Application of Argument Principle for Study a Structure of Roots of  Transcendental Equations

Nail Suleiman Khabeev

Journal of Vibration Testing and System Dynamics

C. Steve Suh (editor), Pawel Olejnik (editor),

Xianguo Tuo (editor)

Application of Argument Principle for Study a Structure of Roots of Transcendental Equations

Journal of Vibration Testing and System Dynamics 5(3) (2021) 233--236 | DOI:10.5890/JVTSD.2021.09.003

Nail Suleiman Khabeev

Department of Mathematics, University of Bahrain, Kingdom of Bahrain

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Abstract

In study of the structure of shock waves in bubbly liquids we need to find integral curve connecting two singular points of the system of differential equations. We linearized the system of basic equations around equilibrium state. The condition of existence of nontrivial solution of the linear equations leads to the characteristic transcendental equation with respect to exponential index. The structure of roots of such equation is studied by using argument principle [1]. It is shown that for compression waves such equation has one real root.

References

[1] Mathews, J.M. and Howell, R.W. (2006), Complex Analysis for Mathematics and Engineering, Jones and Bartlett, London.
[2] Shagapov, V.S., Gimaltdinov, I.K., Khabeev, N.S., and Bailey, S.S. (2003), Acoustic Waves in a Liquid with a Bubble Screen, Shock waves, 13(1), 49-56.
[3] Nigmatulin, R.I., Khabeev, N.S., and Hai, Z.N. (1988), Waves in Liquids with Vapour Bubbles, Journal of fluid mechanics, 185, 85-117.