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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
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