A class of singularly perturbed Robin boundary value problems in critical case

. This paper discusses a class of nonlinear singular perturbation problems with Robin boundary values in critical cases. By using the boundary layer function method and successive approximation theory, the corresponding asymptotic expansions of small parameters are constructed, and the existence of uniformly efficient smooth solutions is proved. Meanwhile, we give a concrete example to prove the validity of our results.


Introduction
Singular perturbation problems with small parameters have been used in many fields such as chemical kinetics [14], semiconductor simulation [16], and radio engineering [5,18,20].The singular perturbed differential equation considered in this paper is obtained by transforming and dimensionless the differential equation controlling the enzyme kinetic reaction [4,15,18].Using boundary layer function method [18], the first or higher order approximate solution of the problem can provide stronger theoretical support for obtaining the approximate value of enzyme concentration, substrate concentration and intermediate enzyme mixture concentration.
It is well known that the reduction equations for singularly perturbed systems usually have isolated roots.However, the degenerate equation of singularly perturbed problem considered in this paper has no isolated root.Instead, it has a series of solutions that depend on one or more parameters.This case will be called the critical case [17].Compared with the singularly perturbed problem in the non-critical case, the singularly perturbed problem in the critical case is not only difficult to find, but also very complicated in the calculation process.Its complexity lies in the need to solve the following three difficulties in the calculation process: first, the zero-degree regular approximation solution is an unknown arbitrary function, which needs to be obtained through the following conditions; secondly, the solution process of the zero-order boundary layer is also very complicated.Finally, appropriate diagonalization should be found for the subsequent k-order boundary layer part to reduce the coupling degree of the equation.
The research methods of singular perturbation critical problems mainly include boundary layer function method [17,18], that is, the solution is separated by fast scale and slow scale.The fast scale is the boundary layer part, and the slow scale is the regular part.The two parts are solved separately to construct the formal asymptotic solution.In addition, unlike the boundary layer function method, in the literature [9], the authors use orthogonal projectors on ker A(t) and ker A(t) ′ (the prime denotes the transposition) for an asymptotic approximation, where A(t) is a singular matrix in front of the unknown function at the right end of the singular perturbation equation.Through the theory of boundary layer function method, Vasil'eva and Butuzov [17] were the first to study initial value problems for singularly perturbed systems in the critical case.Subsequently, Vasil'eva and Adelaida [19], Dontchev and Veliov [3], Wang [21], Karandjulov [8], Kurina and Thi Hoai [10] generalized the results of singularly perturbed in the critical case.As far as we know, only Dirichlet or Neumann boundary value conditions are discussed in the above problems, and Robin boundary value conditions are not studied.In general, Robin boundary value conditions are a combination of Dirichlet and Neumann boundary value conditions.The singularly perturbed Robin boundary value problem in the critical case will result in that the initial value or boundary value corresponding to the differential equation of any order asymptotic term cannot be given directly, but must be obtained indirectly through certain techniques.In the past several decades, authors of [1,2,6,7,11,12,22,23] discussed the singularly perturbed Robin boundary value problem for various noncritical cases.However, until now, from what we understand, there is no literature talking about the singular perturbation problem in critical cases with Robin boundary value conditions seriously so far.Motivated by these issues, we fill in the gaps of this class of problems, giving corresponding asymptotic expansions and numerical examples in this paper.
The structure of the paper is as follows.In Section 2 we discuss singularly perturbed critical cases with Robin boundary value conditions.In the next section, we determine all terms of the asymptotic expansion of the system (2.1)-(2.2) using boundary layer function theory.Based on the successive approximation principle, section 4 proves the existence, uniqueness, and remainder estimation of the solutions to problems (2.1)-(2.2).Section 5 illustrates our results with an example.The last section gives concluding remarks.

Problem formulation
We consider a class of nonlinear singularly perturbed systems in critical case then, add the corresponding Robin initial and boundary value conditions as follows Where x, y, and z are scalar functions, and 0 < µ ≪ 1 is a small parameter.y 0 , x 1 , and z 0 are given known initial boundary values.
The following assumptions are theoretically some basic assumptions of questions (2.1)-(2.2).
[H 1 ] Suppose that the functions A, B, C, D, F, and H are sufficiently smooth for a ≤ t ≤ b, |x| ≤ l, |y| ≤ l, and |z| ≤ l, where l is some real numbers.
[H 3 ] Suppose that the degradation equation of the system (2.1) is we suppose that the system (2.3) has a series of solutions where ω(t) is an arbitrary scalar function and β(ω(t), t) is a function with respect to ω(t).
Then the Jacobian matrix W u at the equilibrium point x = x * , y = y * , z = z * has the following form Therefore, it is easy to get the eigenvalue λ ≡ 0 of the matrix W u and the other two eigenvalues λ 1 = C y < 0, λ 2 = F < 0. According to the eigenvalues corresponding to the equilibrium point, λ ≡ 0 corresponds to the critical case, while the other two eigenvalues λ 1,2 < 0 correspond to the stable equilibrium position.At the same time, the above case is called critically stable case in singular perturbation problems.Therefore, we can use the boundary layer function method [18] in the stable case, and because λ 1,2 < 0, it is easy to find that the boundary function decays exponentially.Given the assumption that the condition [H 2 ] and the initial value condition (2.2) hold, the solutions x(t, µ), y(t, µ) and z(t, µ) generally only produce boundary layers near t = a.
We consider the zero-order regular part of the asymptotic solution of the form of problems (2.1)-(2.2) and obtain the zero-order regular parts ū0 (t) is the same as the degenerate problem (2.3)-(2.4) the root of the system of degradation equations (3.5) is The equations for L 0 u(τ 0 ) are: with the initial and boundary conditions where ω(a) is unknown, and the initial value of L 0 u(0) is arbitrary.We will use this arbitrariness to ensure that L 0 u(τ 0 ) decays exponentially and satisfies L 0 u(+∞) = 0. From (3.7)-(3.8),we have (3.9) At this time, substituting (3.9) into (3.7) the second differential equation and adding the initial boundary value condition (3.8), the following form can be obtained (3.10) [H 4 ] Suppose that equation (3.10) has a root L 0 y(τ 0 ), which is denoted by However, neither ω(a) nor L 0 y(0) is known.Therefore, L 0 y(τ 0 ) is not know.After ω(t) is found, ω(a) and L 0 y(0) can be determined and thus known.For the currently unknown functions ω(t) and β(ω, t), we need to determine them in the first approximation equation of the regular part ū1 (t).
For ū1 (t), we get the system of differential equations (3.12) can be rewritten in the following expression where Ā, B and H are all taken value at the point (β(ω, t), 0, t), F is taken value at the point (0, β(ω, t), t).Using (3.13) and by taking (3.5) into account, we can determine ω(b) = ω 1 .
According to the existence of the solution of the boundary value problem, the second firstorder differential equation in (3.13), and the known boundary value condition ω(b) = ω 1 , there is a solution ω(t) for a ≤ t ≤ b.Hence, both ω(a) and ȳ0 = β(ω, t) can be determined.At this time, ū0 (t) can be completely determined.
Substituting (3.10) into (3.8) the first initial value condition, we can determine L 0 y(0) = φ 0 .Therefore, L 0 y(τ 0 ) can be obtained by [H 4 ].By (3.9) and [H 4 ], we obtain L 0 z(τ 0 ) and L 0 x(τ 0 ).At this time, L 0 u(τ 0 ) are completely determined.For the first approximation of the regular part of the system (2.1)-(2.2),equations (3.12) has only been determined z1 (t), while x1 (t) and ȳ1 (t) need to be determined by the equation system of the regular parts ū2 (t) and the corresponding boundary value conditions.Next, we need to first determine the first-order boundary layer terms L 1 u(τ 0 ) of the asymptotic solution.
The equations for L 1 u(τ 0 ) are: where Here B, Ā, D, H take values at the point (β(ω, a), 0, a), Cx , Cy , Ct take values at the point (ω(a), β(ω, a), a), F take values at the point (0, β(ω, a), a), F t take values at the point (0, β(ω, a) + L 0 y(τ 0 ), a), B, D, Ĥ take values at the point (β(ω, a) + L 0 y(τ 0 ), L 0 z(τ 0 ), a), Cx , Cy , Ct take values at the point (ω(a) + L 0 x(τ 0 ), β(ω, a) + L 0 y(τ 0 ), a) and The initial and boundary value conditions corresponding to Introduce a diagonal transformation we can get where ϕ 4 ( ȳ1 (a), τ 0 ) = ( The initial and boundary conditions of δ 1 , δ 2 and δ 3 are We introduce a new transformation δ 3 so that the equation (3.17) changes to the following form where where From the above two transformations, we find that the initial value condition (3.20) has the following form We find that the right end of the first two equations of equation system (3.20)contains only one unknown function, which greatly reduces the coupling of the right end of the original equation system (3.14).Thus, by the system of equations (3.20) and the initial value condition (3.21), δ 3 (τ 0 ) and δ 4 (τ 0 ) can be written as At the same time, we substitute the expression (3.22) into the last equation of the system of equations (3.20), and the right end contains only the unknown function δ 5 (τ 0 ).At this point, the entire equation is a first-order integral differential equation for δ 5 (τ 0 ).First, we get the integral equation for the initial value condition δ 5 (0) [H 5 ] Suppose that the integral equation (3.23) can be converted to δ 5 (0) = ς(a, ȳ1 (a)).
Next, the coefficients ūk (t) and L k u(τ 0 )(k ≥ 2) of the higher-order asymptotic solutions are similar to the first-order asymptotic solutions ū1 (t) and L 1 u(τ 0 ).At this point, we need to write the equations and initial-boundary value conditions to determine ūk+1 (t) and L k u(τ 0 ).
The equations for L k u(τ 0 ) are: where Q k , I k , and S k are all known composite functions.The initial and boundary conditions of L k u(τ 0 ) are: (3.28) The equations for ūk+1 (t) are: where m k+1 , g k+1 , and f k+1 are all known composite functions.The process of solving these problems is almost identical to the case of k = 1, so we will not repeat it here.In this way, the asymptotic expansion (3.1) can be completely determined.

The existence of the solution and the remainder estimate
We first introduce a curve L 0 in the space of the variables (u, t).The curve L 0 is composed of the following we denote the projection of L 0 onto the space of the variables (u, t) by L0 .
[H 7 ] Suppose that the functions A, B, C, D, F, and H have continuous partial derivatives con- cerning each argument up to order (n + 2) inclusive in some δ-tube of L0 .
As a result of u(t, µ) − U n (t, µ) = O µ n+1 , we obtain the inequality (4.2).This completes the proof.with the initial and boundary conditions
The system for L 1 u(τ 0 ) and ū2 (t), we have (5.7) and first-order initial boundary value conditions τ 0 e −2τ 0 + 5 48 e −4τ 0 . (5.10) It is easy to get that L 1 u(+∞) = 0. Thus, L 1 u(τ 0 ) satisfies an exponential decay estimation.Therefore, we construct the first-order approximate solution u = (x y z) T of the system (5.1)-(5.2).Namely, we have where  Therefore, these results were obtained using the boundary layer function method in [18].The results obtained by Matlab are given in Figures 5.1 and 5.2.Different line types corresponding to exact and approximate solutions have been marked in the figure.These graphs show that an asymptotic solution is closer to the exact one if we use higher-order asymptotics.If we use the small parameter µ to be smaller, the formal asymptotic solution is more approximate to the exact solution.The image of the solution also better illustrates the nature of the exponential decay of the boundary function.

Conclusive remarks
This paper studies a class of nonlinear critical singular perturbation problems with Robin initial boundary value, the results show that how to obtain the zero-order approximate solution and simplify the first-order boundary layer term equation is the key to obtaining the approximate solution of the system.In this paper, the successive approximation method is used to prove the remainder estimation and the existence and uniqueness of the solution.
Finally, in the process of researching the system (2.1)-(2.2), the following situations were discovered.Remark 6.1.When F > 0 and F ̸ = 1 in the interval [a, b] of t, the system (2.1)only produces the right layer, but the solution method is similar to the left boundary layer.And when the F has a zero point in the interval [a, b], the stability will be in the interval [a, b] has changed, thus forming a very complicated nonlinear turning point problem.Since the linear turning point problem is already very complicated, as the nonlinear strength increases, the nonlinear turning point problem will become more complicated.So this type of problem will be very challenging.At this time, the boundary layer function method will no longer be applicable, and it is necessary to find a suitable method to solve it.Remark 6.2.When the coefficients in front of y ′ (t, µ) and z ′ (t, µ) in Robin's initial boundary value condition (2.2) do not contain the small parameter µ when the derivative of t is calculated for the boundary layer function, the µ −1 term will appear.So the right end of the corresponding boundary condition should have the µ −1 term to perform matching, and in the process of using the boundary layer function method to solve the problem, it should be set to the following form µ k+1 zk+1 (t, µ) + µ k L k z(τ 0 , µ) .( But when y and z are substituted into the system (2.1) at this time, A, B, C, D, H, and F can't perform Taylor expansion when µ = 0, which contradicts the boundary layer function method.Therefore, the weak nonlinear problem generally does not produce an infinite initial boundary value problem.Remark 6.3.When the initial boundary values (2.2) are all Robin initial conditions, the form is as follows y(a, µ) − µy ′ (a, µ) = y 0 , x(a, µ) + µx ′ (a, µ) = x 0 , z(a, µ) − µz ′ (a, µ) = z 0 , (6.2) the process of solving the n-order equations of problem (2.1)-(2.2) does not change, but the corresponding initial conditions and boundary value conditions change.Through calculations, we found that the basic process of the solution does not change significantly, but from first finding the value of x(t) at t = b, it becomes possible to find the value of ȳ(t) at t = a, and L 0 y(τ 0 ) at the value of τ 0 = 0. Remark 6.4.For system 2.1, because the functions B, D and H at the right end are preceded by a small parameter µ.Therefore, no matter the form is (z, y, t), (x, y, t) or (x, z, t), system 2.1 is still a critically stable situation, and the corresponding asymptotic solution form and solution method will not be changed.

Figure 5 . 2 :
Figure 5.2: Contrast of the exact with the approximate solutions to (5.1) with boundary condition (5.2) for values of the perturbing parameter µ = 0.01.