Perturbed Differential Equations with Singular Points

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.

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.

The asymptotic expansion of the solution of the ordinary differential equation
Consider the Cauchy problem for a normal ordinary differential equation y 0 ðxÞ ¼ f ðx, y, εÞ, yð0Þ ¼ 0: The function f ðx, y, εÞ is infinitely differentiable on the variables x, y, ε in some neighborhood Oð0, 0, 0Þ. It is correct next. Theorem 1. The solution y ¼ yðx, εÞ of problem (1) exists and unique in some neighborhood point Oð0, 0, 0Þ and yðx, εÞ ∈ C ∞ , for small x, ε.
Corollary. The solution of problem (1) can be expanded in an asymptotic series by the small parameter ε, i.e., yðx, εÞ ¼ X ∞ k¼1 ε k y k ðxÞ: Here and below, the equality is understood in an asymptotic sense. Note 2. Theorem 1 for the case when f ðx, y, εÞ is analytical was given in [1] by Duboshin. Note 3. This theorem 1 is not true if f ðx, y, εÞ is not smooth at ε. For example, the solution of a singularly perturbed equation εy 0 ðxÞ ¼ ÀyðxÞ, yð0Þ ¼ a function yðxÞ ¼ ae Àx=ε and is not expanded in an asymptotic series in powers of ε, because here f ðx, y, εÞ ¼ ÀyðxÞ=ε and f have a pole of the first order with respect to ε.
Note 4. The series 2 is a uniform asymptotic expansion of the function yðxÞ in a neighborhood of x ¼ 0.
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 ½ε α , 1, where 0 < α < 1.
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.
Remark. The division into such classes is conditional, because singularly perturbed equation of Van der Pol in the neighborhood of points y ¼ AE1 leads to an equation of Lighthill type [2, 3].
1.5. Methods of construction of asymptotic expansions of solutions of singularly perturbed differential equations 1. 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 [22]. However, this method is relatively complex for applied scientists.
2. 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 [52] and for nonlinear integral-differential equations (thus for the ordinary differential equations) Imanaliev [24], O'Malley (1971) [38], and Hoppenstedt (1971) [42].
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) [50] after Wasow [69] and Sibuya in 1963 [68] by the method of matching.
This method is constructive and understandable for the applied scientists.
3. The method of Lomov or regularization method [33] is applied for the construction of uniformly valid solutions of a singularly perturbed equation and will apply Fredholm ideas.

4.
The method WKB or Liouville-Green method is used for the second-order differential equations.

5.
The method of multiple scales.

6.
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 [5] Explicit solution of the problem (3) has the form: The corresponding unperturbed equation (ε ¼ 0) has a solutionỹðxÞ ¼ f ðxÞ=x, which is unbounded at x ¼ 0.
If you seek a solution to problem (1) in the form then and a series of Eq. (4) is asymptotic in the segment ð ffiffi ffi ε p , 1, and the point x 0 = ffiffi ffi ε p ¼ μ is singular point of the asymptotic series of Eq. (4). Therefore, the solution of problem (3) we will seek in the form where Y k ðxÞ ∈ C ð∞Þ ½0, 1, π k ðtÞ ∈ C ð∞Þ ½0, μ À1 , x ¼ μt and boundary layer functions π k ðtÞ decreasing by power law as t ! ∞, that is, Substituting Eq. (5) into Eq. (3), we obtain The initial conditions for the functions π kÀ1 ðtÞ, k ¼ 0, 1, … we take in the next form To Y 0 ðxÞ function has been smooth, and we define it from the equation and then from Eq. (7.À1), we have obtained the equation Obviously, this function bounded and is infinitely differentiable on the segment ½0, μ À1 , and This asymptotic expression can be obtained by integration by parts the integral expression for π À1 ðtÞ.
In order to show that the constructed series of [Eq. (5)] is asymptotic series, we consider remainder term R m ðxÞ ¼ yðxÞ À y m ðxÞ, For the remainder term R m ðxÞ, we obtain a problem: We note that if m is odd, then Y 0 m ðxÞ 0.

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 εy 00 ðxÞ À xyðxÞ ¼ f ðxÞ, x∈ ð0, 1Þ, ð9Þ where f ðxÞ ¼ Note 5. It is the general case of this one was considered in Ref. [8,[45][46][47].
If the asymptotic solution of the problems (9)-(10) we seek in the form yðxÞ ¼ y 0 ðxÞ þ εy 1 ðxÞ þ ε 2 y 2 ðxÞ þ …; ð11Þ then we have and the series (11) is asymptotic in the segment ð ffiffi ffi ε The solution of problems (9) and (10) will be sought in the form Here, Y k ðxÞ ∈ C ∞ ½0, 1, π k ðtÞ ∈ C ∞ ½0, 1=μ is boundary layer function in a neighborhood of t ¼ 0 and decreases by the power law as t ! ∞, and the function w k ðtÞ ∈ C ∞ ½0, 1=λ is boundary function in a neighborhood of η ¼ 0 and decreases exponentially as η ! ∞.
This Lemma 1 implies the existence and uniqueness of π À1 ðtÞ ∈ C ∞ ½0, μ À1 solution of the problem: This function bounded and is infinitely differentiable on the segment ½0, μ À1 , and as t ! ∞: This asymptotic expression can be obtained by integration by parts the integral expression for π À1 ðtÞ.
To Y 3 ðxÞ function has been smooth; as above, we define it from the equation then Eq. (15.2) to π 2 ðtÞhase the problem π ″ 2 ðtÞ À tπ 2 ðtÞ ¼ 2f 3 , π 2 ð0Þ ¼ 0, π 2 ð1=μÞ ! 0, μ ! 0: By Lemma 1, we can write an explicit solution to this problem, and this solution bounded and is infinitely differentiable on the segment ½0, μ À1 , and as t ! ∞: Analogously continuing this process, we determine the rest of the functions Y k ðxÞ, π k ðtÞ. Now we will define functions w k ðηÞ from the equality (14) by using the boundary conditions yð1Þ ¼ 0 We state problems One can easily make sure that all these problems (18.0) and (18.k) have unique solutions such that w k ðηÞ ∈ C ∞ ½0, ∞Þ, w k ðηÞ ¼ Oðe Àη Þ with η ! ∞.
Thus, we have proved.
The solution of the unperturbed problem represented as where rðxÞ ¼ p À1 ðxÞf ðxÞ, pðxÞ ¼ exp Extracting in Eq. (23), the main part of the integral in the sense of Hadamard [34], it can be represented as where aðxÞ ¼ xpðxÞ Function aðxÞ ∈ C ∞ ½0, 1.
Theorem 3. Suppose that the conditions referred to the above with respect to qðxÞ and f ðxÞ.
Function z 0 ðxÞis a solution of equation The coefficients z k ðxÞof the series (26) will be determined as the solution of equations Functions π k ðtÞ is the solution of the equations Lπ k π ″ k ðtÞ þ tπ 0 k ðtÞ À qðμtÞπ k ðtÞ À c k μtpðμtÞ with boundary conditions π k ð0Þ ¼ Àz k ð0Þ, π k ðμ À1 Þ ¼ 0.
Next, we use the following lemma.
The proof of Lemma 3 is obvious.
Lemma 4. In order to solve the boundary value problem we have the estimate 0 ≤ Wðμ, tÞ ≤ e À1 lnμ À1 : Proof. This follows from the fact that the solution of this problem existsuniquely by the maximum principle [23,82] and will be represented in the form Proof. Consider the function where γ 1 and γ 2 are positive constants such that It is obvious that From the maximum principle, it follows that jπ k ðμ, tÞj < γ 1 Wðμ, tÞ þ γ 2 XðtÞ: Now the proof of the lemma 5 follows from estimates of Wðμ, tÞ and XðtÞ.

The bisingular problem of Cole equation with a weak singularity
The following problem is considered [9,13,28,29], where x ∈ ½0, 1; a,b are the given constants.
has the general solution This is a nonsmooth function in ½0, 1.
Substituting Eq. (29) into Eq. (27), we have By the method of generalized boundary layer function, we put the term hðx, εÞ ¼ into the equation. We choose functions h k ðxÞ so that y k ðxÞ ∈ C½0, 1.
Firs proof. We can prove this lemma by applying formulas (38) and Theorem 4.
We prove next.
Note that the operator M transforms Fourier series X ∞ k¼1 a k cos kx and X ∞ k¼1 a k sin kx 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 yð0Þ ¼ 1, y 0 ð0Þ ¼ 0: If we seek the solution in the form yðxÞ ¼ y 0 ðxÞ þ εy 1 ðxÞ þ ε 2 y 2 ðxÞ þ … with the initial conditions then for y s ðxÞ, s ¼ 0, 1, … we have next equations Thus, yðxÞ ¼ cos x þ ε 8 3x sin x À 1 4 cos 3x þ 1 4 cos x À Á þ …it is not a uniform expansion of the y (x) on the segment ½À∞, ∞, since the term εx sin x is present here.
If these secular terms do not appear in Eq. (39), it is necessary to make the substitution where the constant α k should be selected so as not to have secular terms in t.

The idea of the Lighthill method
Lighthill in 1949 [67] reported an important generalization of the method of Poincare.
Lighthill proposed to seek the solution of Eq. (41) in the form It is obvious that Eq. (42) has generalized the Poincare ideas (see, the transformation Eq. (40)).
From Eq. (57) for n = 1,2,…, it follows that if we want to define functions x n ðξÞ ðn ¼ 1, 2, …Þ 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 [69], justifying Lighthill method, then in the works Habets [66] and Sibuya, Takahashi [68]. Comstock [65] on the example shows that the condition (58) is not necessary for the existence of solutions on the interval ½0, 1. Further assume that the condition (58) holds. Note that the right-handside of Eq. (57) is linear with respect to x n ðξÞ, and f n function depends from y 0 The solution of Eq. (54) can be written as where gðξÞ ¼ exp Hence, we have Since the differentiation of y 0 ðξÞ increased of its singularity at the point ξ ¼ 0, it is better to choose such that the first brace in Eq. (55) is equal to zero, i.e., Then Eq. (55) takes the form whereã 1 =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 x j ðξÞ e b j ξ Àjq 0 , y j ðξÞ e a j ξ Àðjþ1Þq 0 , j ¼ 1, 2, …: Thus, the series (42) has the asymptotic yðξÞ e ξ Àq 0 ðw 0 þ a 1 εξ Àq 0 þ … þ a n ðεξ Àq 0 Þ n þ …Þ, ξ ! 0, ð66Þ From Eq. (67), it follows that the point x ¼ 0 corresponds to the root of the equation And, under the conditionw 0 > 0, η 0 will be positive. It is obvious that on the interval ½ξ 0 , 1 series (42) or (66) and (67)  Theorem 7 proved by Wasow [69], Sibuya and Takahashi [68] in the case where qðxÞ, rðxÞ are analytic functions on ½0, 1; proved by Habets [66] in the case qðxÞ, rðxÞ ∈ C 2 ½0, 1. Moreover, instead of the condition (3) Wasow impose a stronger condition: a >> 1.
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 y 0 0 ðξÞ 6 ¼ 0, ξ ∈ ð0, 1 is essential in the Lighthill method. Comment 2. Prytula and later Martin [65] proposed the following variant of the Lighthill method. At first direct expansion determined using by the method of small parameter yðxÞ ¼ y 0 ðxÞ þ εy 1 ðxÞ þ ε 2 y 2 ðxÞ þ … ð70Þ and further at second they will make transformation Here unknowns x j ðξÞ are determined from the condition that function y j ðξÞ was less singular function y jÀ1 ðξÞ. 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 yðξÞ ¼ y 0 ðξÞ þ ε{y 1 ðξÞ þ y 0 0 ðξÞx 1 ðξÞ} þ Oðε 2 Þ: 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., x 1 ðξÞ ¼ Ày 1 ðξÞ=y 0 0 ðξÞ. Therefore, yðξÞ ¼ y 0 ðξÞ þ Oðε 2 Þ. Hence, it is clear that we must make the condition of Wasow: y 0 0 ðξÞ 6 ¼ 0 in the method of Prytula or Martin also.
The proof of this theorem is completely analogous to the proof of Theorem 8, even more easily.
Then when we put it in first equality (42), we obtain an explicit solution y ¼ yðx, εÞ.
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).

It is construction explicit form of the solution of the model Lighthill equation
We will consider the problem [57], i.e., (41) again where b is given constant, x ∈ ½0, 1, y 0 ðxÞ ¼ dy=dx . Given functions are subjected to the conditions U: qðxÞ, rðxÞ ∈ C ð∞Þ ½0, 1.
Here, we consider the case q 0 ¼ À1; 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 where t ¼ x=μ, ε ¼ μ 2 , u k ðxÞ ∈ C ð∞Þ ½0, 1 and π k ðtÞ ∈ C ð∞Þ ½0, μ 0 , μ 0 ¼ 1=μ: Note that π k ðtÞ ¼ π k ðt, μÞ , i.e., π k ðtÞ depends also on μ, but this dependence is not indicated.
Corollary. The following inequalities hold: The other function π j ðtÞ, u j ðxÞ, j ¼ 0, 1, 2, … is determined from the inhomogeneous linear equations; therefore, the following lemmas are needed.

Lemma 9.
For any function f ðxÞ ∈ C ð∞Þ ½0, 1, the equation Lξ ¼ f ðxÞ has a unique bounded solution ξðxÞ ∈ C ð∞Þ ½0, 1 expressible as Proof. The proof follows from the fact that the general solution of the equation under consideration is expressed as then we obtain the required result.
This lemma implies that all the functions u k ðxÞ, k ¼ 0, 1, … are uniquely determined and belong to the class C ∞ ½0, 1.
Proof. The fundamental solution of the homogeneous equation (82) is of the form Obviously, kgðt, μÞk ≤ M and g À1 ðt, μÞ ≤ M for t ∈ ½0, μ 0 and μaresmall. The solution of problem (82) can be expressed as The estimate of the integral term in Eq. (83) shows that it is bounded by the constant M. Hence, it also follows that jηðtÞj ≤ Mt À1 ðt > 0Þ . The solution of problem (79.0) is defined by the integral Eq. (83), where kðtÞ ¼ Àu 0 ðtμÞπ À1 ðtÞ ¼ Àu 0 ðtμÞqðμtÞ π À1 ðtÞ t þ π À1 ðtÞ , satisfies the assumptions of the lemma. Therefore, the function π 0 ðtÞ is bounded on ½0, μ 0 . The boundedness of the other functions π k ðtÞ, k ¼ 1, 2, … 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 :¼ b À y 0 ð1Þ > 0holds). Then the solution of problem (76) exists on the closed interval ½0, 1and its asymptotics can be expressed as Eq.
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 vðrÞ.
The first two terms in Eq. (1) is ðk þ 1Þ 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 [70]. The basic idea of this method is as follows.
The initial problem is entered fictitious parameter λ ∈ ½0, 1 with the following properties: 1. λ ¼ 0, the solution of the equation satisfies all initial and boundary conditions; 2. The solution of the problem can be expanded in integral powers of the parameter λ for all λ ∈ ½0, 1.
With the solution of Eq. (90), we can expand in series The coefficients of this series are uniquely determined from the equations