Applied Mathematics
Vol.05 No.13(2014), Article ID:47830,4 pages
10.4236/am.2014.513191

Asymptotic Estimates for Second-Order Parameterized Singularly Perturbed Problem

Mustafa Kudu

Department of Mathematics, Faculty of Art and Science, Erzincan University, Erzincan, Turkey

Email: muskud28@yahoo.com   Received 16 April 2014; revised 20 May 2014; accepted 3 June 2014

ABSTRACT

The boundary value problem (BVP) for parameterized singularly perturbed second order nonlinear ordinary differential equation is considered. The boundary layer behavior of the solution and its first and second derivatives have been established. An example supporting the theoretical analysis is presented.

Keywords:

Parameterized Problem, Asymptotic Bounds, Singular Perturbation, Boundary Layer 1. Introduction

In this paper, we are going to obtain the asymptotıc bounds for the following parameterized singularly perturbed boundary value problem (BVP): (1) (2)

where is a perturbation parameter, are given constants and is a sufficiently smooth function in . Further , the function is assumed to be sufficiently continuously differentiable for our purpose function in and . (3)

By a solution of (1), (2) we mean pair for which problem (1), (2) is satisfied.

An overview of some existence and uniqueness results and applications of parameterized equations may be obtained, for example, in  - . In  -  , some approximating aspects of this kind of problems have also been considered. The qualitative analysis of singular perturbation situations have always been far from trivial because of the boundary layer behavior of the solution. In singular perturbation cases, problems depend on a small parameter in such a way that the solution exhibits a multiscale character, i.e., there are thin transition layers where the solution varies rapidly while away from layers it behaves regularly and varies slowly   . In this note we establish the boundary layer behaviour for of the solution of (1)-(2) and its first and second derivatives. Example that agree with the analytical results is given.

Theorem 1.1. For the solution of the problem (1), (2) satisfies, (4) (5)

where

,

and

(6)

provided and for and .

Proof. We rewrite Equation (1) in the form

, (7)

where,

, , ,―intermediate values.

From (7) for the first derivate, we have

(8)

Integrating this equality over we get

(9)

from which by setting the boundary condition we obtain,

(10)

Applying the mean value theorem for integrals, we deduce that,

(11)

and

(12)

Also, for first and second terms in right side of (10), for values, we have

(13)

It then follows from (11)-(13)

(14)

Next from (9), we see that

Under the conditions and the operator admits the following maximum principle: Suppose be any function satisfiying , and then

Using the maximum principle whith barrier functions we have the inequality

(15)

The inequlities (14), (15) immediately leads to (4), (5). After taking into consideration the uniformly boundnees in of and it then follows from (8) that,

,

which proves (6) for To obtain (6) for, first from Equation (1) we have

,

from which after taking into consideration here and (4)

(16)

Next, differentiation (1) gives

(17)

(18)

with

,

and due to our assumptions clearly,

Consequently, from (17), (18) we have

which proves (6) for.

Example. Consider the particular problem

where, and selected so that the solution is

with

First and second derivatives have the form

Therefore we observe here the accordance in our theoretical results described above.

References

1. Pomentale, T. (1976) A Constructive Theorem of Existence and Uniqueness for the Problem. Zeitschrift für Ange- wandte Mathematik und Mechanik, 56, 387-388. http://dx.doi.org/10.1002/zamm.19760560806
2. Goma, I.A. (1977) Method of Successive Approximations in a Two-Point Boundary Problem with Parameter. Ukrainian Mathematical Journal, 29, 594-599. http://dx.doi.org/10.1007/BF01085968
3. Jankowski, T. (2001) Monotone İterations for Differential Problems. Miskolc Mathematical Notes, 2, 31-38.
4. Jankowski, T. and Lakshmikantham, V. (1997) Monotone Iterations for Differential Equations with a Parameter. Journal of Applied Mathematics and Stochastic Analysis, 10, 273-278. http://dx.doi.org/10.1155/S1048953397000348
5. Jankowski, T. (1999) Generalization of the Method of Quasilinearization for Differential Problems with a Parameter, Dynamic Systems an Applications, 8, 53-72.
6. Rontó, M. and Csikos-Marinets, T. (2000) On the Investigation of Some Non-Linear Boundary Value Problems with Parameters. Miskolc Mathematical Notes, 1, 157-166.
7. Rontó, M. (2000) On Non-Linear Boundary Value Problems Containing Parameters. Archivum Mathematicum, 36, 585-593.
8. Staněk, S. (1997) Nonlinear Boundary Value Problem for Second Order Differential Equations Depending on a Parameter. Mathematica Slovaca, 47, 439-449.
9. Zhang, P. (2011) Existence of Positive Solutions for Nonlocal Second-Order Boundary Value Problem with Variable Parameter in Banach Spaces. Fixed Point Theory and Applications, 43, 1687-1812. http://dx.doi.org/10.1186/1687-1812-2011-43
10. Fěckan, M. (1994) Parametrized Singularly Perturbed Boundary Value Problems. Journal of Mathematical Analysis and Applications, 188, 426-435. http://dx.doi.org/10.1006/jmaa.1994.1436
11. Amiraliyev, G.M., Kudu M. and Duru, H. (2004) Fınıte-Difference Method for Parameterızed Sıngularly Pertur Bed Problem. Journal of Applied Mathematics, 3, 191-199. http://dx.doi.org/10.1155/S1110757X0440103X
12. Amiraliyev, G.M., Kudu M. and Duru, H. (2006) Uniform Difference Method for a Parameterized Singular Pertur Bation Problem. Applied Mathematics and Computation, 175, 89-100. http://dx.doi.org/10.1016/j.amc.2005.07.068
13. Amiraliyeva I.G. and Amiraliyev, G.M. (2009) Uniform Difference Method for Parameterized Singularly Pertur Bed Delay Differential Equations. Numerical Algorithms, 52, 509-552. http://dx.doi.org/10.1007/s11075-009-9295-y
14. Turkyilmazoglu, M. (2011) Analytic Approximate Solutions of Parameterized Unperturbed and Singularly Perturbed Boundary Value Problems. Applied Mathematical Modelling, 35, 3879-3886. http://dx.doi.org/10.1016/j.apm.2011.02.011
15. Nayfeh, A.H. (1981) Introduction to Perturbation Techniques. JohnWiley & Sons, New York.
16. O’Malley Jr., R.E. (1991) Singular Perturbation Methods for Ordinary Differential Equations. Springer-Verlag, New York. http://dx.doi.org/10.1007/978-1-4612-0977-5