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: 10.46698/n3062-4932-2162-c
The Inverse Problem for Singular Perturbed System with Many-Sheeted Slow Surfaces
Kononenko, L. I.
Vladikavkaz Mathematical Journal 2023. Vol. 25. Issue 3.
Abstract: We consider a singularly perturbed system of ordinary differential equations with small parameter, which describes a problem of chemical kinetics. We examine the system by using the method of integral manifolds that serves as a convenient tool for studying multidimensional singularly perturbed systems of differential equations and makes it possible to lower the dimension of the system. The integral manifold consists of sheets; for the small parameter \(\varepsilon=0\), it is a slow surface. We formulate the direct and inverse problems for the system. The direct problem is as follows: given the right-hand sides of the system, find a solution to the system or prove its existence. The inverse problem is to find the unknown right-hand sides of the system of differential equations from some data on a solution of the direct problem. First, we consider the degenerate case, in which the small parameter \(\varepsilon\) equals zero, and some restrictions are imposed on the dimension of the slow and fast variables, on the class of the right-hand sides that are assumed polynomial (with degree 1), and on the number of sheets of the slow surface. Then we pass to the nondegenerate case \(\varepsilon\neq 0\). In the case of a single sheet of the slow surface, the existence and uniqueness theorem was previously proven for a solution of the inverse problem. In this paper, we consider a system whose slow surface consists of several sheets. We prove an existence and uniqueness theorem for a solution to such a system. The proof is based on the result previously obtained for a system whose slow surface consists of a single sheet.
Keywords: inverse problem, ordinary differential equation, small parameter, slow surface, contraction mapping principle, chemical kinetics
For citation: Kononenko, L. I. The Inverse Problem for Singular Perturbed System with Many-Sheeted Slow Surfaces, Vladikavkaz Math. J., 2023, vol. 25, no. 3, pp. 81-88 (in Russian). DOI 10.46698/n3062-4932-2162-c
1. Kononenko, L. I. The Identification Problem for a Nondegenerate
System of Ordinary Differential Equations with Fast and Slow
Variables, Mathematical Notes of NEFU, 2021, vol. 28,
no. 2, pp. 3-15 (in Russian). DOI: 10.25587/SVFU.2021.58.21.001.
2. Gutman, A. E. and Kononenko, L. I.
Formalization of Inverse Problems and its Applications,
Siberian Journal of Pure and Applied Mathematics, 2017, vol. 17, no. 4, pp. 49-56 (in Russian).
DOI: 10.17377/PAM.2017.17.5.
3. Gutman, A. E. and 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/semi.2018.15.006.
4. Mitropolsky, Yu. A. and Lykova, O. B. Integral'nye mnogoobraziya v nelinejnoj mekhanike
[Integral Manifolds in Nonlinear Mechanics], Moscow, Nauka, 1963, 512 p. (in Russian).
5. Vasil'eva, A. V. and Butuzov, V. F. Singulyarno vozmuschennye uravneniya v kriticheskikh
sluchayakh [Singularly Perturbed Equations in Critical Cases],
Moscow, Moscow State University, 1978, 106 p. (in Russian).
6. Goldstein V. M. and Sobolev V. A. Kachestvennyj analiz singulyarno vozmuschennykh sistem
[Qualitative Analysis of Singularly Perturbed Systems], Novosibirsk, Sobolev Institute of Mathematics,
1988 (in Russian).
7. Kononenko, L. I. On the Smoothness of Slow Surfaces of Singularly
Perturbed Systems, Sibirskii Zhurnal Industrial'noi Matematiki, 2002, vol. 5, no. 2,
pp. 109-125 (in Russian).
8. Kononenko, L. I. Slow Surfaces in Problems of Chemical Kinetics,
Mathematical Notes of YSU, 2012, vol. 19, issue 2, pp. 49-67 (in Russian).
9. Tikhonov, A. N.
On Independence of Solutions to Differential Equations on a Small Parameter,
Matematicheskii Sbornik, 1948, vol. 22(64), no. 2, pp. 193-204
(in Russian).
10. Lavrent'ev, M. M., Romanov, V. G. and Shishatskii, S. P. Nekorrektnye zadachi
matematicheskoj fiziki i analiza [Ill-posed Problems of Mathematical Physics and Analysis],
Moscow, Nauka,
1980, 287 p. (in Russian).
11. Romanov, V. G. Inverse Problems for Hyperbolic Systems, Vychislitel'nye metody v
matematicheskoj fizike, geofizike i optimal'nom upravlenii [Numerical Methods
in Mathematical Physics, Geophysics and Optimal Control], Novosibirsk, Nauka,
1978, pp. 75-83 (in Russian).
12. Romanov, V. G. and Slinyucheva L. I. Inverse Problem for Linear
Hyperbolic Systems of the First Order, Matematicheskie problemy
geofiziki [Math Problems Geophysics], Novosibirsk, Izdatelstvo VTs SO AN SSSR, 1972, no. 3,
pp. 187-215 (in Russian).
13. Kozhanov, A. I. Nonlinear Loaded Equations and Inverse Problems,
Computational Mathematics and Mathematical Physics, 2004, vol. 44, no. 4, pp. 657-675.
14. Kabanikhin, S. I. Obratnye i nekorrektnye zadachi [Inverse and Ill-posed Problems],
Novosibirsk, 2010, 458 p. (in Russian).
15. Anikonov, Yu. E. Some Questions in the Theory of Inverse Problems
for Kinetic Equations, Obratnye zadachi matematicheskoj fiziki [Inverse Problems of Mathematical
Physics], Novosibirsk, Akad. Nauk SSSR Sibirsk. Otdel., Vychisl. Tsentr, 1985, pp. 28-41 (in Russian).
16. Golubyatnikov, V. P. An Inverse Problem for the Hamilton-Jacobi
Equation on a Closed Manifold,
Siberian Mathematical Journal, 1997, vol. 38, no. 2, pp. 235-238. DOI: 10.1007/BF02674621.
17. 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.
18. Gutman, A. E. and Kononenko, L. I. Binary Correspondences and the
Inverse Problem of Chemical Kinetics, Vladikavkaz Mathematical Journal, 2018,
vol. 20, no. 3, pp. 37-47. DOI: 10.23671/VNC.2018.3.17981.
19. Reshetnyak, Yu. G. Kurs matematicheskogo analiza [Mathematical Analysis Course],
Novosibirsk, 1999, vol. 1; 2000, vol. 2. (in Russian).