matmessМатематические заметки СВФУMathematical notes of NEFU2411-93262587-876XСеверо-Восточный федеральный университет имени М.К. Аммосова10.25587/SVFU.2023.33.27.005matmess-56Research ArticleМАТЕМАТИКАAn inverse problem of chemical kinetics in a nondegenerate caseAn inverse problem of chemical kinetics in a nondegenerate caseKononenkoL. I.KononenkoL. I.

Larisa I. Kononenko

4 Koptyug Avenue, 630090 Novosibirsk

Larisa I. Kononenko

4 Koptyug Avenue, 630090 Novosibirsk

larak@math.nsc.ruSobolev Institute of MathematicsRussian Federation2023300320233016371Copyright © Kononenko L.I., 20232023Kononenko L.I.Kononenko L.I.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://matmess.elpub.ru/jour/article/view/56

The article contains a review of recent results on solving the direct and inverse problems related to a singularly perturbed system of ordinary differential equations which describe a process in chemical kinetics. We also extend the class of problems under study by considering polynomials of arbitrary degree as the right-hand parts of the differential equations in the case ε ̸= 0. Moreover, an iteration algorithm is proposed of finding an approximate solution to the inverse problem in the nondegenerate case (ε ̸= 0) for arbitrary degree. The theorem is proven on the convergence of the algorithm suggested. The proof is based on the contraction mapping principle (the Banach fixed- point theorem).

The article contains a review of recent results on solving the direct and inverse problems related to a singularly perturbed system of ordinary differential equations which describe a process in chemical kinetics. We also extend the class of problems under study by considering polynomials of arbitrary degree as the right-hand parts of the differential equations in the case ε ̸= 0. Moreover, an iteration algorithm is proposed of finding an approximate solution to the inverse problem in the nondegenerate case (ε ̸= 0) for arbitrary degree. The theorem is proven on the convergence of the algorithm suggested. The proof is based on the contraction mapping principle (the Banach fixed- point theorem).

integral manifoldslow surfacesingularly perturbed systemsmall parameterinverse problemODEReferencesGutman A. E. and Kononenko L. I., “Formalization of inverse problems and its applications [in Russian],” Sib. J. Pure Appl. Math., 17, No. 4, 49–56 (2017). DOI: 10.17377/PAM.2017.17.5.Gutman A. E. and Kononenko L. I., “Formalization of inverse problems and its applications [in Russian],” Sib. J. Pure Appl. Math., 17, No. 4, 49–56 (2017). DOI: 10.17377/PAM.2017.17.5.Gutman A. E. and Kononenko L. I., “The inverse problem of chemical kinetics as a composition of binary correspondences [in Russian],” Sib. Elektron. Mat. Izv., 15, 48–53 (2018). DOI: 10.17377/semi.2018.15.006.Gutman A. E. and Kononenko L. I., “The inverse problem of chemical kinetics as a composition of binary correspondences [in Russian],” Sib. Elektron. Mat. Izv., 15, 48–53 (2018). DOI: 10.17377/semi.2018.15.006.Goldstein V. M. and Sobolev V. A., Qualitative Analysis of Singularly Perturbed Systems [in Russian], Izdat. Inst. Mat., Novosibirsk (1988).Goldstein V. M. and Sobolev V. A., Qualitative Analysis of Singularly Perturbed Systems [in Russian], Izdat. Inst. Mat., Novosibirsk (1988).Kononenko L. I., “Identification problem for singular systems with small parameter in chemical kinetics [in Russian],” Sib. Elektron. Mat. Izv., 13, 175–180 (2016). DOI: 10.17377/semi.2016.13.015.Kononenko L. I., “Identification problem for singular systems with small parameter in chemical kinetics [in Russian],” Sib. Elektron. Mat. Izv., 13, 175–180 (2016). DOI: 10.17377/semi.2016.13.015.Gutman A. E. and Kononenko L. I., “Binary correspondences and the inverse problem of chemical kinetics [in Russian],” Vladikavk. Mat. Zh., 20, No. 3, 37–47 (2018).Gutman A. E. and Kononenko L. I., “Binary correspondences and the inverse problem of chemical kinetics [in Russian],” Vladikavk. Mat. Zh., 20, No. 3, 37–47 (2018).Kononenko L. I., “The identification problem for a nonsingular system of ordinary differential equations with fast and slow variables [in Russian],” Mat. Zamet. SVFU, 28, No. 2, 3–15 (2021).Kononenko L. I., “The identification problem for a nonsingular system of ordinary differential equations with fast and slow variables [in Russian],” Mat. Zamet. SVFU, 28, No. 2, 3–15 (2021).Mitropolsky Yu. A. and Lykova O. B., Integral Manifolds in Nonlinear Mechanics [in Russian], Nauka, Moscow (1963).Mitropolsky Yu. A. and Lykova O. B., Integral Manifolds in Nonlinear Mechanics [in Russian], Nauka, Moscow (1963).Vasil’eva A. V. and Butuzov V. F., Singularly Perturbed Equations in Critical Cases [in Russian], Nauka, Moscow (1978).Vasil’eva A. V. and Butuzov V. F., Singularly Perturbed Equations in Critical Cases [in Russian], Nauka, Moscow (1978).Voropaeva N. V. and Sobolev V. A., Geometric Decomposition of Singularly Perturbed Systems [in Russian], Fizmatlit, Moscow (2009).Voropaeva N. V. and Sobolev V. A., Geometric Decomposition of Singularly Perturbed Systems [in Russian], Fizmatlit, Moscow (2009).Kononenko L. I., “On the smoothness of slow surfaces of singularly perturbed systems [in Russian],” Sib. Zh. Ind. Mat., 5, No. 2, 109–125 (2002).Kononenko L. I., “On the smoothness of slow surfaces of singularly perturbed systems [in Russian],” Sib. Zh. Ind. Mat., 5, No. 2, 109–125 (2002).Tikhonov A. N., “On independence of solutions to differential equations on a small parameter [in Russian],” Mat. Sb., 22, No. 2, 193–204 (1948).Tikhonov A. N., “On independence of solutions to differential equations on a small parameter [in Russian],” Mat. Sb., 22, No. 2, 193–204 (1948).Romanov V. G., “Inverse problems for hyperbolic systems [in Russian],” in: Numerical Methods in Mathematical Physics, Geophysics and Optimal Control, 233, pp. 75–83, Nauka, Novosibirsk (1978).Romanov V. G., “Inverse problems for hyperbolic systems [in Russian],” in: Numerical Methods in Mathematical Physics, Geophysics and Optimal Control, 233, pp. 75–83, Nauka, Novosibirsk (1978).Kabanikhin S. I., Inverse and Ill-posed Problems [in Russian], Sib. Nauch. Izdat., Novosibirsk (2009).Kabanikhin S. I., Inverse and Ill-posed Problems [in Russian], Sib. Nauch. Izdat., Novosibirsk (2009).Kozhanov A. I., “Nonlinear loaded equations and inverse problems,” Comput. Math. Math. Phys., 44, No. 4, 657–675 (2004).Kozhanov A. I., “Nonlinear loaded equations and inverse problems,” Comput. Math. Math. Phys., 44, No. 4, 657–675 (2004).Golubyatnikov V. P., “An inverse problem for the Hamilton–Jacobi equation on a closed manifold,” Sib. Math. J., 38, No. 2, 235–238 (1997).Golubyatnikov V. P., “An inverse problem for the Hamilton–Jacobi equation on a closed manifold,” Sib. Math. J., 38, No. 2, 235–238 (1997).Anikonov Yu. E., Some questions in the theory of inverse problems for kinetic equations [in Russian], in: Inverse Problems of Mathematical Physics, pp. 28–41, Vychisl. Tsentr Sib. Otdel. Akad. Nauk SSSR, Novosibirsk (1985).Anikonov Yu. E., Some questions in the theory of inverse problems for kinetic equations [in Russian], in: Inverse Problems of Mathematical Physics, pp. 28–41, Vychisl. Tsentr Sib. Otdel. Akad. Nauk SSSR, Novosibirsk (1985).

The authors declare that there are no conflicts of interest present.