asymptotic expansions of slow invariant manifolds and reduction of chemical kinetics models
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 1, pp. 81-96 
V. A. Sobolev; E. A. Tropkina. Asymptotic expansions of slow invariant manifolds and reduction of chemical kinetics models. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 1, pp. 81-96. http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_1_a7/
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Voir la notice de l'article provenant de la source Math-Net.Ru

Methods of the geometric theory of singular perturbations are used to reduce the dimensions of problems in chemical kinetics. The methods are based on using slow invariant manifolds. As a result, the original system is replaced by one on an invariant manifold, whose dimension coincides with that of the slow subsystem. Explicit and implicit representations of slow invariant manifolds are applied. The mathematical apparatus described is used to develop N. N. Semenov’s fundamental ideas related to the method of quasi-stationary concentrations and is used to study particular problems in chemical kinetics.

[2] Vasileva A.B., Butuzov V.F., Asimptoticheskie metody v teorii singulyarnykh vozmuschenii, Vyssh. shkola, M., 1990

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

[4] Vasilev V.M., Volpert A.I., Khudyaev S.I., “O metode kvazistatsionarnykh kontsentratsii dlya uravnenii khimicheskoi kinetiki”, Zh. vychisl. matem. i matem. fiz., 13:3 (1973), 683–697 | MR

[5] Goldshtein V.M., Sobolev V.A., Kachestvennyi analiz singulyarno vozmuschennykh sistem, In-t matem. SO AN SSSR, Novosibirsk, 1988

[6] Kaper H.G., Kaper T.J., “Asymptotic analysis of two reduction methods for systems of chemical reactions”, Physica D, 165 (2002), 66–93 | DOI | MR | Zbl

[7] O'Malley R.E., Singular perturbations and hysteresis, eds. M.P. Mortell, A. Pokrovskii, V.A. Sobolev, SIAM, Philadelphia, 2005 | MR

[8] Sobolev V.A., Schepakina E.A., Reduktsiya modelei i kriticheskie yavleniya v makrokinetike, Fizmatlit, M., 2010

[9] Strygin V.V., Sobolev V.A., Razdelenie dvizhenii metodom integralnykh mnogoobrazii, Nauka, M., 1988

[10] Kononenko L.I., Sobolev V.A., “Asimptoticheskie razlozheniya medlennykh integralnykh mnogoobrazii”, Sibirskii matem. zhurnal, 35:6 (1994), 1264–1278 | MR | Zbl

[11] Mitropolskii Yu.A., Lykova O.B., Integralnye mnogoobraziya v nelineinoi mekhanike, Nauka, M., 1975

[12] Sobolev V.A., “Geometriya singulyarnykh vozmuschenii v vyrozhdennykh sluchayakh”, Matem. modelirovanie, 13:12 (2001), 75–94 | MR | Zbl

[13] Sobolev V.A., “Integral manifolds and decomposition of singularly perturbed system”, System and Control Letts., 1984, no. 5, 169–179 | DOI | MR | Zbl

[14] Strygin V.V., Sobolev V.A., “Vliyanie geometricheskikh i kineticheskikh parametrov i dissipatsii energii na ustoichivost orientatsii sputnikov s dvoinym vrascheniem”, Kosmich. issl., 14:3 (1976), 366–371 | MR

[15] Fehrst A., Enzyme structure and mechanisms, 2nd ed., W.F. Freeman Co., New York, 1975

[16] Murray J.D., Mathematical Biology. I. An introduction. II. Spatial Models and Biomedical Applications, Ed. 3rd, Springer, New York, 2002–2003 (Sec. Printings 2004) | MR

[17] Lam S.H., Goussis D.M., “The CSP method for simplifying kinetics”, Internat. J. Chem. Kinetics., 26 (1994), 461–486 | DOI

[18] Maas U., Pope S.B., “Simplifying chemical kinetics: Intrinsic low-dimensional manifolds in composition space”, Combustion and Flame, 88 (1992), 239–264 | DOI

[19] Davis M., Skodje R., “Geometric investigation of low-dimensional manifolds in systems approaching equilibrium”, J. Chem. Phys., 111 (1999), 859–874 | DOI

[20] Leineweber D.B., Efficient reduced SQP methods for the optimization of chemical processes described by large sparse DAE models, VDI Verlag, Dusseldorf, 1999

[21] Reinhardt V., Winckler M., Lebiedz D., “Approximation of slow attracting manifolds in chemical kinetics by trajectory-based optimization approaches”, J. Phys. Chem. Ser. A., 112:8 (2008), 1712–1718 | DOI | MR

[22] Roussel M.R., Fraser S.J., “Geometry of of the steady-state approximation: Perturbation and accelerated convergence method”, J. Chem. Phys., 93 (1990), 1072–1081 | DOI

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

[24] Gu Z.-M., Nefedov N.N., O'Maliey R.E., Jr., “On singular singularly perturbed initial value problems”, SIAM J. Appl. Math., 49:1 (1989), 1–25 | DOI | MR | Zbl