Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

Example of Solution of a Singularly Perturbed Cauchy Problem for a Parabolic Eqiuation in the Presence of "strong" Turning Point

Author(s):

Alexander Georgievich Eliseev

Department of Higher Mathematics, Associate Professor
National Research University MPEI
14 Krasnokazarmennaya St., Moscow, Russia, 111250

yeliseevag@mpei.ru

Abstract:

In the article, on the basis of S.A. Lomov's regularization method, an asymptotic solution of a singularly perturbed Cauchy problem for a parabolic equation in the presence of a "strong" turning point is constructed. The regularization method makes it possible to construct an asymptotic solution uniform on the entire axis. The idea of this paper goes back to the paper where methods for solving a singularly perturbed Cauchy problem in the case of a "simple" turning point of a limit operator with a natural exponent were developed.

Keywords

• "strong" turning point
• asymptotic solution
• parabolic equation
• regularization method
• singularly perturbed Cauchy problem

References:

1. Lomov S. A. Introduction to the general theory of singular perturbations. - M., Nauka, 1981, 400~p
2. Eliseev A. G., Lomov S. A. Theory of singular perturbations in the case of spectral singularities of the limit operator. Mat. collection, 1986, vol. ~131, no. ~4, pp. ~544-557
3. Lomov S. A. Uniform asymptotic expansions of a problem with a turning point. In the book: Dokl. sci. -tech. conf., section math. - M., MPEI, 1969, pp. ~ 42-50
4. Lomov S. A. Asymptotic integration with a change in the nature of the spectrum. \textit{Tr. MPEI}, 1978, issue ~ 357, pp. ~ 56-62
5. Eliseev A. G., Ratnikova T. A. Singularly perturbed Cauchy problem in the presence of a rational " simple" turning point of the limit operator. Diff. eq. and percent. management, 2019, No. ~3, pp. ~63-73
6. Yeliseev A. On the Regularized Asymptotics of a Solution to the Cauchy Problem in the Presence of a Weak Turning Point of the Limit Operator., Axioms, 2020, №~9, 86. - http://doi.org/10.3390/axioms9030086
7. Eliseev A. G., Kirichenko P. V. Solution of a singularly perturbed Cauchy problem in the presence of a < < weak> > turning point of the limit operator. Sibir. electr. math. Izv., 2020, No. ~17, pp. ~51—60 . 8 V. I. Arnold. On matrices depending on parameters. // Uspekhi Mat. Nauk, 1971, volume 26, issue 2(158), 101-114
8. F. G. Mehler. Ueber die Entwicklung einer Function von beliebig vielen Variablen nach Laplaceschen Functionen honerer Ordnung., Jornal fur die Reine und Angewandte Mathematik, 1866, 161-176p

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