 Applied Mathematics, 2011, 2, 1292-1296 doi:10.4236/am.2011.210179 Published Online October 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. 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 E-mail: fatemi.mehran@yahoo.com, shahmorad@tabrizu.ac.ir 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 . 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 ,d1,,, ,xbqaaDyxfxKxt yttKxt yttqm mxab  , (1)  11=1 d,1, ,bmjjij ijiijaya ybHtyttdim where 0,1mq , fx, 1,Kxt , 2,Kxt and iHt, =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 fx, ,jKxt, j = 1, 2 and iHt, =1,im are continuous, ij, ij, i and id, .=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 mqD , we get  12,d ,dxbmq qmqmqmqaaDDyxD fxDKxtytt DKxtytt ,,d, since then we get the equation  =mq qmDDyxDyx  12,d ,xbmaaDy xFxMxtyttMxtytt  (3) M. FATEMI ET AL. Copyright © 2011 SciRes. AM 1293 where    1122d=dd!d,,d!d,,d!qmxmqaqmxtqmxaxFx D fxfxqmxMxtK txqmxMxtKtxqm,d,d. (4) Now, we write Equation (3) in the general from  ,mDyxGxy, (3.1) and accept that is known, then the fundamen-tal solution (see ) is in the form ,Gxy1.1!mxYx xm  (5) where 1,> ,12, ,0,< ,xxxxd,xd (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,bbmaaDyxYxxGxyYx  (7) where,  12,,d,xbaaGxyF 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,d11,,2bmbs nsms xxasayaDyxYxGxyYx xya   ,.bb (8) Hence, the first expressions for the necessary conditions are obtained in the form       11=011=011,211,2mbmbssms xxasambmbssms xxasayaDyxY xaGxyYxaxyaDyxY 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 =xa 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 ==xb. Here, for =sm1, the the expression of fundamental solution for yields the Heaviside =sm1function. For =xb, =b this becomes 12, 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,bbmxxaaDyxYxxGxyYxx 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:    111 1==0,,1,d(1)1,,2bmbssms mxxxasa,,yabDyxY xGxyYxxyab      (10) and so the second group of the necessary conditions are obtained as 1294 M. FATEMI ET AL.       1211=01211=011,211,2mbmbssms xxxasambmbssms xxxasayaD yxYxaGxyYxaxy 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 11d,mmmxxaaDyxYxxGxyYxx d. Here, once integrating by parts on the left side of the obtained expression gives   11=1d,d.bbmmmxxxa abmxaDyxY xDyxYxxGxyY 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:    1111(1)=,,,d1,,2mbbmmmxxmxa ayaDyxY xGxyY xxya ,.bb (12) Therefore, the last group of necessary conditions will be in the form:     11 11=1111=1,d21,d2bbmm mmxxxa abbmm mmxxxa ayaDyxYxaGxyY xaxybDyxY xbGxyY xbx    ,,1m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.           1111121111211 1 11112 12112d,d,1111,d,bmmmmabmmmmmmmmmmm mmammmmmbayaya yaybybybdHtyttyayayaybybybdHtyttyay ayaybYbay bYbaybGxyY xaxya           11212(1)121111()1 11,d,11=,mmmmmbmabmmmmmayay ayaybybYb ay by baybGxyYxaxyaybybYbaybGxyYxax     (14) Copyright © 2011 SciRes. AM 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 1211112112121210,100 111...00 (1)001m mm mmmmmmmmmmmmYba baYba      (15) Then, from system (14), we g et     1,1=1 =11,1=1 =111d,d11d,dbbmmkskssssssaabbmmksmkssssssaayad HtyttGxyYxaxybd HtyttGxyYxax      ,,,,mskmsmk (16) where (,)pq 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:       ()121212,d ,d,d()dd,dd,dd ,dd,bbxbs saaaabbxbbsssaaaaabbbbbsssaaa aaGxyYxaxF xMxtyttMxt yttYxadxdd.FxYxaxY xaxMxtyttYxaxMxtyttFxYx axyttYxaMxtxyttYx aMxtx     Then, we get:  ,dbbsssaaGxyYx axFMtytt d, (17) and so, 21d,,d ,d,bssabbsssaaFFxYxaxMtYxaMxtxYxaMxtxd,t (18) Finally, coming back to (8), we take into account (16) and (17) and write the second kind Fredholm type inte-gral equation  for which the boundary value problem (1)-(2) is reduced to: ,bayA Btyt  (19) where   ,21111,2 10101,111,101011111mk msmmmmsm smskms skkskskmkms bmmmmsm smskmsskkskskadAYbYb FdYaYaF FxYx x      d, (20) Copyright © 2011 SciRes. AM 1296 M. FATEMI ET AL.    ,2111,2 101 01,111,1010112,= 111()1,d ,d.mk msmmm msm smskms skkksk skmkmsmmmmsm smskms skkkskskbbaadBt YbHtYbMtdYaHt YaMtYx MxtxYxMxtx        (21) By the hypothesis of theorem on the functions fx, ,jKxt , j = 1, 2 and iHt, =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  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  S. G. Samko, A. A. Kilbas and O. I. 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