Abstract
Lomov’s regularization method is generalized to resonant, weakly nonlinear, singularly perturbed systems in the case of intersecting roots of the characteristic equation of the limit operator. For constructing asymptotic solutions, the regularization of the original problem by using normal forms developed by the authors is performed. In the absence of resonance, the regularizing normal form is linear, whereas in the presence of resonances, it is nonlinear. In this paper, the resonant case of a weakly nonlinear problem is considered. By using an algorithm of normal forms, we construct an asymptotic solution of any order (with respect to a parameter) and justify this algorithm.
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V. S. Abramov, A. A. Bobodzhanov, and M. A. Bobodzhanova, “Normal forms and asymptotic solutions of nonlinear singularly perturbed problems in the case if intersecting points of the spectrum of the limit operator,” Aspirant Soisk., No. 4, 42–47 (2018).
A. A. Bobodzhanov and V. F. Safonov, “Asymptotics of solutions to nonlinear singularly perturbed systems in the critical case,” Vest. Mosk. Energ. Inst., No. 6, 10–17 (1997).
A. G. Eliseev, “Theory of singular perturbations in the case of a “weak” turning point of the limit operator,” Vest. Mosk. Energ. Inst., No. 6, 31–41 (1997).
A. G. Eliseev and S. A. Lomov, “Theory of singular perturbations in the case of spectral singularities of the limit operator,” Mat. Sb., 131 (173), No. 4, 544–557 (1986).
S. A. Lomov, Introduction to the General Theory of Singular Perturbations [in Russian], Nauka, Moscow (1981).
S. A. Lomov and I. S. Lomov, Foundations of the Mathematical Theory of Boundary Layers [in Russian], Moscow State Univ., Moscow (2011).
V. F. Safonov, “Regularized asymptotic solutions of singularly perturbed problems in the critical case,” Izv. Vyssh. Ucheb. Zaved. Mat., 384, No. 5, 41–48 (1994).
V. F. Safonov and A. A. Bobodzhanov, A Course of Higher Mathematics. Singularly Perturbed Problem and Methods of Regularization [in Russian], Moscow Power Engineering Institute, Moscow (2012).
V. F. Safonov and A. A. Bobodzhanov, Method of Normal Forms for Nnonlinear Resonant Singularly Perturbed Problems [in Russian], Sputnik+, Moscow (2016).
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 172, Proceedings of the Voronezh Winter Mathematical School “Modern Methods of Function Theory and Related Problems,” Voronezh, January 28 – February 2, 2019. Part 3, 2019.
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Bobodzhanov, A.A., Safonov, V.F. Asymptotic Solutions of Resonant Nonlinear Singularly Perturbed Problems in the Case of Intersecting Eigenvalues of the Limit Operator. J Math Sci 268, 1–14 (2022). https://doi.org/10.1007/s10958-022-06175-2
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DOI: https://doi.org/10.1007/s10958-022-06175-2