Here, we generalize the boundary layer functions method (or composite asymptotic expansion) for bisingular perturbed differential equations (BPDE that is perturbed differential equations with singular point). We will construct a uniform valid asymptotic solution of the singularly perturbed first-order equation with a turning point, for BPDE of the Airy type and for BPDE of the second-order with a regularly singular point, and for the boundary value problem of Cole equation with a weak singularity.A uniform valid expansion of solution of Lighthill model equation by the method of uniformization and the explicit solution—this one by the generalization method of the boundary layer function—is constructed. Furthermore, we construct a uniformly convergent solution of the Lagerstrom model equation by the method of fictitious parameter.
- turning point
- singularly perturbed
- bisingularly perturbed
- Cauchy problem
- Dirichlet problem
- Lagerstrom model equation
- Lighthill model equation
- Cole equation
- generalization boundary layer functions
1.1. Symbols ~. Asymptotic expansions of functions
Let a function and be defined in a neighborhood of .
Definition 1. If , then write , and is constant.
If , then write .
If , then write .
Definition 2. The sequence , where defined in some neighborhood of zero, is called the asymptotic sequence in , if
Note 1. Everywhere below denotes a small parameter.
Definition 3. We say that function can be expanded in an asymptotic series by the asymptotic sequence , if there exists a sequence of numbers and has the relation
1.2. The asymptotic expansion of infinitely differentiable functions
Theorem (Taylor (1715) and Maclaurin (1742)). If the function in some neighborhood of , then it can be expanded in an asymptotic series for the asymptotic sequence , i.e.,
, where .
Thus, the concept of an asymptotic expansion was given for the first time by Taylor and Maclaurin,although an explicit definition was given by Poincaré in 1886.
1.3. The asymptotic expansion of the solution of the ordinary differential equation
Consider the Cauchy problem for a normal ordinary differential equation
The function is infinitely differentiable on the variables in some neighborhood . It is correct next.
Theorem 1. The solution of problem (1) exists and unique in some neighborhood point and , for small .
Corollary. The solution of problem (1) can be expanded in an asymptotic series by the small parameter ε, i.e.,
Here and below, the equality is understood in an asymptotic sense.
Note 2. Theorem 1 for the case when is analytical was given in  by Duboshin.
Note 3. This theorem 1 is not true if is not smooth at . For example, the solution of a singularly perturbed equation
function and is not expanded in an asymptotic series in powers of ε, because here and have a pole of the first order with respect to .
Note 4. The series 2 is a uniform asymptotic expansion of the function in a neighborhood of .
For example. Series
It is not uniform valid asymptotic series on the interval [0, 1], but it is a uniform valid asymptotic expansion of the segment , where .
1.4. Singularly perturbed ordinary differential equations
We divide such equations into three types:
Singular perturbations of ordinary differential equations such as the Prandtl-Tikhonov [2–56], i.e., perturbed equations that contain a small parameter at the highest derivative, i.e., equations of the form
where are infinitely differentiable in the variables in the neighborhood of . It is obvious that unperturbed equation ()
is a first order.
Definition 4. Singularly perturbed equation will be called bisingulary perturbed if the corresponding unperturbed differential equation has a singular point, or this one is an unbounded solution in the considering domain.
Equation is a singularly perturbed ordinary differential equation.
Equation Vander Pol
It is a bisingularly perturbed ordinary differential equation with singular points, if .
is a bisingularly perturbed equation, because the unperturbed equation has an unbounded solution .
is a bisingularly perturbed equation also.
Singularly perturbed differential equations such as the Lighthill’s type [57–69], in which the order of the corresponding unperturbed equation is not reduced, but has a singular point in the considering domain.
For example, a Lighthill model equation
where , . For unperturbed equation
point is a regular singular point.
where is a given number and is the dimension space.
1.5. Methods of construction of asymptotic expansions of solutions of singularly perturbed differential equations
The method of matching of outer and inner expansions [13, 19, 28, 29, 37, 49] is the most common method for constructing asymptotic expansions of solutions of singularly perturbed differential equations. Justification for this method is given by Il’in . However, this method is relatively complex for applied scientists.
The boundary layer function method (or composite asymptotic expansion)dates back to the work of many mathematicians. For the first time, this method for a singularly perturbed differential equations in partial derivatives is developed by Vishik and Lyusternik  and for nonlinear integral-differential equations (thus for the ordinary differential equations) Imanaliev , O’Malley (1971) , and Hoppenstedt (1971) .
It should be noted that, for the first time, the uniform valid asymptotic expansion of the solution of Eq. (5) is constructed by Vasil’eva (1960)  after Wasow  and Sibuya in 1963  by the method of matching.
This method is constructive and understandable for the applied scientists.
The method of Lomov or regularization method  is applied for the construction of uniformly valid solutions of a singularly perturbed equation and will apply Fredholm ideas.
The method WKB or Liouville-Green method is used for the second-order differential equations.
The method of multiple scales.
The averaging method is applicable to the construction of solutions of a singularly perturbed equation on a large but finite interval.
Here, we consider a bisingularly perturbed differential equations and types of equations of Lighthill and Lagerstrom.
Here, we generalize the boundary layer function method for bisingular perturbed equations. We will construct a uniform asymptotic solution of the Lighthill model equation by the method of uniformization and construct the explicit solution of this one by the generalized method of the boundary layer functions.
Furthermore, we construct a uniformly convergent solution of the Lagerstrom model equation by the method of fictitious parameter.
2. Bisingularly perturbed ordinary differential equations
2.1. Singularly perturbed of the first-order equation with a turning point
Consider the Cauchy problem 
where , f(x)=, is the constant
Explicit solution of the problem (3) has the form:
The corresponding unperturbed equation ()
has a solution , which is unbounded at .
If you seek a solution to problem (1) in the form
where and boundary layer functions decreasing by power law as , that is,.
The initial conditions for the functions we take in the next form
From Eq. (6), we have
To function has been smooth, and we define it from the equation
and then from Eq. (7.−1), we have obtained the equation
Obviously, this function bounded and is infinitely differentiable on the segment , and
This asymptotic expression can be obtained by integration by parts the integral expression for .
Eq. (7.0) define and . Let , then
Hence, we find
From Eq. (7c) for , we have
Let , then .
From these, we get
From Eq. (7c) for , we have
Let , then
From this, we get
Analogously continuing this process, we determine the others of the functions .
In order to show that the constructed series of [Eq. (5)] is asymptotic series, we consider remainder term ,
For the remainder term , we obtain a problem:
We note that if is odd, then .
The problem (8) has a unique solution
and from this, we have
2.2. Bisingularly perturbed in a homogenous differential equation of the Airy type
Consider the boundary value problem for the second-order ordinary in a homogenous differential equation with a turning point
Without loss of generality, we consider the homogeneous boundary conditions, since using transformation
can lead to conditions (10).
then we have
and the series (11) is asymptotic in the segment . The point x0= is singular point of asymptotic series (11).
where , . Here, , is boundary layer function in a neighborhood of and decreases by the power law as , and the function is boundary function in a neighborhood of and decreases exponentially as .
From Eq. (13), we have
Boundary conditions for functions we take next form
To function has been smooth; therefore, we define it from the equation
then from Eq. (15.1), we have the equation
Let us prove an auxiliary lemma.
Lemma 1. Next boundary value problem
will have the unique solution and this one have next form
Proof. We verify the boundary conditions:
as , so .
as , so .
Now we show that z(t) satisfies Eq. (16). For this, we compute derivatives:
Substituting the expressions for and in Eq. (17), and given that and , we get: .
The general solution of the homogeneous equation is
is the constant.
Considering the boundary condition , we have . And the second condition follows. This implies that .
Therefore, . It is obvious that . Lemma 1 is proved.
This Lemma 1 implies the existence and uniqueness of solution of the problem:
This function bounded and is infinitely differentiable on the segment , and as :
This asymptotic expression can be obtained by integration by parts the integral expression for .
From Eq. (15.0), we define and . Let , then
And by Lemma 1, we have
Analogously, from Eq. (15.1), we define and . Let , then
In view of Lemma 1, we have .
To function has been smooth; as above, we define it from the equation
then Eq. (15.2) to hase the problem
By Lemma 1, we can write an explicit solution to this problem, and this solution bounded and is infinitely differentiable on the segment , and as :
Analogously continuing this process, we determine the rest of the functions .
Now we will define functions from the equality (14) by using the boundary conditions We state problems
Thus, all functions , and in equality (12) are defined, i.e., a formally asymptotic expansion is constructed. Let us justify the constructed expansion. Let
Then for the remainder term, we state the following problem:
According to the maximum principle [23, p. 117, 82], we have .
Hence, we get .
Thus, we have proved.
Example. Consider the problem
The asymptotic solution this problem we can represent in the form .
We have got
2.3. Bisingularly perturbed equation of the second order with a regularly singular point
Here, for simplicity, we consider the case .
The solution of the unperturbed problem
Function is a solution of equation
The coefficients of the series (26) will be determined as the solution of equations
where , with boundary conditions .
Functions is the solution of the equations
with boundary conditions .
Next, we use the following lemma.
Lemma 2. The problem
It has a unique solution .
Lemma 3. A boundary value problem
has solution , where
The proof of Lemma 3 is obvious.
Lemma 4. In order to solve the boundary value problem
we have the estimate
Lemma 5. The estimate
where is constant.
Proof. Consider the function
where and are positive constants such that
It is obvious that
From the maximum principle, it follows that
Now the proof of the lemma 5 follows from estimates of and .
If we introduce the notation
where are constructed above functions, then
2.4. The bisingular problem of Cole equation with a weak singularity
where are the given constants.
The unperturbed equation
has the general solution
This is a nonsmooth function in .
where is the reminder term.
By the method of generalized boundary layer function, we put the term into the equation. We choose functions so that .
We choose indefinite functions hk(x) as follows: . We can represent
We can rewrite y1(x) in the form:
Analogously, we have obtained
Continuing this process, we have
where are corresponding coefficients of the expansion of in powers of (2 ).
From Eq. (30), we have the following equations for the boundary functions :
The solution of problem (33) is represented in the form
We note that will exponentially decrease as .
Lemma 6. The general solution of this equation will have ; here are constants, and
Two linearly independent solutions and ,
Lemma 7. The boundary problem will have only trivial solution.
The proofs of Lemmas 6 and 7 are evident.
Theorem 4. The problem
will have the unique solution and this one has the next form
is the function of Green and.
Lemma 8. Asymptotical expansions of functions () will have the next forms
Proof for Lemma 8.
Firs proof. We can prove this lemma by applying formulas (38) and Theorem 4.
Now we will prove the boundedness of the reminder function . This function will satisfy the next equation:
Therefore, we have .
We prove next.
3. Singularly perturbed differential equations Lighthill type
3.1. The idea of the method of Poincare
Consider the equation
Unperturbed equation has solutions (where are arbitrary constants) with period . We are looking for the periodic solution of the equation with a period of .
Note that the operator transforms Fourier series and in itself. Poincare’s method reduces the existence of periodic solutions of differential equations to the existence of the solution of an algebraic equation.
We will seek a periodic solution of Eq. (39) with the initial condition
If we seek the solution in the form
with the initial conditions
then for we have next equations
Thus, it is not a uniform expansion of the y(x) on the segment , since the term is present here.
If these secular terms do not appear in Eq. (39), it is necessary to make the substitution
where the constant should be selected so as not to have secular terms in .
Thus, the solution of Eq. (39) must be sought in the form
Then Eq. (39) has the form
We will seek the 2π periodic solution of this equation in the form
The function Z1(t) will have the periodical solution we take . Then .
Similarly, from equations
and etc. are uniquely determined.
Theorem 6. Equation (39) has a unique periodic solution, and it can be represented in the form (40).
3.2. The idea of the Lighthill method
Lighthill in 1949  reported an important generalization of the method of Poincare.
Lighthill proposed to seek the solution of Eq. (41) in the form
At first, we consider the example
It has exact solution
It is obvious that for , the solution (43) exists on the interval and
and considering , this expression can be expanded in powers of , and then we have
The series (45) is uniformly convergent asymptotic series only on the segment .
First, we write Eq. (43) in the form
and equating coefficients of the same powers ,we have
From Eq. (47), we have
If we put in Eq. (49), we obtain
Hence, solving this equation, we have
Since differentiation increased singularity of nonsmooth function, we select so that the expression in the right side of Eq. (49) is equal to zero, i.e.,
Hence, we have
Then Eq. (49) takes the form
Hence, we obtain .
Now Eq. (48) for takes the form
Let , and then . Further also choose , as they also satisfy the initial conditions. Thus, we have found that
Putting in Eq. (51) , we have
Now we will present the main idea of the Lighthill method to Eq. (41) under conditions:and . We will write it in the form of
Hence, equating the coefficients of equal powers has
In these equations, the coefficient of the derivative was replaced by Eq. (54) on .
From Eq. (57) for n = 1,2,…, it follows that if we want to define functions from this differential equations, then we must assume that
And this condition cannot be avoided by applying the Lighthill method to Eq. (41). Condition (58) first appeared in , justifying Lighthill method, then in the works Habets  and Sibuya, Takahashi . Comstock  on the example shows that the condition (58) is not necessary for the existence of solutions on the interval . Further assume that the condition (58) holds. Note that the right-handside of Eq. (57) is linear with respect to , and function depends from only.
The solution of Eq. (54) can be written as
Hence, we have
Since the differentiation of increased of its singularity at the point , it is better to choose such that the first brace in Eq. (55) is equal to zero, i.e.,
Hence, using Eq. (60), we obtain
Then Eq. (55) takes the form
where =const. Hence, we have
Now equating to zero the expression in the first brace in the right-hand side of Eq. (56), we have
From this, we get
Now Eq. (56) takes the form
Solving this equation, we have
Next, the method of induction, it is easy to show that
Thus, the series (42) has the asymptotic
From Eq. (67), it follows that the point corresponds to the root of the equation
We have the
Theorem 7 proved by Wasow , Sibuya and Takahashi  in the case where are analytic functions on ; proved by Habets  in the case . Moreover, instead of the condition (3) Wasow impose a stronger condition: .
In the proof of Theorem 7, we will not stop because it is held by Majorant method.
From the foregoing, it follows that Wasow condition is essential in the Lighthill method.
Comment 2. Prytula and later Martin  proposed the following variant of the Lighthill method. At first direct expansion determined using by the method of small parameter
and further at second they will make transformation
Here unknowns are determined from the condition that function was less singular function . We show that using the method Prytula or Martin, also cannot avoid Wasow conditions. Really, substituting Eq. (71) into Eq. (70) and expanding in a Taylor series in powers of ε, we have
Hence, to obtain a uniform representation of the solution to the second order by ε, we must to put to zero the expression in the curly brackets, i.e., . Therefore, . Hence, it is clear that we must make the condition of Wasow: in the method of Prytula or Martin also.
3.3. Uniformization method for a Lighthill model equation
where is the root equation and if the root and on the interval .
Proof. Sufficiency. Let the solution of the problem (72) exists and are a parametric representation of the solution of the problem (72). Then introducing the variable-parameter , we obtain the problem (73).
Necessity. Let it fulfill the conditions of Theorem 8. Then dividing the first equation by second one, we get Eq. (72). Theorem 8 is proved.
We have the following
Theorem 9. Suppose that the first three conditions of Theorem 8. i.e.,(1) ; (2) ; (3) . Then the solution of problem (72) is represented in the form of an asymptotic series (42) and its solution can be obtained from uniformizing equation (73).
The proof of this theorem is completely analogous to the proof of Theorem 8, even more easily.
Only it remains to show that under the conditions of Theorem 9 we can get an explicit solution . Really, since
Therefore, by the implicit function theorem, we can express .
Then when we put it in first equality (42), we obtain an explicit solution .
Comment 3. Explicit asymptotic solution that this problem obtained in Section 3.4.
Example 43. Uniformized equation is
It is easy to integrate this system, and we obtain
Hence, excluding variable ξ, we have an exact solution (44).
Uniformized equation is
Hence if , we obtain
From the equation , we find .
We prove that on the interval .
3.4. It is construction explicit form of the solution of the model Lighthill equation
where b is given constant, . Given functions are subjected to the conditions : .
Here, we consider the case ; this is done to provide a detailed illustration of the idea of the application of the method. We search for the solution of problem (76) in the form
Note that , i.e., depends also on , but this dependence is not indicated.
The initial conditions for the functions are taken as
we have the following equations:
We solve these problems successively. We write problem (79.−1) as
The fundamental solution of the homogeneous equation corresponding to this equation is of the form
Using the expression for , the solution of the inhomogeneous equation for can be written as
After integrating by parts, we reduce the last expression to the following equation:
Let . Let us introduce the notation . This function satisfies the inequality and is a strictly decreasing bounded function on the closed interval . Here and elsewhere, all constants independent of the small parameter are denoted by . Let be the set of functions satisfying the condition
Theorem 10. If b 0 > 0 , then there exists a unique constraint of the solution of problem (79.-1) from the set S μ .
Proof. Equation (81) is equivalent to the equation , where
Suppose that First, let us estimate on the set . We have
Here, we have used the triangle inequality
as well as the inequality
The Fréchet derivative of the operator with respect to at the point is a linear operator:
where is a continuous function on the closed interval . Note that, in view of the denominator of this expression is strictly positive on the closed interval . For , we can obtain the estimate in the same way as the estimate for . Hence, in turn, it follows from the Lagrange inequality that the operator is a contraction operator in the set . Therefore, by the fixed-point principle, Eq. (81) has a unique solution from the class . The theorem is proved.
Corollary. The following inequalities hold:
for all t ∈ [ 0 , μ 0 ] ;
The other function is determined from the inhomogeneous linear equations; therefore, the following lemmas are needed.
Lemma 9. For any function f ( x ) ∈ C ( ∞ ) [ 0 , 1 ] , the equation L ξ = f ( x ) has a unique bounded solution ξ ( x ) ∈ C ( ∞ ) [ 0 , 1 ] expressible as
Proof. The proof follows from the fact that the general solution of the equation under consideration is expressed as
If we choose
then we obtain the required result.
This lemma implies that all the functions are uniquely determined and belong to the class .
Lemma 10. The problem
where the function k ( t ) belongs to C ∞ [ 0 , 1 ] is continuous and bounded, and if | k ( t ) | ≤ M t − 2 , t → ∞ , has a unique uniformly bounded solution η ( t ) = η ( t , μ ) on the closed interval t ∈ [ 0 , μ 0 ] for a small μ .
Proof. The fundamental solution of the homogeneous equation (82) is of the form
Obviously, and for and aresmall. The solution of problem (82) can be expressed as
satisfies the assumptions of the lemma. Therefore, the function is bounded on . The boundedness of the other functions is proved in a similar way, because the right-hand sides of the equations defining these functions satisfy the assumptions of Lemma 10. The estimate of the asymptotic behavior of the series (77) is also carried out using Lemma 10.
Let us introduce the notation
The following statement holds.
Theorem 11. Let b 0 > 0 (for this, it suffices that the condition b 0 : = b − y 0 ( 1 ) > 0 holds). Then the solution of problem (76) exists on the closed interval [ 0 , 1 ] and its asymptotics can be expressed as Eq. (84) and | R n + 1 ( x , μ ) | ≤ M for all x ∈ [ 0 , 1 ] .
Example. Consider the equation
This equation is integrated exactly
where . If , then the solution of problem (1) exists on the closed interval , which is confirmed by Theorem 11. The equation for is of the form
The solution of this problem can be expressed as
The equation for u0(x) has the solution . Further,
where . The asymptotics of the solutions of problem (76) can be expressed as
4. Lagerstrom model problem
The problem 
where is constant, .
It has been proposed as a model for Lagerstrom Navier-Stokes equations at low Reynolds numbers. It can be interpreted as a problem of distribution of a stationary temperature .
The first two terms in Eq. (1) is dimensional Laplacian depending only on the radius, and the other two members—some nonlinear heat loss.
It turns out that not only the asymptotic solution but also convergent solutions of Eq. (1) can be easily constructed by a fictitious parameter . The basic idea of this method is as follows. The initial problem is entered fictitious parameter with the following properties:
, the solution of the equation satisfies all initial and boundary conditions;
The solution of the problem can be expanded in integral powers of the parameter for all .
It is convenient in Eq. (85) to make setting , then
We have the following
Theorem 12. For small , the solution of problem (86) can be represented in the form of absolutely and uniformly convergent series
for the sufficiently small parameter , where
Note that the function also depends on , but for simplicity, this dependence is not specified.
Proof. We introduce Eq. (86) parameter , i.e., consider the problem
Here, we will prove this Theorem 12 in the case β = 0 only for simplicity.
Setting in Eq. (87), we have
It has a unique solution
Therefore, Eq. (88) with zero boundary conditions is the Green’s function
Hence, the problem (87) is reduced to the system of integral equations
In Eq. (89), we make the substitution , and then we have
To prove the theorem, we need next
Lemma 11. The following estimate holds
Given that, we have , we have
Inequality Eq. (91) for is proved similarly.
Further, by integrating by parts, we have
It is from integral expressing of we can obtain the asymptotic behavior such as indicated in the theorem.
With the solution of Eq. (90), we can expand in series
The coefficients of this series are uniquely determined from the equations
Let then by using Eq. (92) we have a Majorant equation: . The solution of this equation can be expanded in powers λ the under condition for all .
If we call we get the proof of the theorem.