IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011 Solving System of Linear Fredholm Integral Equations of Second Kind Using Open Newton-Cotes Formulas G. H. Ibraheem Department of Mathematics, College of Education, Ibn-Al-Haitham, University of Baghdad Received in: 27,October, 2010 Accepted in: 28, February, 2011 Abstract In this paper, the linear system of Fredholm integral equations is solving using Open Newton-Cotes formula, which we use five different types of Open Newton-Cotes formula to solve this system. Compare the results of suggested method with the results of another method (closed Newton-Cotes formula) Finally, at the end of each method, algorithms and programs developed and written in MATLAB (version 7.0) and we give some numerical examples, illustrate suggested method. Keyword: Open Newton – Cotes formula, Closed Newton-Cotes formula, System of linear Fredholm integral equation. Introduction The problem of Newton-Cotes formula arises when the integration cannot be carried out exactly or when the function is known only at a finite number of data. Furthermore, Newton- Cotes rules are primary tool used by engineers and scientists to obtain approximate answers for definite integrals that cannot be solved analytically. [ 1 ] This paper is organized as follows: in Section 2, we introduce a brief introduction to the Open Newton-Cotes and some basic definitions for integral equation. In Section 3, we construct our methods to approximate the solution of linear system of Fredholm integral equations. Numerical examples are given in Section 4. 1- Review and Background 1.1 Some definitions of integral equation Definition 1-1: [ 2 ] Integral equation is an equation in which the unknown function appears under an integral sign. A general form of linear integral equations may be written as follows: IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011 bxadttutxkxfxuxh xb a   )( )(),()()()(  ………………..(1) Where h(x) and f(x)are given function of x, t)k(x, is a function of two variables x and t called kernel of integral equation which are also known, while u(x) is to be determined and  is a scalar parameter [in this paper we take 1 ]. Definition 1-2: [ 2 ] If the function h(x) =1, then the linear integral equation (1) is said to be an equation of the second kind (i.e.) bxadttutxkxfxu xb a   )( )(),()()( ……………………..(2) Definition 1-3: [ 2 ] The integral equation (1) is called Fredholm integral equation (FIE) if b b(x)  , where b is constant such that ab  . Therefore, the integral equations   b a bxadttutxkxf )(),()( bxadttutxkxfxu b a   )(),()()( Represent the one-dimensional Fredholm integral equation of the first and second kind respectively. 1.2 Open Newton-Cotes formula In numerical analysis, the Newton-Cotes (N-C) formulas are a group of formulas of numerical integration based on evaluating the integrand at equally- spaced points. They are named after Isaac Newton and Roger Cotes [ 3 ]. There are two types of N-C formulas: - The (closed) type which uses the function value at all point in the domain. - The (open) type which does not use the function value at the initial and end point of the domain . Numerical integration formulas of the form )()()()( 00 fExfwdxxfdxxf b a n i x x ii n     where )( fE is the error, niforhixxi ,...,1,0*0  , and niwi ,...,1,0 are the weights, is called closed Newton-Cotes formulas Some of the common closed N-C formulas are as follows: - Trapezoidal rule , Simpson 1/3 rule and Simpson 3/8 rule IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011 Numerical integration formulas of the form )()()()( 0 1 1 fExfwdxxfdxxf b a n i x x ii n       where )( fE is the error, 1,...,0,1*)1(0  niforhixxi , and niwi ,...,1,0 are the weights, is called open Newton-Cotes formulas. Some of the common open N-C formulas with their error terms are a follows: [ 1 ] - n=0 M idpoint rule ),()( 3 )(2)( 11 3 0 1 1 xxwheref h xhfdxxf x x     - n=1 ),()( 4 3 )]()([ 2 3 )( 21 3 10 2 1 xxwheref h xfxf h dxxf x x     - n=2 ),()( 45 14 )](2)()(2[ 3 4 )( 31 )4( 5 210 3 1 xxwheref h xfxfxf h dxxf x x     - n=3 ),()( 144 95 )](11)()()(11[ 24 5 )( 41 )4( 5 3210 4 1 xxwheref h xfxfxfxf h dxxf x x     - n=4 ),()( 1404 4 )](11)(14)(26)(14)(11[ 10 3 )( 51 )6( 7 43210 5 1 xxwheref h xfxfxfxfxf h dxxf x x     2- Solution of a system of linear Fredholm integral equations of the second kind Using Open N-C formulas In this section, we use the common formula of Open N-C to solve the system of linear Fredholm integral equations of the second kind. Consider the system of linear Fredholm integral equations of the second kind nrdtxutxkxfxu s n s b a rsrr ,...,2,1,)(),()()( 1     ……(3) where nsrkfNn rsr ,...,2,1,,,,  are assumed to be continuous function. Suppose that the interval [a, b] is divided into n+2 equal subintervals of length 2   n ab h , such that 1,1 ,   n xbxa with nihiαx i ,...,1,0,*  this implies that hax 0 and hbxn  and )( ir xu for nrni ,...,2,1,,...,1,0  can be determined by: nrnidttutxkxfxu n s b a sirsirir ,,2,1,,,1,0,)(),()()( 1     ………….(4) Thus, we are approximating each integral term by the open N-C formulas. 2.1 Using Open N-C: with n=0 (Midpoint Rule) We replace the integral term that appeared in the right hand side of the above equation by the composite midpoint rule which illustrate in the following theorem: [ 4 ] IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011 Theorem: Let ],[2 baCf  . With )22/()(  mabh and hjαxj )1(  12,...,0,1,  mj , the midpoint rule for n=2m subintervals is: )( 6 )(2)( 20 0 2 fh ab xfhdxxf b a m j j      for some ),( ba . If the number of subinterval is even we apply open N-C with (n=1) rule. Therefore 2/,,1,0,,,2,1, 2 nmwhereminru ir   are obtained by solving the equation:   nsnrniuxxkhfu jii s n j jirsrr ,,2,1,,,2,1, 2 ,,1,0,),(2 2 2/ 0 22     )5( where ir u 2 denote the numerical solution at , m=n/2, Transform all the terms involving the solution ir u 2 2/,,1,0,,,2,1, nminr   to left side of the equation (5) and irf 2 to the right side.                                   ),(21),(2),(2),(2),(2),(2 ),(2),(21),(2),(2),(2),(2 ),(2),(2),(21),(2),(2),(2 ),(2),(2),(2),(21),(2),(2 ),(2),(2),(2),(2),(21),(2 ),(2),(2),(2),(2),(2),(21 1011101 1110111111011 0100001101001 111011 111 101 1 1111101111 1111 1011 1 0110100101 1101 1001 1 mmnnmnnmnnmmnmnmn mnnnnnnmnnn mnnnnnnmnnn mmnmnmnmmmm mnnnm mnnnm xxhkxxhkxxhkxxhkxxhkxxhk xxhkxxhkxxhkxxhkxxhkxxhk xxhkxxhkxxhkxxhkxxhkxxhk xxhkxxhkxxhkxxhkxxhkxxhk xxhkxxhkxxhkxxhkxxhkxxhk xxhkxxhkxxhkxxhkxxhkxxhk                                                                                                   nm m m n nm m m n f f f f f f u u u u u u       2 1 1 21 11 2 1 1 21 11 …(6) Remark: equation (6) has unique solution if the determent of the matrix K not equal to zero Also, if the number of subintervals is odd, we get combination between open N-C: n=1 and open N-C: n=0: Midpoint rules. Therefore mninru ir  )2(,,2,1,,,2,1,  where 2/)1(  nm are obtained by solving the equations: 2/)1()2(,,2,1,,,2,1,,,2,1 2 2 3 2 3 )2( 3 2211      nmwheremninsnrfor ukhuk h uk h fu mn j srssrssrsrr jijiiii  ……(7) Transform all the terms involving the solution mninru ir  )2(,,2,1,,,2,1,  where 2/)1(  nm to left side of the equation (7) and ir f to the right side. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011 Finally, each of system in equations (6) and (7) can be written in matrix form as FKU  where K is the matrix of the coefficients, U is the matrix of solution and F is the matrix of non-homogeneous part. To find the approximate solution nru ir ,,2,1,  we find FKU 1 . The Algorithm of Numerical Solution of a S ystem of linear Fredholm Integral Equations Using Open N-C: {n=0} (SONCn0) Step 1: compute 2   n ab h , Nn Step 2:  Compute 2/,,1,0,,,2,1, 2 nminru ir   , using equation (6) when the number of subintervals is even.  Compute mninru ir  )2(,,2,1,,,2,1,  where 2/)1(  nm , using equation (7) when the number of subintervals is odd. Step 3: solve the resulting system by multiplication it with K-1. 2.2 Using Open N-C: n=1 Rule By the same steps of condition on equation (4), use the open N-C where n=1 formula to approximate each integral term in equation (4). If the number of subintervals is (a multiple of three). Therefore mninru ir  )2(,,2,1,,,2,1,  where 3/)2(  nm are obtained by solving the equations: we apply open N-C with (n=1). Therefore nru ir ,,2,1,  mni  )2(,,2,1,  , where 3/)2(  nm are obtained by solving the equations: where 3/)2(  nm   nsnrmniuxxk h fu jii s mn j jirsrr ,,2,1,,,2,1,)2(,,2,1,),( 2 3 )2( 1      …..(8) where ir u denote the numerical solution at , 3/)2(  nmwhere Transform all the terms involving the solution m-2)+(n,…1,2,=i,,,2,1, nru ir  to left side of the equation (8) and i fr to the right side. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011                                                   ),( 2 3 1),( 2 3 ),( 2 3 ),( 2 3 ),( 2 3 ),( 2 3 ),( 2 3 ),( 2 3 1),( 2 3 ),( 2 3 ),( 2 3 ),( 2 3 ),( 2 3 ),( 2 3 ),( 2 3 1),( 2 3 ),( 2 3 ),( 2 3 ),( 2 3 ),( 2 3 ),( 2 3 ),( 2 3 1),( 2 3 ),( 2 3 ),( 2 3 ),( 2 3 ),( 2 3 ),( 2 3 ),( 2 3 1),( 2 3 ),( 2 3 ),( 2 3 ),( 2 3 ),( 2 3 ),( 2 3 ),( 2 3 1 2112111 2221221221121 1211111211111 1211111211111 2122112121122111211 1121111111121111111 mmnnmnnmnnmmnmnmn mnnnnnnmnnn mnnnnnnmnnn mmnmnmnmmmm mnnnm mnnnm xxhkxxhkxxhkxxhkxxhkxxhk xxhkxxhkxxhkxxhkxxhkxxhk xxhkxxhkxxhkxxhkxxhkxxhk xxhkxxhkxxhkxxhkxxhkxxhk xxhkxxhkxxhkxxhkxxhkxxhk xxhkxxhkxxhkxxhkxxhkxxhk                                                                                                   nm m m n nm m m n f f f f f f u u u u u u       2 1 1 21 11 2 1 1 21 11 ….....(9) Also, if the number of subintervals is (a multiple of three +1), we get combination between open N-C : n=0 Midpoint and open N-C: n=1 rules. Therefore mninru ir  )1(,,2,1,,,2,1,  where 3/)1(  nm are obtained by solving the equation: 3/)1()2(,,2,1,,,2,1,,,2,1 2 3 2 )2( 2 11      nmwheremninsnrfor uk h uhkfu mn j srssrsrr jijiii  …….(10) Transform all the terms involving the solution mninru ir  )1(,,2,1,,,2,1,  , where 3/)1(  nm to left side of the equation (10) and ir f to the right side. if the number of subintervals is (a multiple of three +2), we get combination between open N- C : n=0 (Midpoint method) and open N-C : n=1 rules. Therefore mninru ir  )1(,,2,1,,,2,1,  , where 3/nm are obtained by solving the equation: 3/)1(,,2,1,,,2,1,,,2,1 2 3 22 )2( 3 2211 nmwheremninsnrfor uk h uhkuhkfu mn j srssrssrsrr jijiiii       ……….(11) Transform all the terms involving the solution mninru ir  )1(,,2,1,,,2,1,  where 3/nm to left side of the equation (11) and ir f to the right side. Finally, each system in equations (9), (10) and (11) can be written in a matrix form as FKU  where K is the matrix of the coefficients, U is the matrix of solution and F is the matrix of non-homogeneous part. To find the approximate solution nrur i ,,2,1,  we find FKU 1 The Algorithm of Open N-C {n=1} (SONCn1) IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011 Step 1: compute 2   n ab h , Nn Step 2:  Compute m-2)+(n,…1,2,=i,,,2,1, nru ir  where 3/)2(  nm , using equation (9) when the number of subintervals is (a multiple of 3).  Compute mninru ir  )1(,,2,1,,,2,1,  where 3/)1(  nm , using equation (10) when the number of subintervals is (a multiple of 3 +1).  Compute mninru ir  )1(,,2,1,,,2,1,  where 3/nm , using equation (11) when the number of subintervals is (a multiple of 3 +2). Step 3: solve the resulting system by multiplication it with K -1. 2.3 Using Open N-C: n=2 Rule The open N-C: n=2 formula can be used to approximate equation (4) such that: If the number of subintervals is (a multiple of four), we apply the open N-C: n=2 formula to each integral term in equation (4) as the form:   4/)2()2(,,2,1,,,2,1,,,2,1 2222 3 4 )2())2(()1())1((44332211    nmwheremninsnrfor ukukukukukuk h fu mnmnimnmniiiiiii srssrssrssrssrssrsrr   ……(12) Transform all the terms involving the solution m-2)+(n,…1,2,=i,,,2,1, nru ir  where 4/)2(  nm to the left side of the equation (12) and ir f to the right side. Also, if the number of subintervals (n+2) is (a multiple of four +1), we get combination between open N-C: n=0, open N-C: n=1 and open N-C: n=2 rules and mnimru ir  )1(,2,1,,,2,1,  where 4/)1(  nm are obtained by solving the system of equations: 4/)1()1(,,2,1,,,2,1,,,2,1 ]22[ 3 4 2 3 2 3 2 )1())1(()(5544332211    nmwheremninsnrfor ukukukuk h uk h uk h uhkfu mnmnimnmniiiiiiii srssrssrssrssrssrssrsrr   ...(13) Transform all the terms involving the solution m-1)+(n,…1,2,=i,,,2,1, nru ir  , where 4/)1(  nm to the left side of the equation (13) and ir f to the right side. If the number of subintervals (n+2) is (a multiple of four +2), we get combination between open N-C: n=0 and open N-C: n=2 rules and mnimru ir  )1(,2,1,,,2,1,  , where 4/nm are obtained by solving the system of equations. 4/)1(,,2,1,,,2,1,,,2,1 ]222[ 3 4 2 )1())1(()(44332211 nmwheremninsnrfor ukukukukuk h uhkfu mnmnimnmniiiiiii srssrssrssrssrssrsrr      ………….(14) Transform all the terms involving the solution m-1)+(n,…1,2,=i,,,2,1, nru ir  where 4/nm  to the left side of the equation (14) and ir f to the right side. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011 ]Also, if the number of subintervals (n+2) is (a multiple of four +3) we get combination between open N-C: n=1 and open N-C: n=2 rules and mnimru ir  )1(,2,1,,,2,1,  , where 4/)1(  nm are obtained by solving the system of equations. 4/)1()1(,,2,1,,,2,1,,,2,1 ]22[ 3 4 2 3 2 3 )1())1(()(44332211    nmwheremninsnrfor ukukukuk h uk h uhk h fu mnmnimnmniiiiirsii srssrssrssrssrssrr   …..….(15) Transform all the terms involving the solution m-1)+(n,…1,2,=i,,,2,1, nru ir  where 4/)1(  nm to the left side of the equation (15) and ir f to the right side. Finally, each system in equations (12), (13), (14) and (15) can be written in a matrix form as FKU  , where K is the matrix of the coefficients, U is the matrix of solution and F is the matrix of non-homogeneous part. To find the approximate solution nru ir ,,2,1,  , we find FKU 1 The Algorithm of Open N-C:{n=2} (SONCn2) Step 1: compute 2   n ab h , Nn Step 2:  Compute m-2)+(n,…1,2,=i,,,2,1, nru ir  where 4/)2(  nm , using equation (12) when the number of subintervals is (a multiple of 4).  Compute mninru ir  )1(,,2,1,,,2,1,  where 4/)1(  nm , using equation (13) when the number of subintervals is (a multiple of 4 +1).  Compute mninru ir  )1(,,2,1,,,2,1,  where 4/nm  , using equation (14) when the number of subintervals is (a multiple of 4 +2).  Compute mninru ir  )1(,,2,1,,,2,1,  where 4/)1(  nm , using equation (15) when the number of subintervals is (a multiple of 4 +3). Step 3: solve the resulting system by multiplication it with K -1 . 2.4 Using Open N-C: n=3 Rule The open N-C: n=3 formula can be used to approximate equation (4) such that: If the number of subintervals is (a multiple of five) we apply the open N-C: n=3 formula to each integral term in equation (4) as the form:   5/)2()2(,,2,1,,,2,1,,,2,1 111111 24 5 )2())2(()1())1((44332211    nmwheremninsnrfor ukukukukukuk h fu mnmnimnmniiiiiii srssrssrssrssrssrsrr   ……….(16) Transform all the terms involving the solution m-2)+(n,…1,2,=i,,,2,1, nru ir  , where 5/)2(  nm to the left side of the equation (16) and ir f to the right side. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011 Also, if the number of subintervals (n+2) is (a multiple of five +1) we get combination between open N-C: n=1 and open N-C: n=3 rules and mnimru ir  )1(,2,1,,,2,1,  , where 5/)1(  nm are obtained by solving the system of equations. 5/)1()1(,,2,1,,,2,1,,,2,1 ]1111[ 24 5 2 3 2 3 2 3 2 3 )1())1(()(5544332211    nmwheremninsnrfor ukukuk h uk h uk h uk h uhk h fu mnmnimnmniiiiiiii srssrssrssrssrssrssrsrr   ...(17) Transform all the terms involving the solution m-1)+(n,…1,2,=i,,,2,1, nru ir  , where 5/)1(  nm to the left side of the equation (17) and ir f to the right side. If the number of subintervals (n+2) is (a multiple of five +2) we get combination between open N-C: n=0 and open N-C: n=3 rules and mnimru ir  )1(,2,1,,,2,1,  , where 5/nm are obtained by solving the system of equations. 5/)1(,,2,1,,,2,1,,,2,1 ]1111[ 24 5 2 )1())1(()(44332211 nmwheremninsnrfor ukukukukuk h uhkfu mnmnimnmniiiiiii srssrssrssrssrssrsrr      …….(18) Transform all the terms involving the solution m-1)+(n,…1,2,=i,,,2,1, nru ir  where 5/nm  to the left side of the equation (18) and ir f to the right side. Also, if the number of subintervals (n+2) is (a multiple of five +3) we get combination between open N-C: n=1 and open N-C: n=3 rules and mnimru ir  )1(,2,1,,,2,1,  , where 5/)1(  nm are obtained by solving the system of equations. 5/)1()1(,,2,1,,,2,1,,,2,1 ]1111[ 24 5 2 3 2 3 )1())1(()(44332211    nmwheremninsnrfor ukukukuk h uk h uhk h fu mnmnimnmniiiiiii srssrssrssrssrssrsrr   …….(19) Transform all the terms involving the solution m-1)+(n,…1,2,=i,,,2,1, nru ir  , where 5/)1(  nm to the left side of the equation (19) and ir f to the right side. And, if the number of subintervals (n+2) is (a multiple of five +4) we get combination between open N-C: n=2 and open N-C: n=3 rules and mnimru ir  )1(,2,1,,,2,1,  , where 5/)2(  nm are obtained by solving the system of equations. 5/)2()1(,,2,1,,,2,1,,,2,1 ]1111[ 24 5 ]22[ 3 4 )1())1(()(44332211    nmwheremninsnrfor ukukuk h ukukuhk h fu mnmnimnmniiiiiii srssrssrssrssrssrsrr   …(20) Transform all the terms involving the solution m-1)+(n,…1,2,=i,,,2,1, nru ir  , where 5/)2(  nm to the left side of the equation (20) and ir f to the right side. Finally, each system in equations (16), (17), (18), (19) and (20) can be written in a matrix form as FKU  where K is the matrix of the coefficients, U is the matrix of solution and F IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011 is the matrix of non-homogeneous part. To find the approximate solution nru ir ,,2,1,  we find FKU 1 The Algorithm of Open N-C {n=3} (SONCn3) Step 1: compute 2   n ab h , Nn Step 2:  Compute m-2)+(n,…1,2,=i,,,2,1, nru ir  where 5/)2(  nm , using equation (16) when the number of subintervals is (a multiple of 5).  Compute mninru ir  )1(,,2,1,,,2,1,  where 5/)1(  nm , using equation (17) when the number of subintervals is (a multiple of 5 +1).  Compute mninru ir  )1(,,2,1,,,2,1,  where 5/nm , using equation (18) when the number of subintervals is (a multiple of 5 +2).  Compute mninru ir  )1(,,2,1,,,2,1,  where 5/)1(  nm , using equation (19) when the number of subintervals is (a multiple of 5 +3).  Compute mninru ir  )1(,,2,1,,,2,1,  where 5/)2(  nm , using equation (20) when the number of subintervals is (a multiple of 5 +4). Step 3: solve the resulting system by multiplication it with K -1. 2.4 Using Open N-C: n=4 Rule The open N-C: n=4 formula can be used to approximate equation (4) such that: If the number of subintervals is (a multiple of six) we apply the open N-C: n=4 formula to each integral term in equation (4) as the form:   6/)2()2(,,2,1,,,2,1,,,2,1 11141114261411 10 3 )2())2(()1())1((5544332211    nmwheremninsnrfor ukukukukukukuk h fu mnmnimnmniiiiiiii srssrssrssrssrssrssrsrr   ….(21) Transform all the terms involving the solution m-2)+(n,…1,2,=i,,,2,1, nru ir  , where 6/)2(  nm to the left side of the equation (21) and ir f to the right side. Also, if the number of subintervals (n+2) is (a multiple of six +1) we get combination between open N-C: n=0, open N-C: n=3, and open N-C: n=4 rules and mnimru ir  )1(,2,1,,,2,1,  , where 6/)1(  nm are obtained by solving the system of equations. 5/)1()1(,,2,1,,,2,1,,,2,1 ]111411[ 10 3 1111[ 24 5 2 )1())1(()(665544332211 ]    nmwheremninsnrfor ukukuk h ukukukuk h uhkfu mnmnimnmniiiiiiiii srssrssrssrssrssrssrssrsrr   ....(22) Transform all the terms involving the solution m-1)+(n,…1,2,=i,,,2,1, nru ir  , where 6/)1(  nm to the left side of the equation (22) and ir f to the right side. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011 If the number of subintervals (n+2) is (a multiple of six +2) we get combination between open N-C: n=0 and open N-C: n=4 rules and mnimru ir  )1(,2,1,,,2,1,  where 6/nm are obtained by solving the system of equations. 6/)1(,,2,1,,,2,1,,,2,1 ]1114261411[ 10 3 2 )1())1(()(44332211 nmwheremninsnrfor ukukukukuk h uhkfu mnmnimnmniiiiiii srssrssrssrssrssrsrr      .….(23) Transform all the terms involving the solution m-1)+(n,…1,2,=i,,,2,1, nru ir  , where 6/nm  to the left side of the equation (23) and ir f to the right side. Also, if the number of subintervals (n+2) is (a multiple of six +3) we get combination between open N-C: n=1 and open N-C: n=4 rules and mnimru ir  )1(,2,1,,,2,1,  , where 6/)1(  nm are obtained by solving the system of equations. 6/)1()1(,,2,1,,,2,1,,,2,1 ]11141411[ 10 3 2 3 2 3 )1())1(()(44332211    nmwheremninsnrfor ukukukuk h uk h uhk h fu mnmnimnmniiiiiii srssrssrssrssrssrsrr   ….(24) Transform all the terms involving the solution m-1)+(n,…1,2,=i,,,2,1, nru ir  where 6/)1(  nm to the left side of the equation (24) and ir f to the right side. If the number of subintervals (n+2) is (a multiple of six +4) we get combination between open N-C: n=2 and open N-C: n=4 rules and mnimru ir  )1(,2,1,,,2,1,  , where 6/)2(  nm are obtained by solving the system of equations. 6/)2()1(,,2,1,,,2,1,,,2,1 ]111411[ 10 3 ]22[ 3 4 )1())1(()(44332211    nmwheremninsnrfor ukukuk h ukukuhk h fu mnmnimnmniiiiiii srssrssrssrssrssrsrr   ….(25) Transform all the terms involving the solution m-1)+(n,…1,2,=i,,,2,1, nru ir  , where 6/)2(  nm to the left side of the equation (25) and ifr to the right side. Also, if the number of subintervals (n+2) is (a multiple of six +5) we get combination between open N-C: n=3 and open N-C: n=4 rules and mnimru ir  )1(,2,1,,,2,1,  , where 6/)3(  nm are obtained by solving the system of equations. 6/)3()1(,,2,1,,,2,1,,,2,1 ]111411[ 10 3 ]1111[ 24 5 )1())1(()(5544332211    nmwheremninsnrfor ukukuk h ukukukuhk h fu mnmnimnmniiiiiiii srssrssrssrssrssrssrsrr   ....(26) Transform all the terms involving the solution m-1)+(n,…1,2,=i,,,2,1, nru ir  , where 6/)3(  nm to the left side of the equation (26) and ir f to the right side. Finally, each system in equations (21), (22), (23), (24), (25) and (26) can be written in a matrix form as FKU  where K is the matrix of the coefficients, U is the matrix of solution IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011 and F is the matrix of non-homogeneous part. To find the approximate solution nru ir ,,2,1,  we find FKU 1 The Algorithm of Open N-C {n=4} (SONCn4) Step 1: compute 2   n ab h , Nn Step 2:  Compute m-2)+(n,…1,2,=i,,,2,1, nru ir  where 6/)2(  nm , using equation (21) when the number of subintervals is (a multiple of 6).  Compute mninru ir  )1(,,2,1,,,2,1,  where 6/)1(  nm , using equation (22) when the number of subintervals is (a multiple of 6 +1).  Compute mninru ir  )1(,,2,1,,,2,1,  where 5/nm , using equation (23) when the number of subintervals is (a multiple of 6 +2).  Compute mninru ir  )1(,,2,1,,,2,1,  where 6/)1(  nm , using equation (24) when the number of subintervals is (a multiple of 6 +3).  Compute mninru ir  )1(,,2,1,,,2,1,  where 6/)2(  nm , using equation (25) when the number of subintervals is (a multiple of 6 +4).  Compute mninru ir  )1(,,2,1,,,2,1,  where 6/)2(  nm , using equation (26) when the number of subintervals is (a multiple of 6 +5). Step 3: solve the resulting system by multiplication it with K -1. 3- Numerical Examples In this section, we test some of the numerical examples performed to solving this linear system of Fredholm integral equations. The exact solution is used only to show the accuracy of the numerical solution which obtained with our method. Example (1): Consider the problem:          1 0 21 2 2 1 0 211 )()(1 12 19 )( )()( 3 )( 36 17 18 )( dttutuxtxxxu dttutu txx xu which is a system of two linear FIE's, with exact solution is [5]: 1)(,1)( 2 21  xxuxxu Tables (1) and (2) present a comparison between the exact and numerical solution of four types of Open Newton – Cotes and two types of Closed Newton – Cotes for u1 and u2 respectively depending on least square error and running time with h=1/16. Example (2): Consider the problem:       1 0 3 1 0 2 1 0 1 2 3 1 0 3 1 0 2 1 0 1 22 2 1 0 3 2 1 0 2 1 0 1 2 1 )()()(3)()1( 6 11 12 7 )( )()()()2()()( 6 7 3 5 3 2 )( )()()()( 30 29 3 )( dttuxtdttudttuxxxxu dttuxdttutdttuxxxxu dttutdttutdttutx x xxu IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011 which is a system of three linear FIE's, with exact solution is 2 3 2 2 2 1 1)(,)(,)( xxuxxxuxxu  Tables (3)- (5) present a comparison between the exact and numerical solution of four types of Open Newton – Cotes and three types of Closed Newton – Cotes for u1, u2 and u3 respectively depending on least square error and running time with h= 0.1 Example (3): Consider the problem:     1 0 2 2 1 0 1 24 2 1 0 2 2 1 0 1 2 1 )()()( 12 7 5 1 )( )()()1(1 12 25 6 5 )( dttuxtxdttuxtxxxxu dttutxdttutxxxxu which is a system of two linear FIE's, with exact solution is [6 ]: 4 2 2 1 )(,1)( xxuxxu  Tables (6) and (7) present a comparison between the exact and numerical solution of four types of Open Newton – Cotes and two types of Closed Newton – Cotes for u1 and u2 respectively depending on least square error and running time with h=1/18 Conclusion In this paper we suggest open N-C formula to solve system of linear Fredholm integral equations of the second kind and we obtain the following results: 1- The results obtained using open N-C formulas are more accurate than the results obtained using closed N-C formulas in general. 2- Open N-C formulas are more efficient than closed N-C formulas since the open N-C formulas have most results than closed N-C with fewer nodes in the open N-C formulas. 3- The results obtained in open N-C formula when n=4 or multiple of four is most of the results in a short time in other open N-C formulas. References 1.Mathews, J.H. and Fink, K.D. (1999), "Numerical Methods Using MATLAB "; third edition, Prenice-Hall, Inc. 2.Chambers, L.I.G. (1976), “Integral Equation: A Short Course”; International Textbook Company Limited. 3. Lapidus, L. and Seinfeld, J. H. (1971) “Numerical Solution of Ordinary Differential Equations”; Academic Press New-York and London. 4.Burden, R. L. and Faires, J. D. (2006), “Numerical Analysis”; Eighth Edition, An International Thomson Publishing Company. 5.Vahidi, A.S. R. and Mokhtari, M. (2008), “On the Decomposition Method for System of Linear Fredholm Integral Equations of the Second Kind”; Applied Mathematical Sciences, Iran, 2 :(2) 57-62. 6.Al-Dulaimi, M. Y. (2008),”Some Modified Quadrature Rules for Solving System of Fredholm Linear Integral Equations”; M.Sc. thesis, University of Baghdad, Ibn-Al-Haitham College of Education. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011 Table (1) A comparison between the exact and numerical solution of 4 types of Open N – C and 2 types of Closed N– C for u1 Table (2) A comparison between the exact and numerical solution of 4 types of Open N – C and 2 types of Closed N– C for u2 Closed N-C Simp. 3/8 Closed N-C Trap. Open N-C n=4 Open N-C n=3 Open N-C n=2 Open N-C n=1 Open N-C n=0 Exact u1 x 1.06254649 1.06408126 1.0625000 1.06135355 1.06250000 1.05802317 61.0593592 1.0625000 0.62500 1.12505018 1.12668959 1.1250000 1.12376887 01.1250000 1.1250000 0.12500 1.31256126 1.31451459 1.3750000 1.37500000 1.36899583 1.3750000 0.37500 1.50007234 1.50233959 1.5000000 1.49826084 1.49338489 1.5000000 0.50000 1.75008711 1.75277293 1.7500000 1.74792216 1.74216302 1.7500000 0.75000 1.87509449 1.87798960 1.8750000 1.87275281 1.86655208 1.8750000 0.87500 1.038e-008 1.028e-005 1.972e-031 e-0065.437 e-0311.972 e-0057.662 e-0053.789 L.S.E. 0.297000 0.31200 0.18800 0.172000 0.203000 0.12500 0.07800 Time Closed N-C Simp. 3/8 Closed N-C Trap. Open N-C n=4 Open N-C n=3 Open N-C n=2 Open N-C n=1 Open N-C n=0 Exact u2 x 1.00391427 1.00418242 1.00390625 1.00370716 1.00390625 1.00312412 1.00335774 1.0039062 0.62500 1.01564105 1.01617734 1.0156250 1.01522683 1.01562500 1.0156250 0.12500 1.14067315 1.14228204 1.1406250 1.14062500 1.13593222 1.1406250 0.37500 1.25006420 1.25220939 1.2500000 1.24840733 1.24374297 1.2500000 0.50000 1.56259630 1.56581408 1.5625000 1.56011100 1.55311445 1.5625000 0.75000 1.76573735 1.76949143 1.7656250 1.76283783 1.76562500 1.75467520 1.7656250 0.87500 1.648e-008 1.952e-005 0 e-0068.917 e-0324.930 e-0041.376 e-0056.769 L.S.E. 0.297000 0.31200 0.18800 0.172000 0.203000 0.12500 0.07800 Time IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011 Table (3) A comparison between the exact and numerical solution of 4 types of Open N – C and 2 types of Closed N– C for u1 Table (4) A comparison between the exact and numerical solution of 4 types of Open N – C and 2 types of Closed N– C for u2 Closed N-C Simp. 3/8 Closed N-C Trap. Open N-C n=4 Open N-C n=3 Open N-C n=2 Open N-C n=1 Open N-C n=0 Exact u1 x 0.00853472 -0.0123549 0.00985664 0.00939200 0.01634641 0.07564343 0.05805579 0.010000 0.1 0.08810184 0.06074898 0.08980885 0.08918933 0.09823966 0.17608013 0.15296328 0.090000 0.3 0.24766898 0.21385289 0.24976106 0.26013291 0.35651683 0.32787077 0.250000 0.5 0.48723610 0.44695676 0.48971327 0.48878400 0.50202615 0.58277826 0.490000 0.7 0.80680323 0.76006071 0.80966549 0.80858133 0.82391940 0.95739024 0.91768575 0.810000 0.9 1.16e-005 0.00285 1.118 e-007 2.012 e-006 1.937 e-004 0.02172388 0.01159622 L.S.E. 1.172 1.188 0.15600 0.1710000 0.110000 0.110000 0.078000 Time Closed N-C Simp. 3/8 Closed N-C Trap. Open N-C n=4 Open N-C n=3 Open N-C n=2 Open N-C n=1 Open N-C n=0 Exact u2 x 0.11102031 0.12946933 0.11008521 0.11036142 0.10543206 0.05300473 0.06875521 0.11000 0.1 0.39105674 0.40942113 0.39010194 0.39043235 0.38528876 0.33304139 0.34870102 0.39000 0.3 0.75092003 0.76661449 0.75009955 0.74590276 0.70125273 0.71460984 0.75000 0.5 1.19061016 1.20104942 1.19007805 1.19033102 1.18727406 1.16648165 1.19000 0.7 1.71012715 1.71272591 1.71003743 1.71015875 1.70940265 1.70219946 1.70431646 1.71000 0.9 3.214e-008 6.102e-006 1.401 e-009 2.520 e-008 3.568 e-007 6.084 e-005 3.230 e-005 L.S.E. 1.172 1.188 0.15600 0.1710000 0.110000 0.110000 0.078000 Time IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011 Table (5) A comparison between the exact and numerical solution of 4 types of Open N – C and 2 types of Closed N– C for u3 Table (6) A comparison between the exact and numerical solution of 4 types of Open N – C and 2 types of Closed N– C for u1 Closed N-C Simp. 3/8 Closed N-C Trap. Open N-C n=4 Open N-C n=3 Open N-C n=2 Open N-C n=1 Open N-C n=0 Exact u3 x 0.99026659 0.99687951 0.98996416 0.98984800 0.98878499 0.97099871 0.99000000 0.99000 0.1 0.90969268 0.90749750 0.90989248 0.90954400 0.91131067 0.91867773 0.91000000 0.91000 0.3 0.74911876 0.73811548 0.74982080 0.75383634 0.78635675 0.75000000 0.75000 0.5 0.50854484 0.51000000 0.50974912 0.50893600 0.51636201 0.51000000 0.51000 0.7 0.18797093 0.19000000 0.18967744 0.18863200 0.19888768 0.28171479 0.19000000 0.19000 0.9 5.36e-006 0.0012 1.040 e-007 1.871 e-006 7.899 e-005 0.00841160 0.00450909 L.S.E. 1.172 1.188 0.15600 0.17100 0.11000 0.1100 0.07800 Time Closed N-C Simp. 3/8 Closed N-C Trap. Open N-C n=4 Open N-C n=3 Open N-C n=2 Open N-C n=1 Open N-C n=0 Exact u1 x 1.00308849 1.00439048 1.00308641 1.00286469 1.00299173 0.99935102 1.00056628 1.0030864 0.05556 1.04939139 1.05476199 1.04938271 1.04847428 1.04899455 1.03397239 1.0493827 0.22222 1.15125043 1.16093361 1.15123456 1.14960709 1.15053877 1.12344622 1.13248750 1.1512345 0.38889 1.30866561 1.32290534 1.30864197 1.30626312 1.26777251 1.3086419 0.55556 1.52163693 1.54067718 1.52160493 1.52025143 1.46695127 1.48473476 1.5216049 0.72222 1.79016439 1.81424913 1.79012345 1.78614486 1.78841988 1.72098248 1.7901234 0.88889 2.229e-009 7.634e-004 4.146e-029 e-0051.812 e-0063.324 0.00549471 0.00250056 L.S.E. 0.391000 0.359000 0.281000 0.203000 0.188000 0.157000 0.11000 Time IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011 Table (7) A comparison between the exact and numerical solution of 4 types of Open N – C and 2 types of Closed N– C for u2 Closed N-C Simp. 3/8 Closed N-C Trap. Open N-C n=4 Open N-C n=3 Open N-C n=2 Open N-C n=1 Open N-C n=0 Exact u2 x 0.00000979 0.00031686 0.00000952 0.00004540- 0.00001319- 0.00086507- 0.0005813- 0.0000095 0.05556 0.00243999 0.00387053 0.00243865 0.00219055 0.00233587 0.00164220- 0.0024386 0.22222 0.02287473 0.02573211 0.02287189 0.02238811 0.02267126 0.01471108 0.01736184 0.0228718 0.38889 0.09526463 0.09985221 0.09525986 0.09449787 0.08214541 0.0952598 0.55556 0.27207881 0.27869997 0.27207171 0.27162193 0.25312991 0.25928723 0.2720717 0.72222 0.62430494 0.63326303 0.62429507 0.62284907 0.62369401 0.59865224 0.6242950 0.88889 1.425e-010 1.144e-004 8.985e-030 e-0062.485 e-0074.296 e-0047.879 e-0043.588 L.S.E. 0.391000 0.359000 0.281000 0.203000 0.188000 0.157000 0.094000 Time 2011) 2( 24والتطبیقیة المجلدمجلة ابن الھیثم للعلوم الصرفة صیغ عمالمنظومة معادالت فریدهوم التكاملیة الخطیة من النوع الثاني باستحل كوتس المفتوحة –نیوتن غادة حسن إبراهیم جامعة بغداد، أبن الهیثم –كلیة التربیة ، قسم الریاضیات 2010، تشرین األول، 27: استلم البحث في ,2011شباط ، 28: قبل البحث في الخالصة اذ ،كوتس المفتوحـة –صیغ نیوتن عمالالبحث حل منظومة من معادالت فریدهوم التكاملیة الخطیة باست هذا یتضمن . لحل هذا النظام كوتس المفتوحة –من صیغ نیوتن خمس صیغ مختلفة عملنااست .كوتس المغلقة –ق نیوتن ائق أخرى مثل طر ائالبحث مع نتائج طر كذلك قارنا نتائج الطریقة المقترحة في هذا أیضا ) MATLAB (version 7.0)( را في نهایة كل طریقة ذكرنا الخوارزمیة وبرنامج للطریقة المقترحة للحل وبلغة أخی .وضحنا الطریقة المقترحة من خالل األمثلة العددیة كوتس المغلقة ، منظومة معادالت فریـدهولم التكاملیـة –كوتس المفتوحة ، صیغ نیوتن –صیغ نیوتن :الكلمات المفتاحیة .الخطیة