### Non-stable differential turning point of the 2-nd kind for the fourth order system

#### Abstract

*This paper is concerned with the nonhomogeneous system of the singularly perturbed differential equations of the 4th order with the small parameter at the higher derivative. This system is consequence of reduction of the 4th order scalar singularly perturbed differential equations with the differentiable coefficients and the turning point of the second derivative and the Orr–Sommerfeld type equation with the small parameter of the fourth derivative. The reduced equation is the differential equation of the second order. The turning point is the second genre discontinuity in the solution of the reduced equation. The characteristic equation that corresponds to the system of interest has two identically zero roots and two simple roots which are identically double. Due to the method of essential singular functions the new regularizing variable is obtained and the extension of the vector equation is kept. Asymptotic forms of solutions for the homogeneous problem are constructed with the help of Airy functions and their derivatives. Asymptotic forms of solutions for the nonhomogeneous problem are constructed using Scorer functions. The article discusses the variation of the non-stable turning point or the variation when the turning point is on the left of origin.*

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#### References

Wasow W.( 1958). Asymptotic solution of the differential equation of

hydrodynamic stability in a domain containing a transition point. Annals of mathematics,

(2), 222-252.

Langer, R. E.(1958) Formal solutions and a related equation for a class of fourth

order differentialequations of hydrodynamic type. Trans. Amer. Math. Soc.

Lin C.C.(1958). On the instability of laminar flow and its transition to turbulence.

Proceedings of the Symposium on Boundary Layer Research, 144-160.

Lin C.C., Rabenstein A.L.(1969). On the asymptotic theory of a class of ordinary

differential equations of forth order. II Existence of solutions which are approximated by the

formal solutions. Studies in Appl. Math, 48, 311-340.

Awrejcewicz J., Krysko V.(2006) Introduction to Asymptotic Methods. – New

York: Champan Hall. CRC Taylor Group, 242.

Zelenska I. (2015). The system of singularly perturbed differential equations with

turning point of the I kind.. Izv. Vysshikh uchebnykh zavedenii. Mathematics, 3, 63-74. (in

Rus.)

Boliliy V.O.(2002). Non-stable turning point in the differential equation of the third

order. Matematychni studii, 157-168. (in Uk.)

Boliliy V.O., Zelenska I.O. (2013). Asymptotic integration of systems of

differential equations with differential turning point of II kind. Intern. Scient. Journal.

Spectral and Evolution Problems: Proceeding of the Twenty Second Crimean Autumn

Mathematical School-Symposium, 23, 21-31.(in Rus.)

Bobochko V.N., Perestuk M.O.(2006) Asymptotic integration of the Liouville

equation with turning points. Kyiv: Naukova dumka, 310. (in Uk.)

Abramowitz and Stegun (1979). Handbook of Mathematical Functions. Moscow:

Nauka, 832. (in Rus.)