Method of Boundary Layer Function to Solve the Boundary Value Problem for a Singularly Perturbed Differential Equation of the Order Two with a Turning Point

Apparently, for the first a uniformly asymptotic expansion of a solution of a singularly perturbed linear equations with small parameter in higher derivatives(equations of Prandtl-Tikhonov type, shortly P-T type) was constructed G.E.Latta in 1951 [1] and E.Bromberg in 1956 [2] by method of composite expansions . I note that this method of composite expansion in soviet mathematicians is called method of boundary layer functions (briefly MBLF). But a uniformly asymptotic expansion of solutions of nonlinear equations type P-T received by A.B. Vasilievain1960 only [3]-[5] by the method of matching, after by W.Wasow and Y. Sibuya in 1963 [3]. The MBLF for linear singularly perturbed partial differential equations of type P-T was developed by М.I.Vishik and L.А. Lusternik in 1957 [6] and for nonlinear differential equations ( more exactly: for the singulary perturbed integro-differential equations) by M.I. Imanaliev 1964 [7], after by R.E.O’MalleyиF.Hoppensteadt 1971 [3] . More surveys of related material are contained in [3], [8][11]. The MBLF usually is used for the construction of asymptotic solution of the singularly perturbed equations in the case of exponential asymptotical stability of the equation in the fast variable, i.e. under the A.N. Tikhonov theorem’s conditions. Also, for the construction of asymptotic solutions of the singularly perturbed equations the method of matching outer and inner solution is applied [10-16] and the rule of matching of the outer and inner solution was given by Van Dyke. I note that the developing of the method matching was given in [17]-[20]. Here it is proved the possibility of using of the MBLF for construction of the asymptotic of the boundary value problem for equations with the turning point, where the Tihonov’s condition is broken.


Introduction
Apparently, for the first a uniformly asymptotic expansion of a solution of a singularly perturbed linear equations with small parameter in higher derivatives(equations of Prandtl-Tikhonov type, shortly P-T type) was constructed G.E.Latta in 1951 [1] and E.Bromberg in 1956 [2] by method of composite expansions . I note that this method of composite expansion in soviet mathematicians is called method of boundary layer functions (briefly MBLF).
But a uniformly asymptotic expansion of solutions of nonlinear equations type P-T received by A.B. Vasilievain1960 only [3]- [5] by the method of matching, after by W.Wasow and Y. Sibuya in 1963 [3].
The MBLF usually is used for the construction of asymptotic solution of the singularly perturbed equations in the case of exponential asymptotical stability of the equation in the fast variable, i.e. under the A.N. Tikhonov theorem's conditions.
Also, for the construction of asymptotic solutions of the singularly perturbed equations the method of matching outer and inner solution is applied [10][11][12][13][14][15][16] and the rule of matching of the outer and inner solution was given by Van Dyke. I note that the developing of the method matching was given in [17]- [20].
Here it is proved the possibility of using of the MBLF for construction of the asymptotic of the boundary value problem for equations with the turning point, where the Tihonov's condition is broken.

Statement of the Problem
The following problem is considered where 0 1 ε < << -small parameter, -is given constant.
The following condition (that is class analytical functions) is imposed on the known functions. It follows that Here we will consider the case 1 ) 0 ( 0 = = q q for simplicity. I note that the solution of the problem (1)-(2) was constructed earlier by method of boundary layer functions with the application of differential inequality in [20] [17][18] in [19].

Construct of the Solution of the Problem by the Method of the Boundary Layer Functions
The solution of the unperturbed equation (1) ) (1) is represented as follows By distinguishing the main part of the integral in the sense of Hadamar [21] in (4) we may represent this one to next form where 2 0 1 1 depends on µ , but we will not write it for brevity.
We will take initial data for functions with arbitrary functions ( ) f x . Therefore we must change right hand parts equations (8.2n) and respectively right hand part equations(9.2n).We will change them as follows To define the functions ) (t k π (k=0,1,2,…) we need the following Lemma 2.The homogeneous equation has two linearly independent solutions :   (11) is defined the following formula . We will proof that operator S also is contracting operator. We will consider the operator S on interval and[ , ] l µ  . We may take any number greater than 1for l ,but we will take 1 l = for simplicity.
Evaluating ( ) h t we have got The following result is correct Тtheorem. The solution of the problem(1)-(2)can be represented in the (16) and ( , ) x m ξ ε ≤ ,that is the series (7) is an asymptotical one for this solution.

The Example. The Comparison of Two Methods: Method of Boundary Layer Functions and Method of Structural Matching
It is considered the following problem  We will seek the outer solution in the form