ASYMPTOTIC BEHAVIOUR OF SOLUTIONS OF A DELAYED DIFFERENTIAL EQUATION

1995 ◽  
Vol 28 (1) ◽  
Author(s):  
Zdenĕk Svoboda
Keyword(s):  
Differential Equation ◽  
Asymptotic Behaviour ◽  
Delayed Differential Equation ◽  
Asymptotic Behaviour Of Solutions

13.—The Functional Differential Equation y′(x) = ay(λx) + by(x)

1976 ◽  
Vol 74 ◽  
pp. 165-174 ◽  
Author(s):  
Jack Carr ◽  
Janet Dyson
Keyword(s):  
Differential Equation ◽  
Asymptotic Behaviour ◽  
Functional Differential Equation ◽  
Complex Constant ◽  
Asymptotic Behaviour Of Solutions ◽  
Functional Differential ◽  
Image Position

SynopsisThe paper discusses the asymptotic behaviour of solutions of the functional differential equationwhere a is a complex constant, 0<λ<1, and b is a constant such that Re b = 0, but b ≠ 0.

The asymptotic behaviour of solutions of the differential equation

1975 ◽  
Vol 346 (1645) ◽  
pp. 171-207 ◽  
Keyword(s):  
Differential Equation ◽  
Asymptotic Behaviour ◽  
Bessel Function ◽  
Positive Real ◽  
Generalized Hypergeometric Functions ◽  
Asymptotic Behaviour Of Solutions ◽  
Special Cases ◽  
Transition Points ◽  
Qualitative Discussion ◽  
Asymptotic Forms

The asymptotic behaviour of solutions of the fourth order differential equation in which the constants a and b are supposed real and positive, is examined for large real values of z and b. Exact solutions are given in terms of generalized hypergeometric functions and some special cases of a and b are mentioned for which solutions may be expressed in terms of simpler known functions. The differential equation possesses four transition points and for positive values of b one of these transition points lies on the positive real 2-axis. The asymptotic discussion is centred around an integral representation which involves a modified Bessel function of the second kind whose order is purely imaginary for large positive values of b. Asymptotic forms as z-> oo and about the transition point on the positive real 2-axis are given. A qualitative discussion of the zeros of the Bessel function K^{z) for imaginary order and complex argument is presented in the appendix.

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