Nonlocal Four-Point Boundary Value Problem for the Singularly Perturbed Semilinear Differential Equations. (19th September 2010)

• Record Type: Journal Article
• Authors: Vrabel, Robert
• Other Names: Franco Daniel Academic Editor.
• Is Part Of: Boundary value problems. Volume 2011(2011)
• Publication Date: 2010-09-19
• Subjects: Boundary value problems -- Periodicals; Boundary value problems; Electronic journals; Periodicals; 515.35
• DOI: 10.1155/2011/570493
• Languages: English
• ISSNs: 1687-2762

Abstract

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.