IJST (2013) 37A3 (Special issue-Mathematics): 389-396 Iranian Journal of Science & Technology http://ijsts.shirazu.ac.ir Boundary layer problem for system of first order of ordinary differential equations with linear non-local boundary conditions M. Jahanshahi 1 *, A. R. Sarakhsi 1 , S. Asharafi 1 and N. Aliev 2 1 Department of Mathematics, Azarbaijan Shahid Madani University, 5375171379, Tabriz, Iran 2 Department of Mathematics, Baku State University, Baku, Azerbaijan E-mail: jahanshahi@azaruniv.edu Abstract In this paper we study the boundary layer problems in which boundary conditions are non-local. Here we try to find the necessary conditions by the help of fundamental solution to the given adjoint equation. By getting help from these conditions, at first the boundary condition is changed from non-local to local. The main aim of this paper is to identify the location of the boundary layer. In other words, at which point the boundary layer is formed. Keywords: Singular perturbation problems; boundary layer; fundamental solution; necessary conditions 1. Introduction An important subject in applied mathematics is the theory of singular perturbation problems. The mathematical model for this kind of problem is usually in the form of either ordinary differential equations (O.D.E) or partial differential equations (P.D.E) in which the highest derivative is multiplied by some powers of  as a positive small parameter [1-3]. The object theory of singular perturbation is to solve differential equation with some initial or boundary conditions with a small parameter. These problems are essentially at the heart of boundary value and initial value problems [3-10]. Through these studies we can find out whether the boundary conditions become a local type (Dirichlet) and the solution of the boundary layer problem is satisfied in boundary conditions then there is no boundary layer. If the limit solution (when 0   ) is not satisfied in the boundary condition, then there will be a boundary layer. In book [4] after the first and second chapters, the unsolved boundary layer problem is seen, which shows that boundary layer problems with non-local boundary conditions have not been studied carefully. So, in this paper and some other works: M. Jahanshahi & A. R. Sarakhsi [10-14] and N. Aliev & S. Ashrafi [15], [16], we study the boundary layer problems in which boundary conditions are non-local. Here, an attempt is made to find the necessary conditions with the help of the fundamental solution of the given adjoint *Corresponding author Received: 8 August 2012 / Accepted: 2 February 2013 equation. By taking advantage of these conditions, first the boundary conditions are changed for non- local to local and finally as before there will be local case and the reason for the boundary layer lock will be studied [12]. 2. Mathematical statement of problem We consider the following boundary layer problem: . () [ () ( )] () ( ), ( , ), lX X t pt qt X t ft t ab            (1) () () X a X b        (2) where 0   is a small parameter, ) ( ), ( t q t p are the square matrices of n order in which the elements are real continuous functions and ) (t f and ) (t X  are column vectors whose functions are real continuous. ) ( ), ( t q t p and ) (t f coefficients of equation (1) are known functions while ) (t X  is unknown vector function. Data of boundary condition in the problem, that is  and  , is square matrices of n order with real constant elements, and of column vector has n components with real constant elements. Equation (1) easily shows that when 0   , it changes to an algebraic system. Based on this fact, it can be verified whether solutions of linear algebraic system exist in boundary condition (2) or not. As far as we know, if