Geodesic
Binary correspondences and the inverse problem of chemical kinetics
Vladikavkazskij matematičeskij žurnal, Tome 20 (2018) no. 3, pp. 37-47 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We show how binary correspondences can be used for simple formalization of the notion of problem, definition of the basic components of problems, their properties, and constructions. In particular, formalization of the following notions is presented: condition, data, unknowns, and solutions of a problem, solvability and unique solvability, inverse problem, composition and restriction of problems, isomorphism between problems. We also consider topological problems and the related notions of stability and correctness. A connection is indicated between the stability and continuity of a uniquely solvable topological problem. The definition of parametrized set is given. The notions are introduced of parametrized problem, the problem of reconstruction of an object by the values of parameters, as well as the notions of locally free set of parameters and stability with respect to a set of parameters. As an illustration, we consider a singularly perturbed system of ordinary differential equations which describe a process in chemical kinetics and burning. Direct and inverse problems are stated for such a system. We extend the class of problems under study by considering polynomials of arbitrary degree as the right-hand sides of the differential equations. It is shown how the inverse problem of chemical kinetics can be corrected and made more practical by means of the composition with a simple auxiliary problem which represents the relation between functions and finite sets of numerical characteristics being measured. For the corrected inverse problem, formulas for the solution are presented and the conditions of unique solvability are indicated. Within the study of solvability, a criterion is established for linear independence of functions in terms of finite sets of their values. With the help of the criterion, realizability is clarified of the condition for unique solvability of the inverse problem of chemical kinetics. 
@article{VMJ_2018_20_3_a2,
     author = {A. E. Gutman and L. I. Kononenko},
     title = {Binary correspondences and the inverse problem of chemical kinetics},
     journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
     pages = {37--47},
     year = {2018},
     volume = {20},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/VMJ_2018_20_3_a2/}
}
TY  - JOUR
AU  - A. E. Gutman
AU  - L. I. Kononenko
TI  - Binary correspondences and the inverse problem of chemical kinetics
JO  - Vladikavkazskij matematičeskij žurnal
PY  - 2018
SP  - 37
EP  - 47
VL  - 20
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/VMJ_2018_20_3_a2/
LA  - en
ID  - VMJ_2018_20_3_a2
ER  - 
%0 Journal Article
%A A. E. Gutman
%A L. I. Kononenko
%T Binary correspondences and the inverse problem of chemical kinetics
%J Vladikavkazskij matematičeskij žurnal
%D 2018
%P 37-47
%V 20
%N 3
%U http://geodesic.mathdoc.fr/item/VMJ_2018_20_3_a2/
%G en
%F VMJ_2018_20_3_a2
A. E. Gutman; L. I. Kononenko. Binary correspondences and the inverse problem of chemical kinetics. Vladikavkazskij matematičeskij žurnal, Tome 20 (2018) no. 3, pp. 37-47. http://geodesic.mathdoc.fr/item/VMJ_2018_20_3_a2/

[1] Gutman A. E., Kononenko L. I., “Formalization of Inverse Problems and its Applications”, Siberian Journal of Pure and Applied Mathematics, 17:4 (2017), 49–56 (in Russian) | DOI | MR

[2] Gutman A. E., Kononenko L. I., “The Inverse Problem of Chemical Kinetics as a Composition of Binary Correspondences”, Siberian Electronic Mathematical Reports, 15 (2018), 48–53 (in Russian) | DOI | MR | Zbl

[3] Gutman A. E., Koptev A. V., “Finite Dimensionality and Separability of the Stalks of Banach Bundles”, Siberian Mathematical Journal, 55:2 (2014), 246–253 | DOI | MR | Zbl

[4] Kononenko L. I., “Qualitative Analysis of Singularly Perturbed Systems with One or Two Slow and Fast Variables”, Siberian Journal of Industrial Mathematics, 5:4 (2002), 55–62 (in Russian) | MR | Zbl

[5] Kononenko L. I., “Relaxations in Singularly Perturbed Planar Systems,”, Bulletin of the Novosibirsk State University. Series: Mathematics, Mechanics, Informatics, 9:4 (2009), 45–50 (in Russian) | MR | Zbl

[6] Mitropolsky Yu. A., Lykova O. B., Integral Manifolds in Nonlinear Mechanics, Nauka, M., 1963 (in Russian) | MR

[7] Vasil'eva A. V., Butuzov V. F., Singularly Perturbed Equations in Critical Cases, Moscow State University, M., 1978 (in Russian) | MR

[8] Goldstein V. M., Sobolev V. A., Qualitative Analysis of Singularly Perturbed Systems, Sobolev Institute of Mathematics, Novosibirsk, 1988 (in Russian)

[9] Tikhonov A. N., “On Independence of Solutions to Differential Equations on a Small Parameter”, Sbornik: Mathematics, 22(64):2 (1948), 193–204 (in Russian) | MR | Zbl

[10] Kononenko L. I., “Direct and Inverse Problems for a Singular System with Slow and Fast Variables in Chemical Kinetics”, Vladikavkaz Math. J., 17:1 (2015), 39–46 (in Russian) | DOI | MR

[11] Kononenko L. I., “Identification Problem for Singular Systems with Small Parameter in Chemical Kinetics”, Siberian Electronic Mathematical Reports, 13 (2016), 175–180 (in Russian) | DOI | MR | Zbl