On a generalized nonlinear functional equation

2011 ◽  
Vol 74 (10) ◽  
pp. 3305-3310 ◽  
Author(s):  
Lothar Berg ◽  
Stevo Stević

The continuous small amplitude disturbances generated by the oscillations of a piston in a gas contained in a closed-ended tube are discussed. The coupled characteristic equations are integrated exactly for a model equation of state which approximates any stress–strain law with an error at O ([strain] 3 ). When the small amplitude limit is taken and the relative importance of amplitude dispersion and nonlinear interaction is assessed, the disturbances can be determined from solutions to a nonlinear functional equation. The motions are characterized by a similarity parameter, A , and a frequency parameter, ∆ . A simple algebraic scheme for constructing continuous periodic solutions of the functional equation is given, and the A–∆ plane is divided by a transition curve. The various resonances are then evident.


We study analytic solutions of the functional equation h(z) 2 – h(z 2 ) + c = 0, (H) where c > 0 is a parameter, and constant solutions are excluded. It suffices to consider solutions for which h(z) – z -1 is regular in a neighbourhood of z = 0. If 0 < c ≼ ¼, h(z) can be continued as a single-valued analytic function into the unit disc | z | < 1 where its only singularity is the pole at z = 0; the circle | z | = 1 is a natural boundary. On the other hand, if c > ¼, then by analytic continuation h(z) becomes a multiple-valued function with an infinite sequence of quadratic branch points tending to every point of | z | = 1, and no branch of h(z) can be continued beyond this circle. A change of variable transforms (H) into the difference equation g ( Z + 1) – g(Z) = g(Z) 2 + C , where C is a real parameter. The solutions of this equation have properties similar to thóse of (H).


Author(s):  
Valery C. Covachev ◽  
H. Ali Yurtsever

In the present paper a generalization of a theorem of I.B. Risteski (2004) concerning the solution of a nonlinear functional equation is given. The proof is based on a parametric approach by introducing a parameter in an arbitrary set , and on a matrix method for solving linear functional equations. 


Sign in / Sign up

Export Citation Format

Share Document