# Construction of asymptotics of the solution for a system of singularly perturbed equations by the method of essentially singular functions

## DOI:

https://doi.org/10.17721/1812-5409.2023/2.34## Keywords:

small parameter, turning point, singular perturbations, Airy functions, Liouville equation## Abstract

*Singularly perturbed problems with turning points arise as mathematical models for various physical phenomena. The internal turning point problem is a one-dimensional version of the steady-state convection-diffusion problem with a dominant convective term and a velocity field that changes sign in the reservoir. Boundary turning point problems, on the other hand, arise in geophysics and in the modeling of thermal boundary layers in laminar flow. The paper analyzes the results from the asymptotic analysis of singularly perturbed problems with turning points. For a homogeneous system of singularly perturbed differential equations with a small parameter at the highest derivative and a turning point, the conditions for constructing a uniform asymptotic solution are obtained. We consider the case when the spectrum of the limit operator contains multiple and identically zero elements. The asymptotics are constructed by the method of essentially singular functions, which allows using the Airy model operator in the vicinity of the turning point. The construction of asymptotic solutions contains arbitrary constants, which are determined uniquely during the solution of the iterative equations. At the same time, the conditions for the existence of a solution of a system of differentials with a small parameter for the highest derivative and for the presence of a turning point are obtained, provided that the turning point is located on the interval [0; l]. An example of constructing the asymptotic of a homogeneous system of differential equations is given.*

* Pages of the article in the issue*: 184 - 192

** Language of the article**: Ukrainian

## References

EBERHARD, W., FREILING, G., WILCKEN, K. (2001) Indefinite eigenvalue problems with several singular points and turning points. Math. Nachr, 229, pp. 51-71. doi: 10.1002/15222616(200109)229:13. 0.CO;2-4.

LANGER, R.E. (1959) The asymptotic solutions of a linear differential equations of the second order with two turning points. Trans. Amer. Math. Soc. V. 90, pp.113–142.

NIJIMBERE, V. (2019) Asymptotic approximation of the eigenvalues and the eigenfunctions for the Orr-Sommerfeld equation on infinite intervals”. Advances in Pure Mathematics, 9, pp. 967-989. DOI: 10.4236/apm.2019.912049.

LOCKER, J. (2000) Spectral Theory of Non-Self-Adjoint Two-Point Differential Operators, Mathematical Surveys and Monographs. American Mathematical Society, Rhode Island, V. 73, doi: 10.1090/surv/073

BOBOCHKO V., PERESTYUK M. (2002) Asymptotic integration of the Liouville equation with turning points. Kyiv: Scientific opinion.

SAMOILENKO, A., KLYUCHNYK, I. (2009) On the asymptotic integration of a linear system of differential equations with a small parameter with partial derivatives”. Nonlinear oscillations. V. 12(2). pp. 208-234.

SAMOILENKO, A., SAMUSENKO, P. (2020) Asymptotic Integration of Singularly Perturbed Differential Algebraic Equations With Turning Points. Part I. Ukrains’kyi Matematychnyi Zhurnal, V 72(12), pp. 1669-81, doi:10.37863/umzh.v72i12.6261.

ZELENSKA, I. (2015) System of singularly perturbed equations with differential turning point of the first kind. Russ Math. 2015.- 59. – p.55–65. https://doi.org/10.3103/S1066369X1 5030068

SOBCHUK, V., ZELENSKA, I (2022) Construction of the asymptotics of the solution of the 4th-order SZDR system with a differential turning point by the method of essentially singular functions. Scientific Bulletin of the Uzhhorod University, V. 41(2), p. 78-90. DOI: https://doi.org/10.24144/2616-7700.2022.41(2).78-90 (in Ukrainian).

SOBCHUK, V., LAPTIEV, O., ZELENSKA, I. (2023) Algorithm for solution of systems of singularly perturbed differential equations with a differential turning point. Bulletin of the Polish Academy of Sciences: Technical Sciences, V. 71(3). pp. Article number: e145682 DOI: 10.24425/bpasts.2023.145682

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*Bulletin of Taras Shevchenko National University of Kyiv. Physical and Mathematical Sciences*, (2), 184–192. https://doi.org/10.17721/1812-5409.2023/2.34

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Copyright (c) 2023 V. V. Sobchuk, I. O. Zelenska

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