Applied Mathematics, 2013, 4, 1301-1312
http://dx.doi.org/10.4236/am.2013.49176 Published Online September 2013 (http://www.scirp.org/journal/am)
To Theory One Class Linear Model Noclassical
Volterra Type Integral Equation with
Left Boundary Singular Point
Nusrat Rajabov
Tajik National University, Dushanbe, Tajikistan
Email: nusrat38@mail.ru
Received April 22, 2013; revised May 22, 2013; accepted June 1, 2013
Copyright © 2013 Nusrat Rajabov. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
In this work, we investigate one class of Volterra type integral equation, in model case, when kernels have first order
fixed singularity and logarithmic singularity. In detail study the case, when n = 3. In depend of the signs parameters
solution to this integral equation can contain three arbitrary constants, two arbitrary constants, one constant and may
have unique solution. In the case when general solution of integral equation contains arbitrary constants, we stand and
investigate different boundary value problems, when conditions are given in singular point. Besides for considered in-
tegral equation, the solution found cane represented in generalized power series. Some results obtained in the general
model case.
Keywords: Neoclassical Volterra Type Integral Equation; Left Boundary Singular Point; Boundary Value Problems
1. Introduction
Let
:
x
axb  be a set of point on the real axis
and consider an integral equation
  
1
1ln d
xnk
k
k
a
t
xa
x
p
ta ta







tfx
n
, (I)
where is given constants,
1
j
pj
f
x is
given function in and

x
to be found.
In what follows we in detail go into case n = 3. In this
case the Equation (1) accepts the following form


2
12 3
ln lnd
.
x
a
t
xa xa
x
pp pt
tata ta
fx
 
 
 
 

 
(1)
Integral Equation (1) at p2 = 0, p3 = 0 is model second
kind Volterra type singular integral equation with left
boundary singular point, theory construction in [1-5]. In
the case, when in (1) p3 = 0 Equation (1) investigates in
[6].
As [4,5] the solution to this equation is sought in the
class of function
,
x
Cab
,
0a
with fol-
lowing asymptotic behavior
  
t,0a
oxax a

 

. (2)
In this case the integrals in the Equation (1
proper one. Moreover
) are im-

0afa

i.e
si
. right-hand
de is necessarily zero at .
x
a
In this case in Equation (1) 0 it investi-
gates in [1]. In this casfrom signs p
23
pp
depend 1
e, in
p
11
0,0 ,p
solution integral1) is found
in explicit form. In this case at 10p homogeneous
integral Equatn (1) has one solution and general solu-
tion no homogeneous (1) contains one arbitrary constant
and at 10p, integral Equation (1) has unique solution.
In case of, when in (1) 30p, 10p, 20p
Equation (
io
in-
tegral Equation (1) investigates in [6]. In this case in de-
pearacteriic equaon obtand from corresponding chsttiined
solution integral Equation (1) by two arbitrary constants,
one arbitrary constant. Select the case, when integral
Equation (1) has unique solution. To problems investiga-
tion one dimensional and many-dimensional Volterra
type integral equation with fixed boundary and interior
singular points and singular domains in kernels dedicate
[1-7].
Support that solution integral Equation (1) function



3
xC
. Besides, let in Equation (1) function

f


3
x C
two. Then differentiating integral Equa-
mes, we obtained the following third or-
n differential equation
tion (1) three ti
der degeneratio
C
opyright © 2013 SciRes. AM
N. RAJABOV
1302

3
Dx
 

2
123
3,
xx x
x
pDxpDxpx
Df x


(3)

d
d
x
Dxa
x
 . where
Homogeneous differential Equation (3) is correspond-
e following characteristic equation
. (4)
tion
2.1. The Case, When the Roots of th
ing to th
3220pp p


12 3
2. Representation the General Solu
e
Characteristic Equation Real and Different
Let in differential Equation (3) parameters
13
jjp
ion (4) real suchthat, the roots of the characteristic Equat
and different. Its denote by 123
,,

. In this case, im-
mediately testing we see that solution homoge-
neous differential Equation (2) is given by formula
 
12 3
, general
123
x
xa CxaCxaC
 
 , (5)
where
13
j
Cj
n,
arbts. itrary constan
Whe
01
3j
, function
j
x
definable
by formula (5) eneous integraluation (1).
So, function
satisfy homog Eq
x
determined by formula (5) is given
general solutioogeneous integral Equation (1).
For obtained the solution non homogeneous inte
n hom
gral
Eq
uation (1), first time use the variation arbitrary con-
stants methods, we use the general solution of the differ-
ential Equation (3). After transformation, we see that, if
solution integral Equation (1) in this case exist, then we
its my be represented in the following form
 

123
12 333 3
12312 3
0
1123
1d
,,,
x
a
f
t
xa xaxa
x
xa CxaCxa

C fxt
ta tata ta
KCCCfx
 



 
 

 
 
 


 

(6)
where
13
j
Cj arbitrary constants,



0 123 2332 12211313
222
123
1, 1, 1
,, .
,,

  

 
The solution of the type (6) obtained in the case, when
,



3
fx C
0fa, solution integral equation
(1), function
x
exist and belong to Class

3

C
.
esting, ee that, of
Immediately twe s
01 3
jj

,

fx C

3,
0fa with asymptotic b ehavior
  
1
1
,xo xa

123
max ,,at
f,
x
a




(7)
then function (5) satisfied Equation (1).
Be valid the following confirmation.
(1) parameters
Theorem 1. Let in integral Equation
13
jj such that, the roots of the
p algebraic Equ-
ation (4) real, different and positive, function
f
x

C,
0 with asymptotic behavior (7). Then
integral Equation (1) in class of function
fa
xC
x
a
form
vanishing in point is always solvability and its
solution is given by ula (6),
13
j
Cj are ar-
gral Equation
bitrary constants.
Characteristics 1. Let in inte (1) pa-
rameters
13
j
pj
, function
f
x satisfy any con-
dition of theorem 1.Then, from (6) it follows, the solu-
tion integr1)
al Equation (
xC

,
0a
with
following asymptotic behavior
  
1
1
, min

123
,,, at
x
oxa

 x a


 .
If, the roots of the characteristic Equation (4) real, -
ferent and
dif
10
, 20
, 30
, then it follows, from
formula (6) 10C
.
In this case,exist th solutio integral Equation (1),
then it is posllowing form
if en
sible is represent in fo
 

3
4512 3
0
245
1
,,
x
a
123
33 d
23
f
t
taxa xa
x
xaCxaCfxt
x
atatata
KCCfx



 


 

(8)
where -are arbitrary constants.
Th of the type (8) exist, if



 
 


 
4
C,
e solu
5
C
tion
fx C
,
0fa
  

2
21
123
,,
min ,at
fxo xa
xa


 


(9)
with asymptotic behavior
Copyright © 2013 SciRes. AM
N. RAJABOV 1303
So, in this case have the following confirmation.
re 2. (1) parameters
following asymptotic behavior
Theo m Let in integral Equation
13
jj such that, the roots of the algeb-
real, different and also
praic Equ
ation (4) 10
, 20
,
30
,

fx C
,
0fa
with asym
) in c
pt
Then integral Equation (1
otic be-
havior (9).
tion

lass of func-
x

C va nishing inpoint
x
a is al-
ways solvability and its solution is given by formula
(8),
4, 5j are arbitrary constants.
j
Characteristics 2. Let in integral Equation (1) pa-
rameters
C
13
j
pj, function
f
x satisfy any con-
dition of thm 2en, (8) it follows, the solu-
tion integral Equation (1)
eore. Thfrom
xC

,
0a
with
  
2,m

223
i atn,,
x
oxa

 x a

 
Remark 1. Confirmation similar to theorem 2 ob-
ta

ined and in the following cases:
a) 10
, 20
, 30
; b) 10
, 0
2
, 0
3
.
he o of the charatiIf totsterisa) ndrcc eqution (4real a
different, 10
, 20
, 30
, then from integral
representation), follows, that in order that (6
x
is
solution integral Equation (1) in this case, it is necessary
12
0CC
. In this case, if exist solution integral Equa-
tio it will be represented in following form n (1), then
 
123
3 3
23
d

0
26
,,
a
33
6
1x
f
t
ta xata
x
xa Cfxt
x
axatata


 
 
 
 
 

(10)
tant.
The solution of the type (10) exist, if
KC
fx


where C6 are arbitrary cons

 
fx C
,
0fa with asymptotic behavior
,

3
fxox a



33
at
x
a. (11)
So, we proof.
The following confirmation.
uation (1) parameters
Theorem 3. Let in integral Eq
13pj such that, the roo
jts of the -
4) real, different and also 1
algebraic Equ
ation (0
, 20
,
30
,
fx C
,
0fa
with asympt-
integrn
otic be
havior (11). Then
tion

al Euatioq(1) in class of func-
x

C va nishing inpoint
x
a is al-
rmula (10)
given by foways solvability and
wh
its solution is
ere C are arbitrary consta
,
rameters
3nt.
Characteristics 3. Let in integral Equation (1) pa-
13
j
pj
, function
f
x
lution
satisfy any
conditio Then the soof the integral
Equation
n of theorem 3.
(1) in point
x
a
vanishs asymptotic
behavior determined from formula
and it
  
3at .
x
oxax a

 

Remark 2.tion similar tom 3, ob-
tained and in the following cases:
a) 10
Confirma theore
, 20
, 30
; b) 10
, 20
, 30
.
If the roots of the characteristic Equation (4) real, dif-
ferent and
013
jj
 , then from integral repre-
sentation (6er that ) follows, in ord
x
nece
is solution
integral Eqase, it isC1 = C2
=
uation (1) in this cssary
C3 = 0. In this case, if exist solution integral Equation
(1), then its will be represented in form
  
23
3
3 4
d
ft
ta
1
33
12
0
x
a
ta ta1
x
fx xa x

 
 

 
The solution of the type (12) exist, if
t Kfx
x

 
 
 
(12)
aata

  
fx C
,
0r
fa with asymptotic behavio
  
,0at
f
xoxa xa


 (13)
So we proof the following confirmation.
Theorem 4. Let in integral Equation (1) parameters
13
j
pj
ati

such that, the roots of the algebraic Equ-
The function on (4) real, different and positive.
f
x

C,
0fa with asymptotic behavior
tegrn (1) in class of function (13). Then in
al Equatio
xC

vapoint x = a have
give b(12).
ation (1) pa-
nishing in
y formula
unique so-
lution, which
Characteristics 4. Let in integral Equ
rameters
13
j
pj
, function
f
x satisfy any con-
dition of theorem 4. Then the solution of the integral
Equation (1) in point
x
a
vanish and its asymptotic
behavior determined from formula
  
,0at
x
oxa x

 a


.
2.2. The Case, When the Roots of the
Characteristic Equation Real and Equal
Let in integral Equation (1) parameters
13
j
pj
,
aracteristic Equation (4) real
and equal.
such that, the roots of the ch
In this case we have the following confirmation:
Copyright © 2013 SciRes. AM
N. RAJABOV
1304
Theorem 5. Let in integral Equation (1) parameters
13
j
pj such that, the roots of the characteristic
Equation (4) real, equal and positive, that is 12
30
 that a fun
. Assume ction
fx C
,
0fa with the following asymptotic behavior
  
4
4at,,.
f
xoxa xa


 

Then homogeneous integral Equation (1) class of in
function
xC
vanishing in point
x
a
, have
three linear independent solutions the type
 
1
x
xa
 ,

2ln
x
xa xa
 
 
2
ln .
,

3
x
xa xa
 
Non homogeneous integral Equation (1), always solv-
able. Its general solution contain three arbitrary con-
stant and given by formula
 


6
2
,,, ,
a
2
3
22
5123
ln ln
6 lnlnd
xaCxaCfx
12
x
xxaC
f
t
axaxat
tatata ta

 
 

 



(14)
x


KC
CCfx



Were
13
j
Cj -arbitrary constants.
stics 5. In this case, when in integral Equa-
tion (1) parameters
Characteri
13
j
pj , function
f
x
ion inte
sat-
isfy any clut gra
Equation
ondition of theorem 5, then sol
(1) in point
x
a vanish and its asymptotic
  
,
0, .at
xoxa
xa



From integral representation (14) follows. If solution
integral Equation (1) at 123 0


d it’s in form
behavior determined from formula
exist, then
we may be represente
  
22
66 lnln
xxa
6
d .
2a
ft
xa ta
x
fx


 
 
 
t Kfx
tataxa ta






 
 (15)
The solution of the type (15) exist, if
fx C
,
0fa
2.3. The Case, When One Roots of the
Characteristic Equat
the Roots of the Characteristic Equation
Complex and Conjugate
Let in integral Equation (1) parameters
with the following asymptotic behavior
  
, at 0, .
f
xoxaxa

 
 (16)
o in the case, when S123 0

, proof the
following confirmation.
Theorem 6. Let in integral Equation (1) parameters
13
j
pj such thots of the cheris-
tic Equation (4) real, equal and negative, that is 12
at, the all roaract
30
. Assume hat a function
t
fx C
,
0fa with the asymptotic behavior (16). Then, in-
gral Equation (1) in class te
C have
mula (15).
Characteristics 6. In this case, when fulfillment any
condition theorem 6, then solution integral equation in
po
unique solution
and give by for
int
x
a vanish and its asymptotic behavior deter-
mined from formula
  
, at 0, .
x
oxaxa


 

ion Real and Two
13
j
pj
Equation (4) real
plex
1
such that, the one roots of characteristic
and two the roots of the characteristic equation com
conjugate. Correspondingly its denote by
, 2
A
iB
,
3
A
iB
. When 10, 0A
, then by this roots cor-
responding following parmogeneous
integral Equation (1):
ticular solution ho
 
 
 
1
1
2
3
cos ln
sin ln
,
.
A
A
xxa
 
x
xaB xa
x
xaB xa


 

(17)
In this c
ase, if solution integral Equation (1) exist, then
it will be represented in form
  


1
3
11
0
71
23
1sin
,,,
a
1
12
A
A
x
Cxa Ca3
2
sin ln
lncoslnd
CB xafxcos lnxxaB x
f
t
x xa
DBt
ta
 


 
 
 


(18)
xa xa
BD
ta ta
KCCCfx









a
Btata

 

 
 


 
Copyright © 2013 SciRes. AM
N. RAJABOV 1305
where

222
011
20ABBB AB

 ,
44 3DBAA 22
BA
 
11
 
2222
21
23DABABBBA

.
The solution of the type (18) exist, if 10
,
, 0A,
fx C
,
0
fa with the following asymptotic
behavior
 
5
51
, max, at .
f
xoxaA xa


 
 (19)
So in this case we have the following confirmation.
Theorem 7. Let in integral Equation (1) parameters
13
j
pj
tic Equation
(2
such that, one the roots of the character-
(4) real positive, two out of its complex is
conjugate
A
iB
 , 3
A
iB
). Besides let
2
RealA0
.
Assume that a funct


fx C
,
0fa
ion with
asymptotic behavior (19). Then homogeegral
Equation (1), in class
neous int
C
vanishing in point
x
a
,
lvable,
s and
has three linear Independent solution of
Non homogenous integral Equation (1) a
its general solution contain three arbitrar
gi
the
lways so
y constant
type (17).
ven by formula (18), where
13
j
Cj -arbitrary
constants.
Characteristics 7. In the case, when fulfillment any
condition theorem 7, then solution integral Equation (1)
in point
x
a
vanish and its behavior determined from
following asymptotic formula
   
6
61
a, min,At .
x
oxax a


 

From integral representation (18) follow
of the algebraic Equation (4) satisfy condition of the
theorem 7, besides
s, if the roots
10
, 0A. If 10
,
) in t
0A,
is th on hen if exist solution integ (1case,
then its represented in following form
 
ral Equati
 


2
3
,, .
1
23
3
11
0
8
cos lnsin ln
1sin
A
A
x
a
xxaCBxaC B
2
lncos lnd
xa fx
f
t
xa xa
BDBt
ta xa
BD
x
ata ta
K
 










 
 
 


In this case for convergence integrals in right part (20),
necessary
ta ta
 

 
 

 
 
 (20)
C
Cfx

fx C
,
0fa with asym
beha
ptotic
vior
  
7
7
, at .
f
xoxa Axa

 
 (21)
So, we proof. the following confirmation.
Theorem 8. integation (1) parameters
Let inral Equ
1
j
pj
3
sndition theatisfy coorem 7, besides 1
0, 0.A Let 10
, 0A. Function

fx C
,
0fa with
geneous integr
asympt
al Equat
otic be
on (1),
havior (21)
i in class
. Then homo-

C vanishing
in point
x
a, has two linear Independlution ent so
 
2cos ln
A
x
xaB xa
 


,
 
3sin ln
A
x
xaB xa



.
Non homogenous integral Equation (1) always solv-
able and its general solution from

C
class is given
by formula (20), where -arbitrary con-
stants.

23
j
Cj
Characteristics 8. In the case , when fulfillment any
condition theorem 8, then solution integral Equation (1)
in point
x
a
vanish and its behavior determined from
following asymptotic formula
A
  
at
x
oxax a


 .
Now suppose, that the roots of the algebraic Equation
(4) satisfy condition of the theorem 7, besides 10
,
0A. Let 10
, 0A
. Then, if exist s
then its repr
lowing form
olution inte-
gral Equation (1) in this case,esented in fol-
 

1
13
11
1
1 2
cos lnd
0
91
sin ln
,.
A
x
a
f
t
axaxa tax
x
xa CfxBta xa
K








D Bta
Cfx










(22
(22),
ne
D Bt
ta ta

 
 

 

)
cessary
In this case for convergence integrals in right part

fx C,
0fa with asymptotic
havior
  
8
81
, at .
f
xoxa xa


 
 (23)
So, we proof. the following confirmation.
be
Copyright © 2013 SciRes. AM
N. RAJABOV
1306
Theorem 9. Let in integral Equation (1)arameters p
13
j
pj satisfy condition theorem 7, besides 1
0, 0.A Let 10
, 0A. Function
fx C
,
0fa with
geneous integr
asympt
al Equat
otic be
on (1),
havior (23)
i in class
. Then homo-

C
 
1
vanishing
,lin point x = a one soution

1
x
xa
.
 Non ho-
mogenous integral Equation (1) always solvable and its
general solution from class

C is given by formula
(22), where arbitrary constant.
Characcs 9. In the case, when fulfillment any
co
1
C-
teristi
ndition theorem 9, then solution integral Equation (1)
in point
x
a
vanish and its behavior deined from
following asymptotic formu
term
la
  
1,
at
xoxa
xa

In the case, when 10
, 0A, then from integral
representation (18) follows, that, if exidt solution integral
Equation (1) in this case, then it is possible in following
form
 

1
3
11
0
10
1sin lnl
.
A
x
a
ta ta
2
cos nd
ft
xa xa
x
fxBD BDBt
x
axa
Kfx




 












In this case for convergence integrals in right part (24),
it is sufficient
ta tata
 
 
 
 
 
 
(24)
Characteristics 10. In the case, when fulfillment any
condition theorem 10, then solution integral equati
in point

fx C
,
0fa with asymp-
totic behavior
  
, at 0, .
f
xoxaxa

 
 (25)
. the following confir
10. Let in integral Equation
So, we proofmation.
Theorem (1) parameters
13
j
pj satisfy condition theorem 7, besides 1
0, 0.A Let 10
, 0A. Function
Cfx
,
0a with asymptotic behavior (25). T
f
Eq
hen integral
uation (1), in class

C vanishing in point
x
a
,
olution, which given by formula (24). have unique s
on (1)
x
a
vanish and its behavior determined from
following asymptotic formula
  
, 0 at
x
oxax a




3. Property of the Solution
Let fulfillment any condition of the theorem 1. Differen-
tiating the solution of the type (6), immdiate verification,
we can easily convince to correctness of the following
eq
e
uality:
 
 
 
12
112 23
333 1
a
x
x
Dxxa CxaC
x
3
12
3
a
x
a CDfx
3
44
23
23 d,
x
ft
axa xa
4
1
1
a
00
f
xt
atata ta


 
 
 
 
 
(26)
t

 

 


where

d
d
a
x
Dxa
x
 .
In an analogous way differentiating the expression (26), we have

 

 
 
12 3
123
333
2 2
22 2123
112233
0
444
55 5
123
123
00
1d.
aaa
xxx
x
a
Dxxa CxaCxa CDfxDfx
ft
xa xaxa
fx t
tatatata
 
 

 
 

 

 
 


 
 
 


(27)
From Equality (6), (26), (27) we find









32 1
23
0
lim ,
xx
xa
xTx

 
(28)
1
32
3
0
lim a
xa aD Dx





2
12
a
x
Cxx
 
 





 


22
13
21313
0 0
limlim ,
aa
xx x
xa xa
CxaD DxxTx
  
 


 



(29)
13 2

x





 

21 3
31
212
0
lim .
a
x x
xa
DxxTx

 

(30)
32
21
0
lim a
x
xa
CxaDx
 
 
Copyright © 2013 SciRes. AM
N. RAJABOV 1307
Differentiating the solution of the type (8), immediate v
lowing equality:

eri correctness of the fol-fication, we can easily convince to
 

23
123
24
3
444
123
0
1d.
x
a
xaC xa
ft
xa xat
xatata ta


 


 


 
  
 


(31)
From equality (8) and (31) we find
333
123
5
0
a
x
CDfx fx


 
a
x
Dx
ta

 

24
43
00
1
limlim ,
a
xx
xa xa
CxaDxxT
 


 


1
x
(32)
 

35
52
0
xa
0
11
limlim .
a
xx
xa
CxaDxxTx
 

 

(33)
From integral representation (10) it follows that if parameters
13
j
pj
and function
f
x in Equation (1)
satisfy all condition of theorem 3, then the solution of the type (10) has the property

3
6
xa
x
ax

C

.
From integral representation (14) it follows that
(34)
 

 

 
2
12 3
32
1ln2 lnln
3128lnlnd,
2
a
x
D xxaC

 a
x
x
a
xaCxaxaC Dfx
ft
xaxa xa
fx t
tatatata
 

 

 
 
 
 


 

(35)

 
 





222 22
12 3
222342
2ln24ln ln3
8128 2lnlnd.
2
aaa
xxx
x
a
DxxaCxaCxaxaCDfxDfx
ft
xaxaxa
fx t
tatata ta
 


 



 
 
 


 

(36)
as the following properties:
Using the formulas (14), (35) and (36), we easily see that, when fulfillment any condition of theorem 5, then solution
of the type (14) h



 



 




2
2
1
22 6
limln2 ln1ln
22 lnlim
x
xa
x x
xa
Cxaxaxxaxa
Dxlnxa xaxx
D
T
 
 

(37)
 
2lim2ln
xa
Cxa











2
222
22 ln
21 ln22ln
x
xx
xaDxxa


7
lim
xa
x
aDxxax


 


Tx
(38)
x
 












228
3lim2 lim
xx x
xa xa
CxaDxDx xT

 
 

 . (39)
From integral representation (18) it follows that
 
1
11.
A
x
Dx xaCxa

 

 
2
3
1
cos lnD
x
B BxafxDfx
 


1A
x
 
23
0
4
11
221
0
coslnsin lnsinln
1sinlncosln
a
C
ABxaB BxaCA Bxa
xa xaxaxa
BADDBBADDBB
ta tatata
  
 
 

 

 
 
 


 


d, (40)
ft t
ta
 
 





Copyright © 2013 SciRes. AM
N. RAJABOV
1308

 
 

 


1
1
2222
11 2
22
3
42
25
121 12
1
00 0
22
cos ln2sinln
2 sinlncosln
1
A
a
x
A
x
a
xx
a
Dx xaCxaCAB BxaABBxa
CABABBxa BBxa
ADBDDx ax a
fxDfxD fxBtata
AB



 
 



 



 



 




22
12 21
2sinln2cosln d.
ft
xa xa
DADB BABD ADBBt
tata ta
 
 


 
 


 
 
(41)
Using the formulas (18), (40) and (41), we easily see that, when fulfillment any condition of theorem 7, then solution
of the type (18) has the following properties:


 




1222 10
1
0 0
lim ,
x
xa
T x
(42)
11
lim 2
a
xx
xa
CxaBDxABDxBABx
 

 





  


 




 

2
21
0
222
1
222 211
11 1
0
1limsinlncos ln
sin ln2cosln.
1
sinln2cos lnlim,
Aa
x
xa
x
x
xa
CxaDxABxaBBxa
DxABBxaAB Bxax
A
ABBxaBAB BxaT x

 
 

 





 




 
 
 

(43)










 

2
3
0
x
xa 1
222
1
222 212
11 1
0
1lim[cos lnsin ln
cosln2sin ln.
1
cosln2coslnlim.
Aa
x
x
xa
CxaDxABxaBBxa
DxABBxaABBxax
A
ABB xaB ABBxaTx

 
 

 



 




 
 
 

(44)
Differentiating the solution of the type (20), immediate verification, we can easily convince to correctness of the fol-
lowing equality:
 



 
 
1
2
3
12
3
0
4
11221
0
cos lnsinln
sinlncos ln
1sin lncosln
A
x
x
A
x
a
D xxaCABxaBBxa
BD
CABxaBBxafx Dfx
ta xaxaxa
BAD BDBADBDB
xa tatata
 




 
 
 
 
 
 
 
 
d,
ft t
ta

 
(45)
Using the formulas (20) and (45), we easily see that, when fulfillment any condition of theorem 8, then solution of
the type (20) has the following properties:




13
2cos lnsinln
1lim .
x
xa
x
xa
aBBxaxBxaDx
B
Tx
B

 

1limsin ln
A
CxaABx



(46)
 
 


3
14
1limcos lnsin lncos ln
1lim .
A
x
xa
x
xa
CxaABxaBBxaxBxaD x
B
Tx
B


 
 





(47)
Copyright © 2013 SciRes. AM
N. RAJABOV
Copyright © 2013 SciRes. AM
1309
From integral representation (22) it follows that if pa-
rameters
13
j
pj and function
f
x
hen th
in equation
(1) satisfy all con of theorem 9, te solution of
the type (22) has the property
nditio

1
1
xa
x
ax



C
. (48)
4. Boundary Value Problems
When, the general solution constants, arbitrary constants
higher mentioned properties of the solution the integral
Equation (1) give possibility for integral Equation (1) put
and investigate the following boundary value problems:
Problem N1. Is required found the solution of the in-
tegral Equation (1) from class
CΓ, when the roots the
algebraic Equation (4) real, different and positive by
boundary conditions
(49)
where A11, A12, A13-are given constants.
Problem N2. Is required found the solution of the in-
tegral Equation (1) from class



1
11
2
12
3
13 ,
xxa
xxa
xxa
Tx A
Tx A
Tx A






CΓ, when the roots the
algebraic Equation (4) real, different and also 10
,
20
, 30
, by boundary conditions


4
21
5
22 ,
xxa
xxa
Tx A
Tx A




(50)
where A21, A22-are given constants.
Problem N3. Is required found the solution of the in-
tegral Equation (1) from class
CΓ,
fferent
when the roots the
algebraic Equation (4) real, di and also 10
,
20
, 30
by boundary conditions

3
31
xa
x
ax



A
, (51)
ven con
where A31-are gistant.
Problem N4. Is required found the solution of the in-
tegral Equation (1) from class
CΓ, when the roots the
algebraic Equation (4) real, equal and positive, that is
123 0

 by boundary conditions


6
41
A
Tx A


42
8
43 ,
xxa
xxa
Tx A



(52)
where A41, A42, A43-are given constants.
7
xxa
Tx


Problem N5. Is required found the solution of the in-
tegral Equation (1) from class
CΓ, when the one roots
of the algebraic Equation (4) real positive, two out of its
complex-conjugate. Besides 2
Real 0A

51
52
53 ,
xa
xa
xa
A
A
A
constants.
found the solution o
, by bound-
ary conditions
(53)
where A51, A52, A53-are given
Problem N6. Is required f t
tegral Equation (1) from class



9
10
11
x
x
x
Tx
Tx
Tx






he in-
CΓ, when the on
of the algebraic Equation itive, two out of its
complex-conjugate. Besides
e roots
(4) real pos
10
, 2
Real 0A
, by
boundary conditions
(54)
where A61, A62-are given constants.
Problem N7. Is required found the solution of the in-
tegral Equation (1) from class

12
TxA


61
13
62 ,
xxa
xxa
Tx A



CΓ,
(4) real pos
10
when the one roots
of the algebraic Equation itive, two out of its
complex-conjugate. Besides,
, 2
Real 0A
, by
boundary conditions

1
71
xa
x
axA

, (55)


where A71-are given constant.
Solution problem N1. Let fulfillment any condition of
theorem 1. Then using the solution of the type (6) and its
properties (28)-(30) and condition (49), we have

32 1323
1112122
00
,,СAСAС13
0
A
  



Substituting obtained valued C1, C2 and C3 in formula
(6), we find the solution of problem N1 in form


32 1323
1111213
0
,,,.
00
x
K
AA Afx
  


(56)
So, we proof.
Theorem 11. Let in integral Equation (1) parameters
13
j
pj
, function
f
x
m N1 ha
satisfy a
n Probles a uniquesolution which
is given by formula (56).
any condition of
th
on (5
ny condition of
theorem 1. The
Solution problem N2. Let fulfillment
eorem 2. Then using the solution of the type (8) and its
properties (32), (33) and conditi0), we have:
421
1
С
A
0
 ,
522
0
1
С
A
. Substituting this valu, ed C4
C5 in formula (8), we find the solution of problem N2 in
form
N. RAJABOV
1310
 
.
22122
00
11
,,
x
KAAf



So, we proof.
ion (1) parameters
x
(57)
Theorem 12. Let in integral Equat
13
jj, function
p
f
x satisfy condition of
theorem 2. Then problem N2 haue solution which is
given by formula (57).
ti blem N3. Let fulfillment a
s uniq
Soluon prony condition of
theorem 3. Then using the solution of the
its properties (32) and condition (51), we 1
type (10) and
have: С63
A
.
Substitut this valued C6 in fo
lu
ermula (10), we find theso-
tion of problem N3 in form
331
,.
x
KAfx


(58)
So, we proof.
Theorem 13. Let in integral Equation (1) parameters
13
j
pj , function
f
x
3 has
satisfy condition of Theo-
problem N unique so
given by formula (58).
Solution problem N4. Let fulfillment any condition of
theorem 5. Then using solution of the type (14) and its
, we have:
rem 3. Then lution, which is
properties (37)-(39) and condition (52) С1
41
A
, 242
С
A
, 343
С
A
Substituting this valued 1
С,
2
С and 3
С in formula (14)find thesolution of
problem N4 in form
, we
.
 
5414243
,,,
x
KAAAfx

(59)
So, we proof.
Theorem 14. Let in integral Equation (1) parameters
13
j
pj , function
f
x
4 has
satisfy
problem N unique solution, which is
given by formula (59).
any condition of
th
ion (5
condition of theo-
rem 5. Then
Solution problem N5. Let fulfillment
eorem 7. Then using solution of the type (18) and its
properties (42)-(44), and condit3) we have:
151
0
1
С
A
, 252
0
1
С
A
, 35
0
3
1
С
A
. Substituting
this valued C1, C2 and C3 in formula (18) we findthe so-
lution of problem N5 in form
7515253
,,, .
x
KAAAfx


(60)
So, we proof.
Theorem 15. Let in integral Equation (1) parameters
13
j
pj , function
f
x fy condition theorem
7. Then problem N5 have unique solution, which is given
by formula (60).
Solution problem N. Letfuen of
theorem 8. Then usin
satis
6lfillmnt any conditio
g solution of the type (20) and its
properties (46), (47) and condition (54) we have:
ting this valued C2 and
C
261
С1
A
B
, 362
С
3 in formula (20) we find the solution of problem N6 in
form
86162
,, .
x
KAAfx
(61)
So, we proof.
Theorem 16. Let in integral Equation (1) parameters
13
jjp
, function
f
x satisfy condition theorem
8. Then problem N6 have unique solution, which is given
by formula (61).
Solution problem N7. Let fulfillment any condition of
theorem 9. Then using solution of the type (22) and its
properties (48) and condition (55) we have:1
17
С
A
.
we findSubstituting this value C1 in formula (22)
so N7 in form
the
lution of problem
971
,.
x
KAfx


(62)
So, we proof.
Theorem 17. Let in integral Equation (1) parameters
13
jjp
, function
f
x satisfy condition theorem
9. Then problem N7 have unique solution, which is given
by formula (62).
5. Presentan the Sol
Equation (1) in the Generalized Power
hat
tioution of the Integral
Series
Suppose t
f
x
on on
has uniformly convergent power
series expansi
:
 
0
k
k
k
f
xxa

f
, (63)
where constant 0
and fk, 0,1,2,k, are given
nstants. We attempt to ind a solution of (1) in the
form

k
x
co f
 
xa
0k
k
where the coefficients,
, (64)
0,1, 2,
kk
are unknown.
Substituting power series representations of v alue
f
x and
x
into (1), equating the coefficients of
the corresponding function, and for , we obtain
k

 
3
32
12
23
kk
k
0,1, 2, 3,.
,
f
kpkpk p


If
k
 
32
12 3
20kpkp
(65)

kp

 
for in
all 0,1,2,k
, putting the found coefficients back into
(64), we arrive at the particular solution of (1).
 
 
1
A
B
. Substitu
3
2
012 3
.
2
k
k
x
k
x
3k
af
If, for some values
kp
kpk p

 
(66)
1
kk
, 2
kk and 3
kk
, con-
Copyright © 2013 SciRes. AM
N. RAJABOV
Copyright © 2013 SciRes. AM
1311
stants
,13
j
pj

p
in the
satisfy
3
th
sentedit is

k

 
32
12
20kpkp
 
 
,
en the solution to integral Equation (1) can be repre-
form (64) necessary and sufficiently
that 0
j
k
f, 1,2,3j, that is, it is necessary and suf-
ficiently that function satisfies
olvility condition the following three sab

0,
j
k
afx jx





1, 2,3. (67)
xa
In this case the solution of the integral Equation (1) in
the cla
f
x in point
x
a
ss of function can be represented in form (64) is
given by formula

3
k
  
 


 

12
kk
3
k
3
3
123
3
1
32
12 3
3
32
1
12 3
2
2
k
k
k
k
kk
kkk
p
k
11
0
k
k
k
xx
a
k
11kk
2
23
3
2k
f
pk p
k
1
k
x
a
k
f
pkpk p
k
x
a
k

f
p
kpk p



 
 
(68)

 

xa xa


where 1
k
xa
, 2
k
, 3
k
arbitrary constants.
Immediately testing it we see that, if converges radius
of the series (63) is defined by formula 1
Rl
,
1
lim n
nn
f
l
f
, then converges radius of the series
 (66),
rem 18. Let in integral Equation (1), function
(68) are also defined by this formula. So, we prove the
next result.
Theo
f
x
ized
represent in formuniformly-conv
power series type (63) and
for ntegr
erges general-
 
12 3
20kpkpkp

 
,
. Then ial Equation (1) in class of
function
32
0,1,2,k
x
represented in form (64) has a unique-
solution, which is given by formula (66). For values
ary and sufficiently ful-
fillment three solvability condition type
integral Equation (1) in class of functio
fo ays solvability and its general solution
co
(68).
se
e
j
k, 1, 2, 3j,
 
32
12 3
20kpkpkp

 
,
the existence of the solution of Equation (1) can be rep-
resented in form (64) it is necess
k
(67). In this case
n represented in
rm (63) is alw
ntain tree arbitrary constants and is given by formula
6. General Ca
In general case to integral Equation (I) corresponding th
following algebraic equation
12 3nn nn
pp p

4
3! 1 !0.
n
n
p
12 3
4
2!
n
p
 

 

(II)
Example
in the case, when the roots of the Equation (II) real, dif-
ferent and positive have the following confirmation.
Theorem 19. Let in integral Equation (I) parmeters
Some results obtained in the general case to.
1
j
pjn
tion (II) real,
such that, the roots of the algebraic Equa-
different and positive, function
f
x
C
,
0fa
withpthavior
1, xa
asymotic be
 

1
12
,
max,,, at ,
n
fxo
x
a



 


Then integral Equation (I) in class of function
x
C
vanishing in point
x
a is always solvability
and iion i by formula
 
ts soluts given
 

1
3
1
0
1123
1d
,,,,, ,
k
n
xxaC
k
k
k
xn
k
k
a
n
f
t
xa
f
xt
tata
KCCC Cfx










(III)
where

1
k
Ckn
-arbitrary constants,
12
22 2
12
0
111
12
1,1, ,1
,,
,,, .
,,,
n
n
nnn
n
 

 


7. Conclusions
So, in this article we consider new class Volterra type
integral equation, which no submitting exists Fredholm
theory (Theory Volterra type integral equation in class
N. RAJABOV
1312

C,
2
L), that is for this type integral equation,
homogeneous integral equation may have non-zero solu-
tion. In particular in certain cases (Example,
roots of the characteristic Equation (4) or (II) real, dif-
fe
is type integral equation coincides to the theory Fred-
holm integral equation.
By means methods (example [5]) in the theory one
dimensional singular integral equation, problem finding
when all
rent negative or real, equal and negative) the theory
the solution general equation
th
 

1
1,ln d
,
xnm
m
m
a
t
xa
x
Kxt t
ta ta
fx







(IV)
reduces to finding solution Volterra type integral equa-
tion with weak singularity. On this basis, in depend from
roots of the algebraic equation
 

12 3 4
123 4
,,2!,3!, 1!,
nn nnn
KaaK aaKaaK aanK aa
 

 0,
n (V)

,01
m
K
aamn
lution equation contai
, select cases, when general so-
ns 1
arbitrary
constants, and cases when Equation (IV) has unique solu-
tion.
In this case, integral Equation (IV), we represented to
following form
,1,2,,nn n
 
1
1,ln
xnm
m
t
xa

d ,
m
a
x
Kaa


(VI)
where
tFx
ta ta

 

1d
mt
xa
1,,ln
x
mm
m
n
a
F
xfxKxt at



(VII)
K at
a ta


.
According to the mentioned above, writing the solu-
tion integral Equation (VI) in depend to the roots of the
characteristic Equation (V) or (II), after substituting for
F
x from formula (VII) we arrive at the solution of the
new type integral equation. At specific condition to func-
tions
,,
mm
K
xtKaa and
f
x
tegral equ
this integr equ-
ination wit
al
h weak ation
ul
arity
will be Volterra type
arity in poi
tegra
iint
singnt = a. In this basis the problem inves-
tigation inl Equation (IV), reduce to problem inves-
tigation Volterra type integral equation with weak singu-
ln po
x
a.
REFERENCES
stem of Linear Integral Equations of
and Super-singular Kernels,”
n-Classical Problems of Mathematical
Conference, Samarkand, Uzbekistan, Kluwer, Utrecht,
Boston, 11-15 September 2000, pp. 103-124.
[4] e Linear
I Inter
thematics, Physics
and Chemistry, Vol. 147, Kluwer Academic Publishers,
2004, pp. 317-326.
[5] N. Rajabov, “Volterra Type Integral Equation with
dary and Interior Fixed Singularity and Super-Sing
Kernels and Their Application,” LAPLAMBERT Aca-
demic Publishing, 2011, 282 p.
Rajabov, “Aboa Type Integral
uations with Bernels,” In Ad-
vances in Applied Mathematics and Approximation The-
ory: Contributions from AMAT2012, Springer, 2012, pp.
341-360.
[7] N. Rajabov, “To Theory One Class Modeling Linear Vol-
terra Type Integral Equation with Boundary Singular Ker-
nels,” Theses of Reports of the 4th International Confer-
ence “Function Spaces. Differential Operators. General
Topology. Problems of Mathematical Education”, PFUR
Publishers, Moscow, 2013, pp. 221-222.
[1] N. Rajabov, “About One Three-Dimensional Volterra
Type Integral Equation with Singular Boundary Surfaces
in Kernels,” Russian of Sc. Dokl, Vol. 409, No. 6, 2006,
pp. 749-753.
[2] N. Rajabov, “On a Volterra Integral Equation,” Doclady
Mathematics, Vol. 65, No. 2, 2002, pp. 217-220.
[3] N. Rajabov, “Sy
Volterra Type with Singular
Ill-Posed and No
Physics and Analysis. Proceedings of the International
N. Rajabov, “About One Class of Volterra Typ
ntegral Equations with anior Fixed Singular or Su-
per-singular Point,” Topics in Analysisand its Applica-
tions, NATO Science Series, II, Ma
Boun-
ularity
[6] N.ut New Class of Volterr
Eqoundary Singularity in K
Copyright © 2013 SciRes. AM