Open Access Research Article

Nonlocal Four-Point Boundary Value Problem for the Singularly Perturbed Semilinear Differential Equations

Robert Vrabel

Author Affiliations

Institute of Applied Informatics, Automation and Mathematics, Faculty of Materials Science and Technology, Hajdoczyho 1, 917 01 Trnava, Slovakia

Boundary Value Problems 2011, 2011:570493  doi:10.1155/2011/570493

Published: 19 September 2010

Abstract

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.