Institute of Applied Informatics, Automation and Mathematics, Faculty of Materials Science and Technology, Hajdoczyho 1, 917 01 Trnava, Slovakia
Copyright © 2011 Robert Vrabel. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper deals with the existence and asymptotic behavior of the solutions
to the singularly perturbed second-order nonlinear differential equations. For example,
feedback control problems, such as the steady states of the thermostats,
where the controllers add or remove heat, depending upon the temperature detected
by the sensors in other places, can be interpreted with a second-order ordinary
differential equation subject to a nonlocal four-point boundary condition.
Singular perturbation problems arise in the heat transfer problems with large
Peclet numbers. We show that the solutions of mathematical model, in general,
start with fast transient which is the so-called boundary layer phenomenon, and
after decay of this transient they remain close to the solution of reduced problem
with an arising new fast transient at the end of considered interval. Our analysis
relies on the method of lower and upper solutions.