Research Article | Open Access
Asymptotic Estimates of the Solution of a Restoration Problem with an Initial Jump
The asymptotic behavior of the solution of the singularly perturbed boundary value problem is examined. The derivations prove that a unique pair exists, in which components and satisfy the equation and boundary value conditions . The issues of limit transfer of the perturbed problem solution to the unperturbed problem solution as a small parameter approaches zero and the existence of the initial jump phenomenon are studied. This research is conducted in two stages. In the first stage, the Cauchy function and boundary functions are introduced. Then, on the basis of the introduced Cauchy function and boundary functions, the solution of the restoration problem is obtained from the position of the singularly perturbed problem with the initial jump. Through this process, the formula of the initial jump and the asymptotic estimates of the solution of the considered boundary value problem are identified.
One of the fundamental theorems of singular perturbations theories is Tikhonov’s theorem [1, 2] on the limit transition that establishes the limit equations, expressing the relations between the solutions of a degenerate problem and an original singularly perturbed initial problem, and this theorem allows us to obtain the leading member of asymptotics.
For a wide class of the singularly perturbed problems, effective asymptotic methods were developed, allowing for the development of uniform approximations with any level of precision. The methods of Višik and Lyusternik  and Vasilyeva  were the first methods to be developed, which are called methods of boundary functions. Butuzov  developed a method of angular boundary functions, which was a significant development for the boundary functions method.
Each of these methods has a certain area of applicability; that is, they are successful in solving some problems and are invalid when solving other problems. For example, there were some fundamental difficulties in the realization of the boundary functions method in problems with initial jumps. The beginning of the mathematical solution of the initial jump phenomenon was considered in the studies of Višik and Lyusternik  and Kasymov , in which the method of zone integration for nonlinear singularly perturbed initial tasks with unbounded initial data when a small parameter approaches zero. The research efforts of Višik, Lyusternik, and Kasymov were continued in [8, 9] and other studies.
Simultaneously, there were problems in practice that extended beyond the scope of traditional research, in which ready-made asymptotic methods were inapplicable and required modification or generalization. For example, in the study of Neǐmark and Smirnova , a new rationale of the physical and mathematical nature of the Painleve paradox was introduced, which enriched the possible types of movements and dynamics of the system as a whole. At this point, we saw the races and contrasting structures. Butuzov and Vasiliyeva studied mathematical solutions of the question of contrast structures , and the phenomenon of the initial jump requires additional mathematical research.
Boundary value problems for ordinary differential equations containing parameters in the right-hand side and in the boundary conditions were examined [12, 13]. In these studies, a restoration problem of the right-hand side of the differential equations and boundary conditions is solved with the well-known structure of the differential equation and additional information.
The following natural generalization in this direction is to study the solutions of singularly perturbed boundary value problems with an additional parameter having the initial jump phenomenon. Such a study has not yet been reported. This study considers such problems. It studies the problems of solution building, restoring the right-hand side of the equation and boundary conditions of the perturbed problem, the limit solution passing of the perturbed problem solution to the unperturbed problem solution.
2. Statement of the Problem
Let , where belongs to . We consider with nonseparated boundary conditions where is a small parameter, is an unknown parameter, are known constants, and . Let the matrix have a rank equal to , where is the maximum number of linearly independent rows of matrix . To the certainty, let the first rows of the matrix be linearly independent. Then, the linear forms , as the functions of and are linearly independent of each other.
Consider that(a),(b)a constant is independent of and is (c)inequalities are valid, where is a determinant of ()th order, which is obtained from the rectangular matrix as follows: by deleting the first line, and is the fundamental system of the solutions of the following homogeneous unperturbed (degenerate) equation: Corresponding to (1),(d) is valid,where is the ()th-order determinant obtained from by replacing the ()th row with the row, and is the Wronskian of the solution system , and the ()th-order determinant obtained from by replacing the ()th row with the row.(e)Consider where is the ()th-order determinant, which originates from by deleting the th row, and .
The problem lies in determining the pair , where and satisfy (1) and the boundary conditions in (2), in building the asymptotic estimates of (1) and (2) problem solutions and in the study of the initial jump phenomenon.
The following study is conducted according to a specific rule. In the first stage, we build the recovery problem solutions (1) and (2) from the position of a singularly perturbed problem with the initial jump on the basis of initial and boundary functions. In the next stage, we study the asymptotic behavior of the boundary value problem solutions (1) and (2).
3. Fundamental Solution System
Along with (1), we consider the following corresponding homogeneous perturbed equation: We seek a fundamental solution system of (10) in the following form: where represents unknown functions to be found and .
By substituting (11) into (10) and by matching the coefficients of like powers of on both sides of the resulting relation, we obtain a sequence of equations for all terms in expansion (11). For our aim, however, it suffices to consider the zero approximation. So, for the zero approximation (for ), we have the following problems:
Unique solutions exist for the problems (12) on the interval , and they form the fundamental solution system for the following homogeneous equation: where is presented in the following form:
Lemma 1. Let conditions (a) and (b) be satisfied. Then, the fundamental solution system of the singularly perturbed equation (10) permits the following asymptotic representations: as .
Let us introduce the Wronskian determinant for the fundamental solution system of (9) and expand it in entries of the th column: where , is ()th order.
With regard to (15), we obtain for sufficiently small for , where is obtained from the rectangular matrix by deleting the th row. In particular, the determinant is the Wronskian of the fundamental solution system of (13). With regard to (15) and (17), the expression (16) acquires the following form: Hence, considering that the last summand is dominant, when for the Wronskian determinant, it implies the asymptotic representation where is given by (14).
4. Cauchy Function and Boundary Functions
Following previous work , let us introduce the Cauchy function.
Theorem 3. Let conditions (a) and (b) be satisfied. Then, for sufficiently small , the Cauchy function with exists, is unique, and is expressed as follows: where is the th order determinant obtained from the Wronskian by replacing the th row of the fundamental solution system with of (10).
Lemma 4. If conditions (a) and (b) are satisfied, then, for sufficiently small , the Cauchy function with can be represented in the following form: where is the determinant obtained from the Wronskian by substituting the ()th row with the row.
Proof. By expanding in entries of the th column, we obtain
From (15), the minors for sufficiently small can be represented in the following form: where and is the determinant obtained from the rectangular matrix by deleting the th row and replacing the ()th row with the row.
Then, from (23) and (15), relation (24) acquires the following form:
This readily implies that
From (26) with regard to (19) and (21), we obtain the desired estimates (22).
Definition 5. Functions are referred to as boundary functions of the boundary value problems (1) and (2), if they satisfy the homogeneous equation (10) and the boundary conditions
Consider the determinant where with (15), the entries can be represented in the following form: This, together with the estimate (where is an arbitrary positive integer), implies that where
Now, let us expand in the entries of the last column: Here, from (15), the minors can be expressed in the following form: where is the ()th-order determinant introduced in Section 1.
By substituting (30) and (33) into (32), we obtain This, together with condition (c), implies that the determinant (28) permits the asymptotic representation:
The following theorem is valid.
Theorem 6. Let conditions (a)–(c) be satisfied. Then, for sufficiently small , the boundary functions exist on the interval , are unique, and are given as follows: where is the determinant obtained from by substituting the th row with the fundamental solution system of (10).
Lemma 7. If conditions (a)–(c) are satisfied, then the boundary functions permit the asymptotic representations: on the closed interval as , where and are the determinants introduced in Section 1, , and is the ()th-order determinant obtained from by replacing the ()th row with the row.
Proof. From (36), we define the derivative of the function . Let us expand in the entries of the th column:
Here in the minors , the first subscript represents the number of the row containing the system of functions . From (30), the determinants can be presented in the following form: where the determinants are obtained from by substituting the first row with the row.
With regard to (15), (30), and (40), the expression acquires the following form: This, together with the condition (c), gives the following representation:
Taking into account (35), (42), and (36), we obtain relation (37) for .
Now, we expand in the entries of the th column:
Here the first subscript in the determinants indicates the row containing the system of the functions , and the second subscript shows that they are the minors of the entry of the determinant at the intersection of the th row and th column. From (30), the minors can be presented in the following form: where is as defined in Section 1, is obtained from by replacing the ()th row with the row, and is obtained from by replacing the th with the row.
In view of (30) and (44) out of (43), the expression takes the following form:
Here the expression is dominant for small . Then, considering condition (c), we get the following asymptotic representation for :
Substituting (35) and (46) into formula (36), we obtain the asymptotic formula (38) for the boundary functions . The proof of the lemma is complete.
5. Analytic Representation and Estimates of the Solution
Consider the singularly perturbed boundary value problems (1) and (2). The unique solution of the problems (1) and (2) is found in the following form: where are the boundary functions, is the Cauchy function, are unknown consonants, and is the required parameter. By a straightforward verification, one can show that the function , defined by formula (47), is identical to (1). For the determination of ,, we substitute (47) into the boundary conditions (b). Then, with regard to the boundary (27), we obtain the following system of linear equations with respect to , as follows: where
We study the coefficient in (49) for and the right-hand side of this equation as a function of . Then, with regard to (27), (35), (37), and (38), we obtain From condition (d), . Therefore, for sufficiently small ,
Thus, the following theorem is valid.
Theorem 8. Let conditions (a)–(d) be satisfied. Then, for sufficiently small in a sufficiently small neighborhood of the point , a unique value of can be determined such that the pair is the unique solution for the boundary value problems (1) and (2) on the interval , where is given by formula (58) and is given by formula (54).
Theorem 9. Let conditions (a)–(e) be satisfied. Then, for sufficiently small , the solution of the boundary value problems (1) and (2) and its derivatives on the interval can be estimated as follows: where and are the constants independent of .
Proof. Formulae (22), (37), and (38) together with (5) imply the following estimates:
By estimating solution (58) and taking into account (60), we obtain (59), which completes the proof of the theorem.
Now, let us formulate the boundary conditions for the following unperturbed (degenerate) equation: obtained from (1) within . On the basis of Theorem 9, we must conclude that in (58), the coefficient of approaches zero when and the coefficients of have the order . Therefore, the boundary conditions for the solution of the unperturbed (61) are defined with help of the boundary conditions (2), containing :
By analogy with (17) and (28) for the problems (51) and (52), we introduce the initial function and boundary functions: where is the determinant (6), is the determinant from Lemma 7, is the determinant from Lemma 4, and is the Wronskian determinant of the fundamental solution system of homogeneous degenerate equation (13), which is obtained from by replacing the ()th row with the row.
Apparently, is the Cauchy function satisfying the homogeneous equation with respect to and initial conditions