Integral Boundary Value Problem for Second-Order Linear Integro-Differential Equations With a Small Parameter
1
KALIMOLDAYEV MAKSAT,
1,2KALIZHANOVA АLIYA,
2,3KOZBAKOVA AINUR,
3
KARTBAYEV TIMUR,
3AITKULOV ZHALAU,
1ABDILDAYEVA ASSEL,
1
AKHMETZHANOV MAXAT,
2KOPBOSYN LEILA.
1
Institute of Information and Computational Technologies Almaty, Pushkin str.125, KAZAKHSTAN
2
Al-Farabi Kazakh National University Almaty, al-Farabi str.70, KAZAKHSTAN
3
Almaty University of Power Engineering and Telecommunications Almaty, Baitursynov str.126, KAZAKHSTAN
kalizhanova_aliya@mail.ru, ainur79@mail.ru, kartbaev_t@mail.ru, jalau@mail.ru , maks714@mail.ru, leila_s@list.ru.
Abstract: - This work is devoted to the asymptotic solutions of integral boundary value problem for the Inter- linear second order differential equation of Fredholm type. Studying an integral boundary value task, obtaining solution assessment of the set singular perturbed integral boundary value problem and difference estimate between the solutions of singular perturbed and unperturbed tasks; determination of singular perturbed integral boundary value problem solution behavior mode and its derivatives in discontinuity (jump) of the considered section and determination of the solution initial jumps values at discontinuity and of an integral member of the equation, as well, creation of asymptotic solution expansion assessing a residual member with any range of accuracy according to a small parameter by means of Cauchy task with an initial jump, at that selection of initial conditions due to singular perturbed boundary value problem solution behavior mode and its derivatives in the jump point. In the paper there applied methods of differential and integral equations theories, boundary function method, method of successive approximations and method of mathematical induction.
Key-Words: - Singular, differential equations, asymptotic solutions, integral boundary problem, small parameter.
1 Introduction
Theme actuality. Overall interest of mathematicians in singular perturbed equations determined with the fact, that they function as mathematical models in many applied tasks, connected with processes of diffusion, heat and mass transfer, in chemical kinetics and combustion, in the problems of heat distribution in slender bodies, in semiconductor theories, quantum mechanics, biology and biophysics and many other branches of science and engineering. Singular perturbed equations is an important class of differential equations.
2 Asymptotic Solutions of Integral Boundary Value Problem
2.1 Setting up a problem
Let us consider the following Fredholm-type integro- partial differential equation
[
K tx yx K tx y x]
dxt F
y t B y t A y y L
∫
+ ′+
=
=
′+
′′+
≡
1
0
1
0(, ) ( , ) (, ) ( , )
) (
) ( ) (
ε ε
ε ε
(2.1) with integral boundary conditions
[ ]
[
() (, ) () (, )]
,) , 1 (
, ) , ( ) ( ) , ( ) ( )
, 0 (
1
0
1 0
1 1
0
1 0
0
dt t y t c t y t c a y
dt t y t b t y t b a y
∫
∫
+ ′ +
=
′ + +
=
ε ε
ε
ε ε
ε
(2.2)
where
ε > 0
- a small parameter, ai,
i= 0 , 1
−certain acquainted permanents, ),
( ), ( ), ( ), ( ),
(t Bt F t b t c t
A i i Ki(t,x), i=0,1 − certain acquainted functions, defined in the domain
) 1 0 , 1 0
( ≤ ≤ ≤ ≤
= t x
D .
Solution of y(t,
ε
) singular perturbed integral boundary value problem (2.1), (2.2) atε
small parameter vanishing will not tend to a solutiony(t)of an ordinary unperturbed (singular) task, obtained from (2.1), (2.2) at
ε = 0
:[
K t x y x K t x y x]
dxt F y t B y t A y L
∫
+ ′+
+
=
′+
≡
1
0
1 0
0
) ( ) , ( ) ( ) , (
) ( ) ( ) (
with boundary condition at t
= 0
or at t= 1
, but tends to the solution y0(
t)
, changed, unperturbed equation[
K t x y x K t x y x]
dx tt F t y L
∫
′ + ′+
+
∆ +
=
1
0
0 1 0 0
0 0
) ( ) , ( ) ( ) , (
) ( ) ( ) (
(2.3)
with changed boundary condition at the point t
= 0
;[
b t y t b t y t]
dt ay = +∆ +
∫
1 +0
0 1 0 0 0 0
0(0) () () () () (2.4) or with changed boundary condition at the point
= 1
t :[
c t y t c t y t]
dt ay = +∆ +
∫
1 +0
0 1 0 0 1 1
0(1) () () () () (2.5) Assume, that:
I. Functions A(t),B(t),F(t),bi(t),ci(t) и 1
, 0 , ) , (t x i=
Ki are sufficiently smooth in the domain D=(0≤t≤1,0≤x≤1);
II. Function
A
(t
) at the segment [0,1] satisfies inequation:1 0 , 0 )
(t ≥ ≡const> ≤t≤
A γ ;
III. Number λ=1 at sufficiently small
ε
is not a proper value of the kernelJ(t,s,ε):[ ]
,
) , ( ) , ( ) , ( ) , ( ) , ) ( ( 1
) , ( ) , , (
1 1
) 1 ( 1
1 0
1
) 1 (
+ Ο +
+
+ + ′
=
=
+ Ο +
=
− ∫
−∫
∫
s s
dx x A s
dx x A
e
dx s x K x t K s x K x t K s t t K A
e s
t J s t J
ε ε
ε ε ε
where the function K( st, )expressed with a formula
) . (
) exp (
) ,
(
−
=
∫
ts
x dx A
x s B
t K
IV. True an inequation:
( )
[
(
( ) (, ) () (, ))
( ) 0,) (
) 0 , ( ) ( ) 0 , ( ) ( 1
1
1 0
1
1
0
1 0
0 0
≠
+ + ′
+
′ + +
−
≡
∆
∫
∫
ds s dt s t K t b s t K t b s b
s K s b s K s b
s
σ
where function
σ
(t) has a view∫
+
= 1
0
) ( ) , ( ) ( )
(t ϕ t Rt sϕ s ds
σ ,
and function R( st, )−kernel resolvent J
( s
t, )
, and functionϕ
(t) may be represented as[
(, ) ( ,0) (, ) ( ,0)]
. )( ) 1 (
1
0
1
0 t x K x K t xK x dx
t K
t = A
∫
+ ′ϕ
V. True an equation:
, ) 0
( ) 0 , ( ) , ( ) (
) 0 , ) ( , ) ( 0 ( 1
) (
) 0 , ( ) , ( ) (
) 0 , ( ) 0 ( ) 1 ( ) , ( ) ( ) 0 , ( ) (
) (
) 0 , ( ) , ( ) (
) 0 , ) ( , ) ( 0 ( ) 1 ( ) , ( ) 0 , ( ) (
) 0 (
) 0 ( )
( ) 0 , ( ) , ( ) (
) 0 , ) ( , 1 ) ( 0 ( ) 1 ( ) , 1 ( ) 0 , 1 (
0
1
0 1 1
1
0 1 1
0 1
0 1
0
1
0 1 1
0 1
0 0
1 1
0
1
0 1 1
1
0 0
1
≠
+
+ ′
+
+
+
′ + + ′
−
−
+
+
+
−
−
−
+
+
+
≡
∆
∫ ∫
∫
∫
∫
∫ ∫
∫
∫
∫ ∫
∫
dt ds x dx A
x K x s R s A
s s K t A K
x dx A
x K x t R t A
t K ds A s s t K t t K N t c
dt ds x dx A
x K x s R s A
s s K t A K ds s s t K t K N t c
A ds c x dx A
x K x s R s A
s s K A K ds s s K K N
t t
t t
σ σ
σ σ
where
( )
) . (
) 0 , ( ) , ( ) (
) 0 , (
) , ( ) ( ) , ( ) ( ) ) ( 0 ( 1 ) 0 (
) 0 ( 1 1
1
0
1 1
1
0 1
1 0
1 1
0 0
+
×
×
+ + ′
+ +
=∆
∫
∫ ∫
ds x dx
A x K x s R s A
s K
dt s t K t b s t K t b s A b A N b
s
With reference to 1.7, section 1 it can be seen, that solution y(t,
ε
) of integral boundary value problem (1.1), (1.2) at the point t= 0
is limited and its first derivative y′(t,ε
) at the point t= 0
has unlimited growth of the order
Ο
ε
1 at
ε → 0
. Hence, for creating the boundary value problem solution asymptotics (2.1), (2.2), let’s preliminarily consider an auxiliary Cauchy problem with an initial jump, i.e., consider an equation (2.1) with initial conditions at the point t= 0
:[
() (, ) () (, )]
, ), 0 (
1
0
1 0
0 b t yt b t y t dt
a
y ε = +
∫
ε + ′ ε (2.6), ) , 0
( ε
ε =α y′
where
α = α ( ) ε
− regularly dependent onε
permanent, represented as:
( )
ε =α0+εα1+ε2α2 +α (2.7)
Let us define
α ( ) ε
in such a way, that the solution ), , (t
α ε
y of the problem (2.1), (2,6) was the solution of boundary value task (2.1), (2.2), i.e., to fulfill the second condition (2.2):
[
c t yt c t y t]
dt ay = +
∫
1 + ′0
1 0
1 () (, , ) () (, , )
) , , 1
( α ε α ε α ε
.(2.8)
2.2 Creating asymptotic solution of Cauchy problem with an initial jump
Solution y(t,
ε
)= y(t,α
,ε
) of Cauchy problem (2.1), (2.6) we will search as the sum:y(t,ε)= yε(t)+wε(τ), (2.9) where
τ =
t/ ε
- boundary-layer independent invariable, yε(t )
-solution’s regular part, defined at the section [0, 1] and wε(τ )
-boundary-layer part of the solution, defined atτ ≥0.Preliminarily multiply equations (2.1) by ε and further insert formula (2.9) into equation (2.1).
Thereupon we obtained:
[ ]
, )
, 1 ( )
, (
) ( ) , ( ) ( ) , ( )
( ) ( ) (
) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (
1
0
1 0
1
0
1 0
2
x dx w x t x K
w x t K
dx x y x t K x y x t K t F w B
w A w t y t B t y t A t y
∫
∫
+
+
+
′ +
+
= +
+ +
+
′ +
′′ +
ε ε
ε ε
ε ε τ ετ ε
τ ετ τ ε
ε ε
ε ε
ε ε
ε
ε ε
ε ε
ε
where the point
( )
⋅ − a derivative per τ beyond integral members and a derivative per х in integral members. If we make replacement∞
=
≤
≤
=
=ε ε ε1
0 ,
, dx ds s
s
x , then from here it
follows that:
[ ]
[ (, ) () (, ) ()] . ) ( ) , ( ) ( ) , ( ) (
) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (
0
1 0
1
0
1 0
2
ds s w s t K s w s t K
dx x y x t K x y x t K t F
w B w A w t y t B t y t A t y
∫
∫
∞
+ +
′ + + +
=
= +
+ +
′ +
′′ +
ε ε
ε ε
ε ε
ε ε ε
ε
ε ε
ε ε
ε ε
τ ετ ε τ ετ τ ε
ε ε
(2.10) And now let’s write out separately the equations with factors, dependent on t, and separately equation with factors dependent on
τ
. Then from (2.10) we obtain following equations separately for) (t
yε and separately forwε
(t )
:
[ ]
[
(, ) ( ) (, ) ( )]
;) ( ) , ( ) ( ) , (
) ( ) ( ) ( ) ( ) ( ) (
0
1 0
1
0
1 0
ds s w s t K s w s t K
dx x y x t K x y x t K
t F t y t B t y t A t y
∫
∫
∞ +
+
+
′ +
+
+
=
′ +
′′ +
ε ε
ε ε
ε ε
ε
ε ε
ε ε
(2.11)
. 0 ) ( ) ( ) ( ) ( )
(τ + ετ ε τ +ε ετ ε τ =
ε A w B w
w (2.12)
Let’s insert (2.9) into initial conditions (2.6), (2.7):
[ ]
( ) ( )
[ ]
[ ].
) 1 0 1 ( ) 0 (
, ) ( ) (
) ( ) ( ) ( ) 1 (
0 , ,
) 1 ( ) ( ) ( ) ( ) ( ) ( ) 0 ( ) 0 (
2 2 1 0 0
1 0
1
0
1 0 0 1
0
1 0
1 0 0
+ + +
=
′ +
+ +
′ + + +
=
= = ≤ ≤ =∞
=
=
+
+ + ′ +
= +
∫
∫
∫
∞
α ε εα εα ε
τ τ ετ τ ετ ε
τ ε τ ε ε τ
ετ ε ε ε
ε ε
ε ε
ε ε
ε ε
ε ε ε
ε
w y
d w b w b
dt t y t b t y t b a d
t dt
t dt w t b w t b t y t b t y t b a w
y (2.13)
Solutions yε
(t )
and wε( τ )
of equations (2.11) and (2.12) we will search as following series in terms of a small parameterε
:
. ) ( )
( ) ( ) (
, ) ( )
( ) ( ) (
2 2 1 0
2 2 1 0
+ +
+
=
+ +
+
=
t w t w t w t w
t y t y t y t y
ε ε
ε ε
ε
ε (2.14)
Functions A(ετ), B(ετ), bi(ετ), and Ki(t,εs) expand in series of Taylor development:
( ) ,
! 2
) 0 (
! 1 ) 0 ) ( 0 ( )
(ετ = + A′ ετ++ A′′ ετ 2+ A
A
( ) ,
! 2
) 0 (
! 1
) 0 ) ( 0 ( )
( ′′ 2+
+
′ + +
= ετ ετ
ετ B B
B B
, 1 , 0
! , 2
) )(
0 (
! 1 ) 0 ) ( 0 ( ) (
2 + =
+ ′′
′ + +
= b b i
b
bi i i i ετ
ετ
ετ (2.15)
. 1 , 0 )
! ( 2
) 0 , (
! 1 ) 0 , ) ( 0 , ( ) ,
( ′′ 2+ =
+
′ + +
= K t s K t s i
t K s t
Ki ε i i ε i ε
Inserting transformations in due form (2.14), (2.15) in the equations (2.11), (2.12) we obtain
[
]
( )
( () () () )] ; 2!
) 0 , ( ) ) ( 0 , ( ) 0 , ( )
( ) ( ) (
! 2
) 0 , ( ) ) ( 0 , ( ) 0 , ( ) ) ( ) (
) ( )(
, ( ) ) ( ) ( ) ( )(
, ( ) ( ) ) (
) ( ) ( )(
( ) ) ( ) ( ) ( )(
( ) ) ( ) ( ) ( (
2 2 1 0
1 2 1 1 2 2 1 0
0
0 2 0 0 2
2 1
1
0
0 1 2 2 1 0 0 2
2
1 0 2
2 1 0 2
2 1 0
ds s w s w s w
t K t s K s t K s w s w s w
t K t s K s t K dx x y x y
x y x t K x y x y x y x t K t F t y
t y t y t B t y t y t y t A t y t y t y
+ + +
×
×
′′ +
+
′ + + + + +
×
×
′′ +
+
′ + +
+
′ +
′ +
+
′ + + + + +
= + +
+ + + +
′ +
′ +
′ + +
′′
+
′′
+
′′
∫
∫
∞
ε ε
ε ε ε
ε
ε ε ε ε
ε
ε ε ε
ε ε
ε ε
ε
ε (2.16)
. 0 ) ) ( ) ( ) ( (
) )
! ( 2 ) 0 (
! 1 ) 0 ) ( 0 ( ( ) ) ( ) ( ) ( (
) )
! ( 2 ) 0 (
! 1 ) 0 ) ( 0 ( ( ) ( ) ( ) (
2 2 1 0
2 2
2 1 0
2 2
2 0
= + + +
×
×
′′ +
′ + + + + + +
×
×
′′ +
′ + + + + + +
τ ε τ ε τ
τε τε ε
τ ε τ ε τ
τε τε τ
ε τ ε τ
w w w
B B B
w w w
A A A
w w w
(2.17)
Similarly insert transformations (2.14), (2.15) into initial conditions (2.13). Thereupon we obtain:
