Applied Mathematics, 2011, 2, 1292-1296
doi:10.4236/am.2011.210179 Published Online October 2011 (http://www.SciRP.org/journal/am)
Existence and Uniqueness of Solution for a
Fractional Order Integro-Differential Equation
with Non-Local and Global Boundary
Conditions
Mehran Fatemi1, Nihan Aliev1, Sedaghat Shahmorad2
1Department of Mat hem at i cs, Baku State University, Baku, Azerbaijan
2Department of Applied Mathematics, University of Tabriz, Tabriz, Iarn
Received February 24 , 20 1 1; r evised September 5, 2011; accept e d September 13, 2011
Abstract
In this paper, we prove an important existence and uniqueness theorem for a fractional order Fredholm –
Volterra integro-differential equation with non-local and global boundary conditions by converting it to the
corresponding well known Fredholm integral equation of second kind. The considered problem in this paper
has been solved already numerically in [1].
Keywords: Fractional Order Integro-Differential Equation, Non-Local Boundary Conditions, Fundamental
Solution
1. Introduction
Let’s consider a problem under boundary condition con-
taining non-local and global terms for a fractional order
integro-differential eq uation
 
  
12
,d ,d
1,,, ,
xb
q
aa
DyxfxKxt yttKxt ytt
qm mxab
 
 

,
(1)





 
11
=1 d,
1, ,
b
mjj
ij ijii
ja
ya ybHtyttd
im
 




where


0,1mq ,


f
x,

1,
K
xt ,


2,
K
xt and


i
H
t, =1,im are continuous, real-valued functions,
ij

, ij

, i

and i, d=1,im, =1,jm are real con-
stants, and boundary conditions (2) are linearly independent.
2. Existence and Uniqueness of Solution
Theorem. Let the functions


f
x,

,
j
K
xt, j = 1, 2 and


i
H
t, =1,im are continuous, ij

, ij

, i

and i d,
.=1ij

(2)
,m are real constant, the boundary conditions (2)
are linearly independent and condition (15) is satisfied.
Then the boundary value problem (1)-(2) has unique
solution.
Proof: Acting in Equation (1) by fractional order de-
rivative operator mq
D

[2], we get
 
12
,d ,d
xb
mq qmqmqmq
aa
DDyxD fxDKxtytt DKxtytt
 





,
,
d,
since then we get the equation
 
=
mq qm
DDyxDyx

  
12
,d ,
xb
m
aa
Dy xFxMxtyttMxtytt 

(3)

M. FATEMI ET AL.
1293
where
 


 


 

11
22
d
=d
d!
d
,,
d!
d
,,
d!
qm
x
mq
a
qm
x
t
qm
x
a
x
Fx D fxf
xqm
x
MxtK t
xqm
x
MxtKt
xqm

,
d,
d.























(4)
Now, we write Equation (3) in the general from
 
,
m
DyxGxy,

(3.1)
and accept that is known, then the fundamen-
tal solution (see [3]) is in the form

,Gxy




1
.
1!
m
x
Yx x
m






 
 (5)
where

1,> ,
12, ,
0,< ,
x
x
x
x













d,x
d
(6)
is Heaviside’s unit function.
Now, we try to get some basic relations. The first of
these relations is Lagrange’s formula. We multiply both
ides of Equation (3) by fundamental solution (5) and
integrate the obtained expression on (see [4,5])
to get

,ab
  
d,
bb
m
aa
DyxYxxGxyYx   

(7)
where,
 
12
,,d,
xb
aa
GxyF xMxt yttMxtytt 

(7.1)
integrating by parts on the left hand side of expression (7)
and taking into account that (5) is a fundamental solution
of (3.1), give the first basic relation in the form
 

  



11
=
=0
,,
1,d1
1,,
2
b
mb
s n
s
ms xxa
sa
ya
DyxYxGxyYx xya


 




 

 



,
.
b
b
(8)
Hence, the first expressions for the necessary conditions are obtained in the form
 

  
 

  
11
=0
11
=0
11,
2
11,
2
mb
mb
ss
ms xxa
sa
mb
mb
ss
ms xxa
sa
yaDyxY xaGxyYxax
yaDyxY xbGxyYxbx





  



 




d,
d,
(9)
It is easy to see that the second expression in (9) turns
into an identity. Indeed, as it is seen from (5)-(6), the
integral at the right side of the second condition contains
the value of the function


,x

  which is zero for
=b

. For =
x
a the the summation in the second ex-
pression contains the Heaviside function which is zero
for =0, 1ms

.
Finally, the first summand contains positive degrees of

x

 for =0, 2sm these terms become zero at
==
x
b

. Here, for =sm1

, the the expression of
fundamental solution for yields the Heaviside =sm1
function. For =
x
b, =b

this becomes 1
2, therefore,
the second one of necessary conditions (9) turns into
identity.
Now, we construct the second basic expression to get
the second group of necessary conditions. For that, we
multiply both sides of (3) by the derivative of (5) and
integrate on


,ab [6,7]:
 
d,
bb
mxx
aa
DyxYxxGxyYxx 

 

d.
Integrating by parts on the left side of the obtained ex-
pression and taking into account (5) and (6), we get the
second basic relation as follows:
 

  



11
1 1
=
=0
,,
1,d(1)
1,,
2
b
mb
ss
ms m
xx
xa
sa
,
,
y
ab
DyxY xGxyYxx
y
ab


 

 



 
 
 


 (10)
and so the second group of the necessary conditions are obtained as

1294 M. FATEMI ET AL.
  

 
  

 
121
1
=0
121
1
=0
11,
2
11,
2
mb
mb
ss
ms xx
xa
sa
mb
mb
ss
ms xx
xa
sa
yaD yxYxaGxyYxax
y bDyxYxbGxyYxbx








  



  




d,
d.
(11)
Similar to the second expression of (9), we can show
that the second expression of (11) turns into identity. If
we continue this process, in order to get the m-th basic
relation, we multiply (3) the


1m

bb
-th order derivative
of (5) and integrate on to get:

,ab


 


11
d,
mm
mxx
aa
DyxYxxGxyYxx 

 

d.
Here, once integrating by parts on the left side of the
obtained expression gives


  



1
1
=
1
d
,d.
b
b
m
mm
xx
xa a
bm
x
a
DyxY xDyxYxx
GxyY xx





 



m
Thus, if we take into account that (5) is the fund amen-
tal solution, the last relation (m-th) will be as follows:


 

  

1
11
1
(1)
=
,,
,d
1,,
2
m
b
b
mm
mxx
m
xa a
ya
DyxY xGxyY xxya


 








,
.
b
b




 (12)
Therefore, the last group of necessary conditions will be in the form:

 

 



 

 


11 1
1
=
111
1
=
1,d
2
1,d
2
b
b
mm m
mxx
xa a
b
b
mm m
mxx
xa a
yaDyxYxaGxyY xax
ybDyxY xbGxyY xbx
 

 

 
 


,
,
1
m

d,
(13)
here, as above, the second necessary condition turns in to
identity.
Now, we join to the given m linearly independent
boundary condition (7), the necessary conditions in (9),
(11) and etc. (13) that are not identities, and write the
system of 2m linear algebraic equations obtained with
respect to the boundary values of the unknown function
in the following way.
 

 

 
 

 

 
 

 

 


 

11
11121111211 1 1
11
12 12
1
12
d,
d,
1111
,d,
b
mm
mm
a
b
mm
mmmmmmmmm mm
a
mmm
mm
b
a
yaya yaybybybdHtytt
yayayaybybybdHtytt
yay ayaybYbay bYbayb
GxyY xax
ya
 
 




  
  
  









 

 


 


 

 



 


112
12
(1)
12111
1()1 1
1,d,
11=,
mmm
mm
b
m
a
b
mmmmm
a
yay ayaybybYb ay b
y baybGxyYxax
yaybybYbaybGxyYxax


















 
 



 




 








(14)

M. FATEMI ET AL.
1295
For solving the system (14) by the Cramer’s rule, it is
necessary that its basic determinant differ from zero. Accept that the following condition is satisfied
 

 
  
11 12111121
1212
12
1
0,
100 111
...
00 (1)001
m m
m mmmmmmm
mmm
m
m
Yba ba
Yba
 
 


 



 


 

(15)
Then, from system (14), we g et

  






  





1,1
=1 =1
1,1
=1 =1
11
d,d
11
d,d
bb
mm
ksks
sss
ss
aa
bb
mm
ksmks
sss
ss
aa
yad HtyttGxyYxax
ybd HtyttGxyYxax





 


 
 
 
 

 





,
,
,
,
msk
msmk 
(16)
where (,)
p
q
 denotes the cofactor of the elements at the
intersection of p-th row and q-th column of the determi-nant

. Calculate the following expression:
 
 

 

 
 

 


()
12
12
12
,d ,d,d()
dd,dd,
dd ,dd,
bbxb
s s
aaaa
bbxbb
ss
s
aaaaa
bbbbb
ss s
aaa aa
GxyYxaxF xMxtyttMxt yttYxadx
d
d.
F
xYxaxY xaxMxtyttYxaxMxtytt
F
xYx axyttYxaMxtxyttYx aMxtx

 


 
 


 
Then, we get:


 
,d
bb
sss
aa
GxyYx axFMtytt

 d, (17)
and so,








21
d,
,d ,d,
bs
sa
bb
ss
saa
FFxYxax
M
tYxaMxtxYxaMxt










x
d,t
(18)
Finally, coming back to (8), we take into account (16)
and (17) and write the second kind Fredholm type inte-gral equation [8] for which the boundary value problem (1)-(2) is reduced to:

,
b
a
yA Btyt   
 (19)
where
 















 
,2
11
11
,2 1
0101
,
11
1,1
0101
11
11
mk ms
mmmm
sm sm
skms s
kk
sksk
mkms b
mmmm
sm sm
skmss
kk
sksk
a
d
AYbYb F
d
YaYaF FxYx x
 



 





 




  


 





d,

(20)

1296 M. FATEMI ET AL.
 



 













 
,2
11
1
,2 1
01 01
,
11
1,1
0101
12
,= 11
1()1
,d ,d.
mk ms
mmm m
sm sm
skms s
kkk
sk sk
mkms
mmmm
sm sm
skms s
kkk
sksk
bb
aa
d
Bt YbHtYbMt
d
YaHt YaMt
Yx MxtxYxMxtx
 







 


 




 


 

 
 



 (21)
By the hypothesis of theorem on the functions


f
x,

,
j

K
xt , j = 1, 2 and


i
H
t, =1,im the integral
Equation (19) has unique solution and so in all con-
ducted operations we can come back and we conclude
that the solution of (19) is the unique solution of
boundary value problem (1)-(2).
3. References
[1] D. Nazari and S. Shahmorad, “Application of Fractional
Differential Transform Method to the Fractional Order
Integro-Differential Equations with Nonlocal Boundary
Conditions,” Journal of Computational and Applied
Mathematics, Vol. 234, No. 3, 2010, pp. 883-891.
doi:10.1016/j.cam.2010.01.053
[2] S. G. Samko, A. A. Kilbas and O. I. Marichev, “Frac-
tional Integrals and Derivatives,” Theory and Applica-
tions, Cordon and Breach, Yverdon, 1993.
[3] C. J. Tranter, “Integral Transforms in Mathematical Phys-
ics,” Methuen, London and New York, 1949.
[4] V. S. Vladimirov, “Equation of Mathematical Physics,”
Mir Publication, Moscow, 1984.
[5] G. E. Shilov, “Mathematical Analysis. The Second Spe-
cial Course,” Nauka, Moscow, 1965.
[6] S. M. Hosseini and N. A. Aliev, “Sufficient Conditions
for the Reduction of a BVP for PDE with Non-Local and
Global Boundary Conditions to Fredholm Integral Equa-
tions (on a Rectangular Domain),” Applied Mathematics
and Computation, Vol. 147, No. 3, 2004, pp. 669-685.
[7] F. Bahrami, N. Aliev and S. M. Hosseini, “A Method for
the Reduction of Four Imensional Mixed Problems with
General Boundary Conditions to a System of Second
Kind Fredholm Integral Equations,” Italian Journal of
Pure and Applied Mathematics, No. 17, 2005, pp. 91-
104.
[8] N. Aliev and M. Jahanshehi, “Solution of Poissoins Equa-
tion with Global, Local and Non-Local Boundary Condi-
tions,” International Journal of Mathematical Education
in Science and Technology, Vol. 33, No. 2, 2002, pp.
241-247. doi:10.1080/00207390110097551