Aboundaryjumpsphenomenonintheintegralboundaryvalue ... · 2020. 6. 24. ·...

DOI 10.31489/2020M2/46-58

UDC 517.983

N.U. Bukanay1 , A.E. Mirzakulova1,2, M.K. Dauylbayev1,3, K.T. Konisbayeva 1,2

1Al-Farabi Kazakh National University, Almaty, Kazakhstan;2Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan;

3Institute of Information and Computational technologies, Almaty, Kazakstan;(E-mail: [email protected], [email protected], 1,[email protected])

A boundary jumps phenomenon in the integral boundary valueproblem for singularly perturbed differential equations

The article is devoted to the study of the asymptotic behavior of solving an integral boundary value problemfor a third-order linear differential equation with a small parameter for two higher derivatives, providedthat the roots of the "additional characteristic equation" have opposite signs. In the work are constructedthe fundamental system of solutions, boundary functions for singularly perturbed homogeneous differentialequation and are provided their asymptotic representations. An analytical formula of solution for a givensingularly perturbed integral boundary value problem is obtained. Theorem about asymptotic estimates ofsolution is proved. For a singularly perturbed integral boundary value problem, the growth of the solutionand its derivatives at the boundary points of this segment is obtained when the small parameter tends tozero. It is established that the solution of a singularly perturbed integral boundary value problem has initialjumps at both ends of this segment. In this case, we say that there is a phenomenon of boundary jumps,which is a feature of the considered singularly perturbed integral boundary value problem. Moreover, theorders of initial jumps were different. Namely, at the point t = 0 , there is a phenomenon of the initialjump of the first order, and at the point t = 1, the order of the initial jump was equal to zero. The resultsobtained allow us to construct uniform asymptotic expansions of solutions of nonlinear singularly perturbedintegral boundary value problems.

Keywords: singularly perturbed differential equation, asymptotic estimates, boundary functions, smallparameter.

Introduction

Equations containing a small parameter in the highest derivatives are called singularly perturbedequations. Such equations are mathematical models of many applied problems. A significant contribut-ion to the development of the theory of singularly perturbed equations was made by L. Schlesinger [1],G.D. Birkhoff [2], P. Noaillon [3], W. Wasow [4], A.N. Tikhonov [5, 6], M.I. Vishik, L.A. Lyusternik[7, 8], N.N. Bogolyubov, U.A. Mitropolsky [9], A.B. Vasilieva and V.F. Butuzov [10], Trenogin, V.A.[11], R.E. O’Malley [12], W. Eckhaus [13], K.W. Chang and F.A. Howes [14], J. Kevorkian and J.D. Cole[15], P.V. Kokotovic [16], S.A. Lomov [17], M.I. Imanaliev [18], K.A. Kassymov [19–21] and others.

Initial problems with singular initial conditions for a second-order nonlinear ordinary differentialequation with a small parameter were first studied by M.I. Vishik and L.A. Lyusternik [8] andK.A. Kassymov [20]. They showed that the solution of the original problem with initial values leads tothe solution of a degenerate equation with altered initial conditions when a small parameter approacheszero. Such problems became known as Cauchy problems with initial jumps. The most general casesof the Cauchy problem for singularly perturbed nonlinear systems of ordinary differential and integro-differential equations, as well as for differential equations in partial derivatives of a hyperbolic type,was studied by K.A. Kassymov. Then, singularly perturbed initial and boundary value problems withinitial jumps have been studied in [22–30]. In this paper, we consider general integral boundary valueproblems for linear ordinary differential equations of the third order with a small parameter with twohighest derivatives, when the roots of an additional characteristic equation have opposite signs. It isshown that there is a phenomenon of boundary jumps. Boundary value problems without integralboundary conditions for singularly perturbed differential and integro-differential equations have beenconsidered in [31–33].

46 Вестник Карагандинского университета

A boundary jumps phenomenon...

Statement of the problem and preliminaries

Consider the singularly perturbed differential equation

Lεy ≡ ε2y′′′ + εA0(t)y′′ +A1(t)y′ +A2(t)y = F (t), (1)

with integral boundary conditions

h1y(t, ε) ≡ y(0, ε)−1∫

0

1∑i=0

ai(x)y(i)(x, ε)dx = α0,

h2y(t, ε) ≡ y′(0, ε)−1∫

0

1∑i=0

bi(x)y(i)(x, ε)dx = α1, (2)

h3y(t, ε) ≡ y(1, ε)−1∫

0

1∑i=0

ci(x)y(i)(x, ε)dx = β,

where ε > 0 is a small parameter, α0, α1, β are known constants independent of ε.We will need the following assumptions:C1) Ai(t) ∈ C2[0, 1], i = 0, 2, F (t) ∈ C[0, 1].C2) The roots µi(t), i = 1, 2 of "additional characteristic equation"µ2(t) + A0(t)µ(t) + A1(t) = 0

satisfy the following inequalities µ1(t) ≤ −γ1 < 0, µ2(t) ≥ γ2 > 0.C3)

∆ ≡ y20(1)a1(1)

y30(1)− 1∫0

1∑i=0

ci(x)y(i)30 (x)dx

+y20(1) (1− c1(1))·1− 1∫

0

1∑i=0

ai(x)y(i)30 (x)dx

6= 0.We consider homogeneous singularly perturbed equation associated with (1)

Lεy ≡ ε2y′′′ + εA0(t)y′′ +A1(t)y′ +A2(t)y = 0. (3)

The system of fundamental solutions of the homogeneous singular perturbed differential equation(3) is as follows

y(q)1 (t, ε) =

1

εqexp

1ε

t∫0

µ1(x)dx

(µq1(t)y10(t) +O(ε)) , q = 0, 2,

y(q)2 (t, ε) =

1

εqexp

−1ε

1∫t

µ2(x)dx

(µq2(t)y20(t) +O(ε)) , q = 0, 2, (4)y

(q)3 (t, ε) = y

(q)30 (t) +O(ε), q = 0, 2,

here µ1(t), µ2(t) are roots of the additional characteristic equation µ2(t) +A0(t)µ(t) +A1(t) = 0 , thefunctions yi0(t), i = 1, 3 are defined by these problems

y′i0(t) +3µi(t)µ

′i(t) +A0(t)µ

′i(t) +A2(t)

µ2i (t)yi0(t) = 0, yi0(0) = 1, i = 1, 2,

Серия «Математика». № 2(98)/2020 47

N.U. Bukanay, A.E. Mirzakulova, et al.

A1(t)y′30(t) +A2(t)y30(t) = 0, y30(0) = 1.

The asymptotic formula of Wronskian consisting of a system of fundamental solutions is expressedas follows

W (t, ε) =1

ε3exp

1ε

t∫0

µ1(x)dx−1

ε

1∫t

µ2(x)dx

· (5)· (µ1(t)µ2(t)(µ2(t)− µ1(t))y10(t)y20(t)y30(t) +O(ε)) 6= 0.

Let’s enter the following functions

K0(t, s, ε) =P0(t, s, ε)

W (s, ε), K1(t, s, ε) =

P1(t, s, ε)

W (s, ε), (6)

where P0(t, s, ε) , P1(t, s, ε) are the third order determinant obtained from the Wronskian W (s, ε)by replacing the third row with y1(t, ε), 0, y3(t, ε) and 0, y2(t, ε), 0 respectively. Sum of K0(t, s, ε) andK1(t, s, ε) is the Cauchy function. Therefore, these functions have the following properties

1. With respect to the variable t satisfy equation (3), i.e.

LεK0(t, s, ε) = 0, LεK1(t, s, ε) = 0, t ∈ [0, 1], t 6= s.

2. When t = s satisfy the conditions

K0(s, s, ε) +K1(s, s, ε) = 0, K′0(s, s, ε) +K

′1(s, s, ε) = 0, K

′′0 (s, s, ε) +K

′′1 (s, s, ε) = 1.

By applying formulas (5), (6), for functions K0(t, s, ε),K1(t, s, ε) are valid the following asymptoticrepresentations as ε→ 0

K(i)0 (t, s, ε) = ε

2

y(i)30 (t)A1(s)y30(s)

− µi1(t)y10(t)

εiµ1(s)(µ2(s)− µ1(s))y10(s)exp

1ε

t∫s

µ1(x)dx

+O(ε) ,

t ≥ s, i = 0, 2. (7)

K(i)1 (t, s, ε) = ε

2

µi2(t)y20(t)εiµ2(s)(µ2(s)− µ1(s))y20(s)

exp

−1ε

s∫t

µ2(x)dx

+O(ε) , t ≤ s, i = 0, 2.

Let functions Φi(t, ε), i = 1, 3 are solutions of the following problem

LεΦi(t, ε) = 0, i = 1, 3, hkΦi(t, ε) = δki, k = 1, 3, (8)

where δki is Kronecker symbol.Functions Φi(t, ε), i = 1, 3 are called boundary functions and can be represented in the form

Φi(t, ε) =∆i(t, ε)

∆(ε), i = 1, 3, (9)

where

∆(ε) =

∣∣∣∣∣∣h1y1(t, ε) h1y2(t, ε) h1y3(t, ε)h2y1(t, ε) h2y2(t, ε) h2y3(t, ε)h3y1(t, ε) h3y2(t, ε) h3y3(t, ε)

∣∣∣∣∣∣ ,∆i(t, ε) is the determinant obtained from ∆(t, ε) by replacing the i-th row by the fundamental systemof solutions y1(t, ε),y2(t, ε),y3(t, ε) of the equation Lεy = 0. By taking account formulas (2), (4), weget asymptotic representation for determinant ∆(ε):

∆(ε) =1

ε(µ1(0)∆ +O(ε)), (10)

where ∆ has the form as in condition (C3).



For boundary functions Φ(j)i (t, ε), j = 0, 2, i = 1, 3 from (9) in view (4), (10) we obtain asymptoticrepresentation as ε→ 0:

Φ(j)1 (t, ε) =

1

εj−1exp

1ε

t∫0

µ1(x)dx

µj1(t)y10(t)M11µ1(0)∆

−

− 1εj

exp

−1ε

1∫t

µ2(x)dx

µj2(t)y20(t)(y30(1)−

1∫0

1∑i=0

ci(x)y(i)30 (x)dx

)∆

+

+y20(1) (1− c1(1)) y(j)30 (t)

∆+O

ε+ 1εj−2

exp

1ε

t∫0

µ1(x)dx

+ 1εj−1

exp

−1ε

1∫t

µ2(x)dx

,Φ

(j)2 (t, ε) =

1

εj−1exp

1ε

t∫0

µ1(x)dx

µj1(t)y10(t)µ1(0)

+1

εj−1exp

−1ε

1∫t

µ2(x)dx

µj2(t)y20(t)M22µ1(0)∆

−

−εM23y(j)30 (t)

µ1(0)∆+O

ε2 + 1εj−2

exp

1ε

t∫0

µ1(x)dx

+ 1εj−2

exp

−1ε

1∫t

µ2(x)dx

, j = 0, 2, (11)Φ

(j)3 (t, ε) =

1

εj−1exp

1ε

t∫0

µ1(x)dx

µj1(t)y10(t)M31µ1(0)∆

+

+1

εjexp

−1ε

1∫t

µ2(x)dx

µj2(t)y20(t)(

1−1∫0

1∑i=0

ai(x)y30(x)(i)dx

)∆

+

+a1(1)y20(1)y

(j)30 (t)

∆+O

ε+ 1εj−2

exp

1ε

t∫0

µ1(x)dx

+ 1εj−1

exp

−1ε

1∫t

µ2(x)dx

,where

M11 =

∣∣∣∣∣∣∣∣−b1(1)y20(1) y′30(0)−

1∫0

1∑i=0

bi(x)y(i)30 (x)dx

y20(1)(1− c1(1)) y30(1)−1∫0

1∑i=0

ci(x)y(i)30 (x)dx

∣∣∣∣∣∣∣∣ ,

M22 =

∣∣∣∣∣∣∣∣1 + a1(0) 1−

1∫0

1∑i=0

ai(x)y(i)30 (x)dx

c1(0) y30(1)−1∫0

1∑i=0

ci(x)y(i)30 (x)dx

∣∣∣∣∣∣∣∣ ,M23 =

∣∣∣∣ 1 + a1(0) −a1(1)y20(1)c1(0) y20(1)(1− c1(1))∣∣∣∣ ,

M31 =

∣∣∣∣∣∣∣∣−a1(1)y20(1) 1−

1∫0

1∑i=0

ai(x)y(i)30 (x)dx

−b1(1)y20(1) y′30(0)−1∫0

1∑i=0

bi(x)y(i)30 (x)dx

∣∣∣∣∣∣∣∣ .



From (11) we obtain the following asymptotic estimations

|Φ(j)1 (t, ε)| ≤ C +C

εj−1e−γ1

tε +

C

εje−γ2

1−tε , j = 0, 2,

|Φ(j)2 (t, ε)| ≤ Cε+C

εj−1e−γ1

tε +

C

εj−1e−γ2

1−tε , j = 0, 2, (12)

|Φ(j)3 (t, ε)| ≤ C +C

εj−1e−γ1

tε +

C

εje−γ2

1−tε , j = 0, 2.

Main result. We seek the solution of the problem (1), (2) in the form

y(t, ε) =

3∑i=1

CiΦi(t, ε) +1

ε2

t∫0

K0(t, s, ε)F (s)ds−1

ε2

1∫t

K1(t, s, ε)F (s)ds, (13)

where Φi(t, ε), i = 1, 3 are boundary functions, K0(t, s, ε), K1(t, s, ε) are auxiliary functions expressedby formula (6), Ci, i = 1, 3 are unknown constants.

Now, we determine the unknown constants Ci, i = 1, 3 in (13). For determining these constantswe substitute (13) into (2). Then, taking into account (8), we find that

C1 = α0 − h1P (t, ε), C2 = α1 − h2P (t, ε), C3 = β − h3P (t, ε) (14)

where

P (t, ε) =1

ε2

t∫0

K0(t, s, ε)F (s)ds−1

ε2

1∫t

K1(t, s, ε)F (s)ds. (15)

The effect on the operator h1 to function P (t, ε) is characterized by the following expression

h1P (t, ε) ≡ P (0, ε)−1∫

0

1∑i=0

ai(x)P(i)(x, ε)dx = − 1

ε2

1∫0

K1(0, s, ε)F (s)ds−

−1∫

0

a0(x)

1ε2

x∫0

K0(x, s, ε)F (s)ds−1

ε2

1∫x

K1(x, s, ε)F (s)ds

dx−−

1∫0

a1(x)

1ε2

x∫0

K ′0(x, s, ε)F (s)ds−1

ε2

1∫x

K ′1(x, s, ε)F (s)ds

dx == − 1

ε2

1∫0

K1(0, s, ε)F (s)ds−1∫

0

F (s)

1∫s

1

ε2

1∑i=0

ai(x)K(i)0 (x, s, ε) dx−

−s∫

0

1

ε2

1∑i=0

ai(x)K(i)1 (x, s, ε)dx

ds = − 1ε2

1∫0

(K1(0, s, ε) +

+

1∫s

1∑i=0

ai(x)K(i)0 (x, s, ε)dx−

s∫0

1∑i=0


F (s)ds.50 Вестник Карагандинского университета


Then from (14) the constant C1 defined by the formula

C1 = α0+1

ε2

1∫0

K1(0, s, ε) + 1∫s

1∑i=0

ai(x)K(i)0 (x, s, ε)dx−

∫ s0

1∑i=0


F (s)ds. (16)Using formula (7) to (16), we get for the constant C1 the following asymptotic estimation as ε→ 0:

C1 = α0 +

1∫0

a1(s) + 1∫s

1∑i=0

ai(x)y

(i)30 (x)dx

y30(s)

F (s)A1(s)

ds+O(ε). (17)

In this way, the effect on the operators h2, h3 to the function P (t, ε), we define the constantsC2, C3:

C2 = α1 +F (0)

µ22(0)(µ2(0)− µ1(0))+

1∫0

b1(s) + 1∫s

1∑i=0

bi(x)y

(i)30 (x)

y30(s)dx

F (s)A1(s)

ds+O(ε), (18)

C3 = β −1∫

0

y30(1)y30(s)

+ c1(s)−1∫s

1∑i=0

ci(x)y

(i)30 (x)

y30(s)dx

F (s)A1(s)

ds+O(ε). (19)

Substituting (7) into (15), we have the asymptotic representation of the function P (j)(t, ε), j = 0, 2as ε→ 0:

P (j)(t, ε) =µj−21 (t)− µ

j−22 (t)

εj−1(µ2(t)− µ1(t))F (t) +

t∫0

y(j)30 (t)F (s)

A1(s)y30(s)ds−

− µj1(t)y10(t)F (0)

εj−1µ21(0)(µ2(0)− µ1(0))e

1ε

t∫0

µ1(x)dx+

µj2(t)y20(t)F (1)

εj−1µ22(1)y20(1)(µ2(1)− µ1(1))e− 1ε

1∫tµ2(x)dx

+O(ε), j = 0, 2.

(20)Thus, the following theorem holds.Theorem 1. Let the conditions (C1)-(C3) are valid. Then integral boundary value problem (1), (2)

on the interval [0, 1] has an unique solution and expressed by the formula

y(t, ε) =3∑i=1

CiΦi(t, ε) + P (t, ε), (21)

where Φi(t, ε), i = 1, 3 are boundary functions, P (t, ε) is defined by the formula (15), Ci, i = 1, 3 havethe form (14) and are expressed by the asymptotic formulas (17), (18), (19).

Theorem 2. If conditions (C1)-(C3) are valid, then solution for integral boundary value problem(1), (2) hold the following asymptotic estimates as ε→ 0 :

|y(j)(t, ε)| ≤ C(|α0|+ ε|α1|+ |β|+ max

0≤t≤1|F (t)|

)+

C

εj−1|µj−21 (t)− µ

j−22 (t)| max

0≤t≤1|F (t)|+

+C

εj−1

(|α0|+ |α1|+ |β|+ max

0≤t≤1|F (t)|

)e−γ1

tε+

+C

εj

(|α0|+ ε|α1|+ |β|+ max

0≤t≤1|F (t)|

)e−γ2

1−tε , j = 0, 2, (22)

where C > 0 is a constant independent of ε.



Proof. By applying formulas (17)–(19), (12), (20) in (21), we get asymptotic representations of thesolution of the problem (1), (2) as ε→ 0 :

y(j)(t, ε) =

α0 + 1∫0

a1(s) + 1∫s

1∑i=0

ai(x)y

(i)30 (x)

y30(s)dx

F (s)A1(s)

ds+O(ε)

·

·

µj1(t)y10(t)M11εj−1 · µ1(0)∆ e1ε

t∫0

µ1(x)dx+

µj2(t)y20(t)

(y30(1) +

1∫0

1∑i=0

ci(x)y(i)30 (x)dx

)εj ·∆

e− 1ε

1∫tµ2(x)dx

+

+y20(1)y

(j)30 (t)(1− c1(1))

∆+O

ε+ 1εj−2

e1ε

t∫0

µ1(x)dx+

1

εj−1e− 1ε

1∫tµ2(x)dx

++

α1 + F (0)µ22(0)(µ2(0)− µ1(0))

+

1∫0

b1(s) + 1∫s

1∑i=0

bi(x)y

(i)30 (x)

y30(s)dx

F (s)A1(s)

ds+O(ε)

··

−µj1(t)y10(t)M21εj−1 · µ1(0)∆

e1ε

t∫0

µ1(x)dx+µj2(t)y20(t)M22

εj−1 · µ1(0)∆e− 1ε

1∫tµ2(x)dx

− εM23y(j)30 (t)

µ1(0)∆+

+O

ε2 + 1εj−2

e1ε

t∫0

µ1(x)dx+

1

εj−1e− 1ε

1∫tµ2(x)dx

+ (23)+

β − 1∫0

y30(1)y30(s)

− c1(s)−1∫s

1∑i=0

ci(x)y

(i)30 (x)

y30(s)dx

F (s)A1(s)

ds+O(ε)

·

·

µj1(t)y10(t)M31εj−1 · µ1(0)∆ e1ε

t∫0

µ1(x)dx+

µj2(t)y20(t)

(1−

1∫0

1∑i=0

ai(x)y(i)30 (x)dx

)εj ·∆

e− 1ε

1∫tµ2(x)dx

+

+a1(1)y20(1)y

(j)30 (t)

∆+O

ε+ 1εj−2

e1ε

t∫0

µ1(x)dx+

1

εj−1e− 1ε

1∫tµ2(x)dx

+ t∫0

y(j)30 (t)F (s)

A1(s)y30(s)ds+

+µj−21 (t)− µ

j−22 (t)

εj−1(µ2(t)− µ1(t))F (t)− µ

j1(t)y10(t)F (0)

εj−1 · µ21(0)(µ2(0)− µ1(0))e

1ε

t∫0

µ1(x)dx+

µj2(t)y20(t)F (1)

εj−1 · µ21(1)y20(1)(µ2(1)− µ1(1))e− 1ε

1∫tµ2(x)dx

+O(ε).

From asymptotic representations (23), we obtain asymptotic estimations (22). Theorem 2 is proved.The theorem 2 implies that the solution of the problem (1), (2) at point t = 0 has the phenomenon

of the first order initial jump and at point t = 1 has the phenomenon of the zero order initial jump, i.e.

y(0, ε) = O(1), y′(0, ε) = O(1), y′′(0, ε) = O

(1

ε

)



andy(1, ε) = O(1), y′(1, ε) = O

(1

ε

), y′′(1, ε) = O

(1

ε2

).

In this case, we say that the solution of the boundary value problem (1), (2) has the phenomenon ofthe boundary jumps.

Conclusion

In this paper, we consider a three-point boundary value problem for a third-order linear differentialequation with a small parameter at two highest derivatives when the roots of the "additional characte-ristic equation" have negative signs. Theorem about asymptotic estimates of solution is proved. It isestablished that the solution of this integral boundary value problem has the phenomenon of boundaryjumps. This means that the points of the initial jump are not only the left, but also the right point ofthe segment. The results allow us to construct uniform asymptotic expansions of solutions of boundaryvalue problems with boundary jumps with any degree of accuracy with respect to a small parameter.

Acknowledgement

The authors were supported in parts by the MESRK grants АР05132587 «Boundary value problemsfor singularly perturbed differential equations with a continuous and piecewise constant argument»(2018–2020) and AP05132573 «Cellular neural networks with continuous/discrete time and singularperturbations» (2018–2020) of the Committee of Science, Ministry of Education and Science of theRepublic of Kazakhstan.

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3 Noaillon Developpements asymptotiques dans les equations differentielles lineaires a parametervariable / Noaillon // Mem. Soc. Sci. Liege. — 1912. — Vol. 3, No. 11. — P. 197.

4 Wasow W. Singular perturbations of boundary value problems for nonlinear differential equationsof the second order / W. Wasow // Comm. On Pure and Appl. Math. — 1956. — No. 9. — P. 93-113.

5 Тихонов А.Н. О зависимости решений дифференциальных уравнений от малого параметра/ А.Н. Тихонов // Матем. сб. — 1948. — Т. 22, Вып. 2. — C. 193-204.

6 Тихонов А.Н. О системах дифференциальных уравнений, содержащих параметры /А.Н. Тихонов // Матем. сб. — 1950. — Т. 27, Вып. 69. — C. 147-156.

7 Vishik M.I. Regular degeneration and boundary layer for linear differential equations with smallparameter multiplying the highest derivatives / M.I. Vishik, L.A. Lyusternik // Usp. Mat. Nauk.— 1957. — No. 12. — P. 3-122; Amer. Math. Soc. Transl. — 1962. — Vol. 20, No. 2. — P. 239-364.

8 Vishik M.I. On the initial jump for non-linear differential equations containing a small parameter/ M.I. Vishik, L.A. Lyusternik // Doklady Akademii Nauk SSSR. — 1960. — Vol. 132, No. 6. —P. 1242-1245.

9 Bogoliubov N.N. The method of integral manifolds in nonlinear mechanics, In Contributions toDifferential Equations / N.N. Bogoliubov, Yu.A. Mitropolsky // Interscience-John Wiley. — 1963.— No. 2. — P. 123-196.



10 Vasil’eva A.B. The boundary function method for singular perturbation problems / A.B. Vasil’eva,A.B. Butuzov, L. Kalachev // SIAM Studiesin Applied Mathematics. — Philadeplhia, 1995.

11 Trenogin V.A. Development and applications of the asymptotic method of Liusternik and Vishik/ V.A. Trenogin // Usp. Mat. Nauk. — 1970. — No. 25. — P. 123-156; Transl. in Russ. Math.Surv. — No. 25. — P. 119-156.

12 O’Malley R. Singular perturbations methods for ordinary differential equations / R. O’Malley.— Berlin, Heidelberg: Springer-Verlag, 1991.

13 Eckhaus W. Fundamental concepts of matching / W. Eckhaus // SIAM Rev. — 1994. — No. 36.— P. 431-439.

14 Chang K. Nonlinear Singular Perturbation Phenomena: Theory and Application / K. Chang,K. Howes. — New York: Springer-Verlag, 1984.

15 Kevorkian J. Singular perturbation methods in applied mathematics / J. Kevorkian, J.D. Cole.— Berlin, Heidelberg: Springer-Verlag, 1981.

16 Kokotovic P.V. Applications of singular perturbation techniques to control problems /P.V. Kokotovic // SIAM Rev. — 1984. — No. 26. — P. 501-550.

17 Lomov S.A. Introduction to the general theory of singular perturbations / S.A. Lomov. —American Mathematical Society, Providence, 1992.

18 Иманалиев М.И. Асимптотические методы в теории сингулярно возмущенных интегро-дифференциальных систем / М.И. Иманалиев. — Фрунзе: Илим, 1972.

19 Касымов К.А. Об асимптотике решения задачи Коши с большими начальными условиямидля нелинейного обыкновенного дифференциального уравнения, содержащего малый пара-метр / К.А. Касымов // Успехи математических наук.— 1962.—Т. 17, Вып. 5.—P. 187–188.

20 Касымов К.А. О задаче с начальным скачком для нелинейных систем дифференциальныхуравнений, содержащих малый параметр / К.А. Касымов // Докл. АН СССР. — 1968. —Т. 179, Вып. 2. — P. 275-278.

21 Касымов К.А. Асимптотика решения задачи с начальными скачками для системы диф-ференциальных уравнений гиперболического типа с малым параметром при произ-водной / К.А. Касымов // Докл. АН СССР. — 1971. — Т. 196, Вып. 2. — С. 274-277.

22 Абильдаев Е.А. Асимптотические оценки решений сингулярно возмущенных краевых задачс начальными скачками для линейных дифференциальных уравнений / Е.А. Абильдаев,К.А. Касымов // Дифференциальные уравнения. — 1992. — Т. 28, Вып. 10. — P. 1659-1668.

23 Kassymov K.A. Estimates of solutions of the Cauchy problem with an arbitrary-order initial jumpfor linear singularly perturbed integro-differential equations / K.A. Kassymov, M.K. Dauylbaev// Differential Equations. — 1999. — Vol. 35, No. 6. — P. 822-830.

24 Dauylbaev M.K. The asymptotic behavior of solutions to singularly perturbed nonlinear integro-differential equations / M.K. Dauylbaev // Siberian Mathematical Journal. — 2000. — Vol. 41,No. 1. — P. 49-60.

25 Kassymov K.A. Asymptotic behavior of solutions of linear singularly perturbed general separatedboundary-value problems with initial jump / K.A.Kassymov, D.N. Nurgabyl // Ukrainian Mathe-matical Journal. — 2003. — Vol. 55, No. 11. — P. 1777-1792.

26 Kassymov K.A. Asymptotic estimates of solution of a singularly perturbed boundary valueproblem with an initial jump for linear differential equations / K.A.Kassymov, D.N. Nurgabyl //Differential Equations. — 2004. — Vol. 40, No. 5. — P. 641-651.

27 Kassymov K.A. Asymptotic estimates for the solutions of boundary-value problems with initialjump for linear differential equations with small parameter in the coefficients of derivatives /K.A.Kassymov, D.N. Nurgabyl, A.B. Uaissov // Ukrainian Mathematical Journal. — 2013. —Vol. 65, No. 5. — P. 694-708.



28 Nurgabul D. Asymptotic estimates for the Solution of a Restoration Problem with Initial Jump/ D. Nurgabul // Journal of Applied Mathematics.– Vol.2014. — Article ID 956402.

29 Dauylbayev M.K. The initial jumps of solutions and integral terms in singular BVP of linearhigher order integro-differential equations / M.K. Dauylbayev, N. Atakhan // Miskolc Math.Notes. — 2015. — Vol. 16, No. 2. — P. 747-761.

30 Akhmet M. A singularly perturbed differential equation with piecewise constant argument ofgeneralized type / M. Akhmet, M. Dauylbayev, A. Mirzakulova // Turkish Journal of Mathematics.— 2018. — Vol. 42, No. 4. — P. 1680 - 1685.

31 Касымов К.А. Асимптотические оценки решения краевой задачи для сингулярно возмущен-ных линейных дифференциальных уравнений третьего порядка / К.А. Касымов, Ж.У. Ша-рипова // Вестн. КазГУ им. С.М. Кирова. Сер. мат. — 1993. — Вып. 1. — P. 146-150.

32 Касымов К.А. Представление решения краевой задачи для линейного дифференциальногоуравнения с малым параметром при старших производных / К.А. Касымов, Д.А. Жакып-бекова, Д.Н. Нургабыл // Вестн. КазНУ им. аль-Фараби. Cер. мат., мех., инф. — 2001. —Вып. 3. — P. 73–78.

33 Dauylbaev M.K. Boundary-value problems with initial jumps for singularly perturbed integro-differential equations / M.K. Dauylbaev, A.E. Mirzakulova // Journal of Mathematical Sciences.— 2017. — Vol. 222, No. 3. — P. 214-225.

Н.Ұ. Буканай, А.Е. Мирзакулова, М.Қ. Дауылбаев, К.Т. Қонысбаева

Сингулярлы ауытқыған дифференциалдық теңдеулерге арналғанинтегралдық шеттiк есептегi шекаралық секiрiстер құбылысы

Мақала қосымша сипаттаушы теңдеудiң түбiрлерi қарама-қарсы болған жағдайдағы екi жоғарғытуындыларының алдында кiшi параметрi бар үшiншi реттi сызықты дифференциалдық теңдеу үшiншекаралы секiрiстi жалпы интегралды шеттiк есебiн зерттеуге арналған.Жұмыста қосымша сипаттау-шы теңдеудiң түбiрлерi қарама-қарсы болған жағдайдағы сингулярлы ауытқыған бiртектi дифферен-циалдық теңдеудiң iргелi шешiмдер жүйесi құрылған. Iргелi шешiмдер жүйесi арқылы сингулярлыауытқыған бiртектi дифференциалдық теңдеудiң Ki(t, s, ε), i = 0, 1 көмекшi функциялары және ше-каралық функциялары берiлген. Және олардың асимптотикалық сипаттары мен бағалаулары кел-тiрiлген. Берiлген сингулярлы ауытқыған жалпы интегралды шеттiк есеп шешiмiнiң аналитикалықформуласы алынған. Шешiмнiң асимптотикалық бағалауы туралы теорема дәлелденген. Сингулярлыауытқыған жалпы интегралды шеттiк есеп шешiмi кесiндiнiң екi жақ шетiнде де бастапқы секiрiскеие болатыны анықталған. Зерттеу нәтижесiнде есеп шешiмiнiң сол жақ және оң жақ нүктелерiндеәртүрлi реттi бастапқы секiрiс құбылыстарын және алынған нәтижелердiң қорытындысында берiлгеншеттiк есептiң шешiмiнiң t = 0 нүктесiнде бiрiншi реттi, ал t = 1 нүктесiнде нөлiншi реттi бастапқысекiрiстерi бар екендiгi анықталды. Алынған нәтижелер сызықты емес сингулярлы ауытқыған инте-гралды шеттiк есептер шешiмдерiнiң бiркелкi асимптотикалық жiктелуiн құруға мүмкiндiк бередi.

Кiлт сөздер: сингулярлы ауытқыған дифференциалдық теңдеу, асимптотикалық бағалау, шекаралықфункциялар, кiшi параметр.



Н.У. Буканай, А.Е. Мирзакулова, М.К. Дауылбаев, К.Т. Конысбаева

Явление граничных скачков в интегральной краевой задаче длясингулярно возмущенных дифференциальных уравнений

Статья посвящена исследованию асимптотического поведения решения интегральной краевой зада-чи для линейного дифференциального уравнения третьего порядка с малым параметром при двухстарших производных при условии, когда корни «дополнительного характеристического уравнения»имеют противоположные знаки. В работе построена фундаментальная система решений сингуляр-но возмущенного однородного дифференциального уравнения с учетом противоположности знаковкорней «дополнительного характеристического уравнения». Затем с помощью фундаментальной си-стемы решений строятся вспомогательные функции и граничные функции сингулярно возмущенно-го однородного дифференциального уравнения. Получены асимптотические представления и оценкивспомогательных и граничных функций. Получена аналитическая формула решения рассматривае-мой сингулярно возмущенной интегральной краевой задачи. Доказана теорема об асимптотическихоценках решения. Для сингулярно возмущенной интегральной краевой задачи получен рост решенияи его производных в граничных точках данного отрезка при стремлении малого параметра к нулю.Установлено, что решение сингулярно возмущенной интегральной краевой задачи имеет начальныескачки на обоих концах данного отрезка. В этом случае мы говорим, что имеет место явление гра-ничных скачков, что является особенностью рассматриваемой сингулярно возмущенной интегральнойкраевой задачи. Причем порядки начальных скачков оказались разными. А именно: в точке t = 0имеет место явление начального скачка первого порядка, а в точке t = 1 порядок начального скачкаоказался равным нулю. Полученные результаты позволяют построить равномерные асимптотическиеразложения решений нелинейных сингулярно возмущенных интегральных краевых задач.

Ключевые слова: сингулярно возмущенное дифференциальное уравнение, асимптотические оценки,граничные функции, малый параметр.

References

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cation. New York: Springer Verlag.15 Kevorkian, J., & Cole, J.D. (1981). Singular perturbation methods in applied mathematics. Berlin,

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24 Dauylbaev, M.K. (2000). The asymptotic behavior of solutions to singularly perturbed nonlinearintegro-differential equations. Siberian Mathematical Journal, 41(1), 49-60.

25 Kassymov, K.A., & Nurgabyl, D.N. (2003). Asymptotic behavior of solutions of linear singularlyperturbed general separated boundary-value problems with initial jump. Ukrainian MathematicalJournal, 55(11), 1777-1792.

26 Kassymov, K.A., & Nurgabyl, D.N. (2004). Asymptotic estimates of solution of a singularlyperturbed boundary value problem with an initial jump for linear differential equations. Differen-tial Equations, 40(5), 641-651.

27 Kassymov, K.A., Nurgabyl D.N., & Uaissov A.B. (2013). Asymptotic estimates for the solutions ofboundary-value problems with initial jump for linear differential equations with small parameterin the coefficients of derivatives. Ukrainian Mathematical Journal, 65(5), 694-708.

28 Nurgabul D. (2014). Asymptotic estimates for the Solution of a Restoration Problem with InitialJump Journal of Applied Mathematics, Vol.2014, Article ID 956402.

29 Dauylbayev, M.K., & Atakhan, N. (2015). The initial jumps of solutions and integral terms insingular BVP of linear higher order integro-differential equations. Miskolc Math. Notes, 16( 2),747-761.



30 Akhmet, M., Dauylbayev, M., & Mirzakulova, A. (2018). A singularly perturbed differentialequation with piecewise constant argument of generalized type. Turkish Journal of Mathematics,42(4), 1680-1685.

31 Kassymov, K.A., & Sharipova, Zh.U. (1993). Asimptoticheskie otsenki resheniia kraevoi zadachidlia sinhuliarno vozmushchennykh lineinykh differentsialnykh uravnenii treteho poriadka. VestnikKazHU imeni S.M. Kirova. Seriia matematicheskaia, 1, 146-150 [in Russian].

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33 Dauylbaev, M.K., & Mirzakulova, A.E. (2017). Boundary-value problems with initial jumps forsingularly perturbed integro-differential equations. Journal of Mathematical Sciences, 222( 3),214-225.


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