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

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Published: Mar 12, 2017
Keywords:
linear system, small parameter, turning point, space of the nonresonance solutions, Airy-Langer model operator, Orr–Sommerfeld type equation

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V. О. Boliliy
І. О. Zelenska

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.

Article Details

Issue
Vol. 389 No. 1 (2016)
Section
Materials Physics

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