Binary correspondences and the inverse problem of chemical kinetics

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system’s guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	Dear authors! Submission of all materials is carried out only electronically through Online Submission System in personal account. DOI: 10.23671/VNC.2018.3.1798 Binary correspondences and the inverse problem of chemical kinetics Gutman, A. E. , Kononenko L. I. Vladikavkaz Mathematical Journal 2018. Vol. 20. Issue 3. Abstract: 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. Keywords: binary correspondence, inverse problem, solvability, composition, stability, correctness, differential equation, chemical kinetics, linear independence. Language: English Download the full text For citation: Gutman A. E., Kononenko L. I. Binary correspondences and the inverse problem of chemical kinetics, Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math. J.], vol. 20, no. 2, pp. 37-47. DOI 10.23671/VNC.2018.3.1798 + References 1. Gutman A. E., Kononenko L. I. Formalization of Inverse Problems and its Applications, Sibirskij zhurnal chistoj i prikladnoj matematiki [Siberian Journal of Pure and Applied Mathematics], 2017, vol. 17, no. 4, pp. 49-56 (in Russian). DOI: 10.17377/PAM.2017.17.5. 2. Gutman A. E., Kononenko L. I. The Inverse Problem of Chemical Kinetics as a Composition of Binary Correspondences. Sibirskie elektronnye matematicheskie izvestiya [Siberian Electronic Mathematical Reports], 2018, vol. 15, pp. 48-53 (in Russian). DOI: 10.17377/\allowbreak semi.2018.15.006. 3. Gutman A. E., Koptev A. V. 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Singulyarno vozmuschennye uravneniya v kriticheskikh sluchayakh [Singularly Perturbed Equations in Critical Cases], Moscow, Moscow State University, 1978 (in Russian). 8. Goldstein V. M., Sobolev V. A. Kachestvennyj analiz singulyarno vozmuschennykh sistem [Qualitative Analysis of Singularly Perturbed Systems], Novosibirsk, Sobolev Institute of Mathematics, 1988 (in Russian). 9. Tikhonov A. N. On Independence of Solutions to Differential Equations on a Small Parameter, Matematicheskij Sbornik [Sbornik: Mathematics], 1948, vol. 22 (64), no. 2, pp. 193-204 (in Russian). 10. Kononenko L. I. Direct and Inverse Problems for a Singular System with Slow and Fast Variables in Chemical Kinetics, Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math. J.], 2015, vol. 17, no. 1, pp. 39-46 (in Russian). DOI: 10.23671/VNC.2015.1.7291. 11. Kononenko L. I. Identification Problem for Singular Systems with Small Parameter in Chemical Kinetics, Sibirskie elektronnye matematicheskie izvestiya [Siberian Electronic Mathematical Reports], 2016, vol. 13, pp. 175-180 (in Russian). DOI: 10.17377/semi.2016.13.015. ← Contents of issue


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