IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (2) 2010 Solutins of Systems for the Linear Fredholm-Volterra Integral Equatio ns of the Second Kind H. H. Omran Departme nt of Mathematics, College of Education, Ibn Al-Haitham,Unive rsity of Baghdad. Abstract In this p aper, we p resent some numerical methods for solving sy stems of lin ear Fredholm- Volterra inte gral equations of the second kind. These methods namely are the Repeated Trapezoidal M ethod (RTM) and the Rep eated Simp son's 1/3 M ethod (RSM). Also some numerical examp les are p resented to show the efficiency and the accuracy of the presented work. 1- Introduction Integral equations have received a consider able interest in the mathematical literature, because of their many filed of app lications in diff erent areas of sciences (see, for examp le [1]-[4]). M any authors gave some numerical solutions for differ ent typ es of Fredholm integr al equations and Vo lterra integral equations (see, for examp le [3]-[9]). In this p aper, we show how t he numerical methods which are based on the Rep eated Trap ezoidal M ethod (RTM ) and Repeated Simp son’s 1/3 M ethod (RSM ) can be used to solve the following system of linear Fredholm- Volterra integr al equation of the second kind: bm i i ij ij j j 1 a u ( x) f (x ) L (x , y)u ( y)dy       xm ij ij j j=1 a K (x , y)u ( y)dy,   i 1, 2, ..., m. (1.1) where a x b  , ij , ij are real numbers, fi, ijL , Kij, are given continuous functions and iu are t he unknown functions that must be determined. If ij  0 for each i,j=1,2,…,m. then equation (1.1) is called system of linear Fredholm inte gral equations. Also, if ij  0 for each i,j=1,2,…,m then equation (1.1) is called sy stem of lin ear Volterra integral equations. T he solution exist s for t hese sp ecial typ es of equation (1.1), (see, [10]-[15]). 2- The Repeated Trapezoidal Method: Consider the sy st em of Fredholm-Volterra integral equation given by equation (1.1). To solve this equation on the finite interval [a, b], we divide it into n smaller intervals of width h, where h  (b  a)/n. The r-th p oint of subdivision is denoted by xr, such that rx a rh,  r  0, 1, …, n. If we app roximate the integrals that app eared in equation (1.1) by the (RTM ) which will y ield the following sy st em of linear equations: m r 1 i,0 i, 0 ij,0,0 j,0 ij,0,s j,s ij,0,n n j 1 s 1 h u f L u 2 L u L u , 2                    ij,r,s m r 1 i, r i, r ij,r ,0 ij, r, 0 j, 0 j 1 s 1 h u f L K u h L 2               ij,r ,s j,s ij, r, r ij,r ,r j,r h K u 2L K u 2    n 1 ij, r,s j,s ij, r, n j,n s r 1 h h L u L u , r 0,1, ..., n 1 2          IHJPAS IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (2) 2010   m i, n i,n ij,n, 0 ij, n,0 j, 0 j 1 h u f L K u 2           n 1 ij,n ,s ij,n ,s j,s ij,n , n ij,n , n j,n s 1 2 L K u L K u      , i 1, 2, ..., m, r 0,1,..., n.  (2.2) By solving the linear sy stem given by equation (2.2) which consists of m(n+1) equations and m(n+1) unknowns, the app roximated solution of (1.1) is obt ained. 3- The Repeated Si mpson’s 1/3 Method: Consider the sy stem of the linear Fredholm-Vo lterra inte gral equation of second kind given by equation (1.1). Here we use (RSM ) to find the solution of equation (1.1). To do this, we divide the finite interval [a, b] into 2n smaller intervals of width h, where h  (b  a)/2n. The solution of (1.1) in the even nods 2 rx is given by bm i 2r i 2r ij 2r j j 1 a u ( x ) g ( x ) L (x , y) u (y)     2rxm ij 2r j j 1 a K (x , y) u (y )dy,    i 1, 2, ..., m, r 0,1,..., n.  (3.1) and in the odd nods 2r 1x  is given by: bm i 2r 1 i 2r 1 ij 2r 1 j j 1 a u ( x ) g (x ) L ( x , y )u ( y)       2r 1xn ij 2r 1 j j 1 a K (x , y)u ( y)dy,      i 1, 2, ..., m, r 0,1,..., n.  (3.2) By using the (RSM ) formula to ap p roximate the integrals t hat ap p eared in equations (3.1) - (3.2) one can get t he following sy st em of equations: m n 1 i, 0 i, 0 ij,0,0 j,0 ij,0,2s 1 j, 2s 1 j 1 s 1 h u f L u 4 L u 3              n 1 ij,0,2s j, 2s ij, 0,n j, n s 1 2 L u L u , i 1, 2,..., m,            m i, 2r i,r ij, 2r, 0 ij, 2r,0 j,0 j 1 h u f L k u 3         r 1 ij,2 r, 2s 1 ij, 2r, 2s 1 j,2 s 1 s 1 4 L k u           r 1 ij,2 r, 2s 1 ij, 2r, 2s 1 j, 2s 1 ij, 2r,2r s 1 4 L k u 2L          n ij, 2r,2r j,2r ij, 2r,2s 1 j, 2s 1 s r 1 k u 4 L u      n 1 ij,2 r, 2s j, 2s ij, 2r, 2n j,2 n s r 1 2 L u L u , r 1, 2, , n 1,               m i, 2r 1 i, 2r 1 ij, 2r 1,0 ij,2r 1,0 j,0 j 1 h u g L K u 3            r ij, 2r 1,2s 1 ij, 2r 1,2s 1 j, 2s 1 s 1 4 L K u          r 1 ij, 2r 1,2s ij,2r 1,2s j,2s s 1 2 L K u       ij, 2r 1, 2s ij, 2r 1, 2s j,2 s 5 2L K u 2          n n 1 ij, 2r 1,2 r 1 ij,2 r 1,2r 1 j, 2r 1 ij,2r 1,2s 1 j, 2r 1 ij,2r 1,2s j, 2s s r 2 s r 1 3 4L K u 4 L u 2 L u 2                          ij, 2r 1,2 r 1 2j 1L u   ,r 0, 1, …, n  1, i 1, 2, ..., m , IHJPAS IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (2) 2010   m i, 2n i,2 n ij, 2n,0 ij, 2n,0 j,0 j 1 h u g L K u 3        n ij, 2n ,2s 1 ij, 2n,2s 1 j,2s 1 s 1 4 L K u        n 1 ij,2n , 2s ij,2 n,2s j,2s s 1 2 L K u       ij,2 n,2n ij, 2n, 2n j,2nL K u , i 1, 2, ..., m. (3.3) By solving the linear sy stem given by equation (3.3) which consists of m(n+1) equations and m(n+1) unknowns, the numerical solution of (1.1) is obtained. 4-Numerical examples: In this section, we p resent some examples and their absolute errors to show the high accuracy of the solution obtained by (RTM ) and (RSM ) for solving sy stems of linear Fredholm- Volterra integral equ ations of t he second kinds. The results for these examp les, ar e co mputed by using M atlab Version 2007. Exam ple (1): Consider the system of Fredholm-Volterra integral equation: 2 3 4 6 1 5 9 3 5 1 1 u (x ) x x x x x 6 4 2 6 4 5         1 1 1 2 0 0 ( y x) u (y) dy (x 5 y)u (y) dy     x x 1 2 0 0 ( x y )u (y )dy (xy 1)u ( y)dy, 0 x 1,      2 3 4 5 7 2 4 5 5 5 1 1 1 u (x) x x x x x x 5 12 4 6 4 4 5         1 1 2 1 2 0 0 ( xy 2) u (y )dy ( x y 2)u ( y)dy 2       x x 2 2 1 2 0 0 ( x y ) u (y) dy (x y x)u ( y)dy, 0 x 1.      with the exact solutions: u1(x)x+1 and u2( x)=x 3 . We solved this sy stem with (RTM ) and (RSM ). Tables 1 and 2 shows t he a selection absolute error of approximated solutions for example (1) for h  0.1, 0.02, 0.01 and figure (1) shows that t he comp arison between the exact solutions and the numerical solutions via (RTM ) and (STM ) for h = 0.1 Example (2): Consider the system of Fredholm-Volterra integral equation: 2 x x 1u (x ) (4 e cos(1) 2 sin(1))x 2e x 2 x cos( x) 2e sin(x ) 1           1 2 2 1 2 0 x y u ( y) u ( y) dy    x 1 2 0 ( x y ) u (y ) u ( y) dy , 0 x 1,     x 2 x 2u (x) (cos(x) e ) x (e sin(x) e 1)x sin(x) sin(1) cos(1) 1             1 1 2 0 ( x y) u ( y) u (y ) dy     x 1 2 0 xy u (y ) u ( y) dy, 0 x 1   . IHJPAS IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (2) 2010 with the exact solutions: u1(x)  e x and u2( x)= sin(x) By using (RTM ) method and (RSM ), the problem can be solved, and a selection absolute error for h  0.1, 0.02, 0.01, are list ed in table (3) and (4). Figure (2) shows comp arison between the exact solutions and the numerical solutions of that example (2) for h = 0.1. 5- Conclus ions and Recommen dations: The sy stems of linear Fredholm- Volterra integr al equations are usually difficult to solve analytically. In many cases, it is requir ed to obtain the nu marical solutions, for this p urp ose the p resented methods can be p rop osed. From numerical examp les it can be seen that the p rop osed numerical methods are efficient and accurate to estimate the solution of these equations, Also, we show that when the values of h decr eases, t he absolute errors decrease to small values and the (RSM ) then more accur ate results than (RTM). 6-References 1. Jerri, A. J. (1985). “Introduction to Integral Equations with App lications", second ed., John Wiley and Sons. 2. Collins, P. J. (2005)."Differential and Integral Equations", Oxford University Press Inc., New York. 3. Linz, P. (1985)."Analytic and Numerical Solution of Integral Equations for Volterra Equations", SIAM Stud. App l. M ath. 4. Poly anin, A. D. and M anzhirov, A.V. (1998). "Handbook of integral equations", CRC Press LLC. 5. At kinson, K. E. (1997). “The Numerical So lution of Integral Equations of the Second Kind", Cambridge University Press. 6. Brunner, H. (2004). "Collocation M ethods for Volterra Inte gral and Related Functional Diff erential Equations", Cambridge University Press. 7. Golberg, M . A. (1979). "Solution M ethods for Integral Equations Theory and App lications", Plenum Press, New York and London. 8. Kyt e, P. K. and Puri, P. (2002). “Computational M ethods for Linear Integr al Equations", Birkhauser, Bost on. 9. M ajeed, S. J. and Omran, H. H. (2009). "Numerical M etods for Solving Lin ear Fredholm- Volterra Integral Equations", J. Al-Nahrain University , 11, No. 3, pp 131-134. 10. M aleknejad, K. and Sh ahrezaee, M . (2004)." Using Run ge–Kutta method for numerical. solution of the sy stem of Volterra integral equation", J. App l. M ath. and Comup t., 149, No. 2, pp 399-410. 11. M aleknejad, K., Agh azadeh, N. and Rabbani, M . (2006). " Numerical solution of second kind Fredholm integr al equations sy stem by using a Taylor-series exp ansion method", J. App l. M ath. and Comup t., 175, No. 2, pp 1229-1234. 12. Rabbani, M ., M aleknejad, K. and N. A ghazadeh, (2007). "Numerical computational solution of the Volterra inte gral equations sy stem of the second kind by using an exp ansion method", J. App l. M ath. and Comupt., 187, No. 2, pp 1143-1146. 13. Babolian, E., Biazar, J. and Vahidi, A. R. (2004)."On the decomposition method for sy stem of linear equations and sy st em of linear Volterra inte gral equations", J. App l. M ath. and Comupt., 147, No. 1, pp 19-27. 14. Vahidi, A. R. and M okhtari, M . (2008)."On the decomposition method for syst em of linear Fredholm inte gral equations of the Second Kind", J. Ap p l. M ath. Scie., 2, No. 2, p p 57-62. 15. M ajeed, S. J. (2009), “M odified Trapezoidal M ethod for Solvin g Sy st em of Linear Integral Equations of the Second Kind", J. Al-Nahrain University ,11, No. 4, pp 131-134. IHJPAS IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (2) 2010 Tabl e (1): The Absolu te Errors at Some Me sh Points of Exam ple (1) by Using (RTM). xr E1r E2r E1r E2r E1r E2r h=0 .1 h=0 .02 h=0 .01 0 5.06286×10 -3 5.06877×10 -4 2.07180×10 -4 2.0 6630×10 -5 5.18322×10 -5 5.1 6887×10 -6 0.1 4.70 150×10 -3 8.77 017×10 -4 1.92551×10 -4 3.5 6507×10 -5 4.84789×10 -5 8.9 1723×10 -6 0.2 4.47 285×10 -3 1.44 803×10 -3 1.83361×10 -4 5.8 7847×10 -5 4.60420×10 -5 1.4 7031×10 -5 0.3 4.38 597×10 -3 2.21 049×10 -3 1.79978×10 -4 8.9 6918×10 -5 4.50450×10 -5 2.2 4331×10 -5 0.4 4.46 029×10 -3 3.16 647×10 -3 1.83197×10 -4 1.2 8468×10 -4 4.58372×10 -5 3.2 1313×10 -5 0.5 4.73 251×10 -3 4.33 054×10 -3 1.94515×10 -4 1.7 5718×10 -4 4.86701×10 -5 4.3 9494×10 -5 0.6 5.26 697×10 -3 5.73 390×10 -3 2.16546×10 -4 2.3 2730×10 -4 5.41830×10 -5 5.8 2094×10 -5 0.7 6.17 244×10 -3 7.43 297×10 -3 2.53692×10 -4 3.0 1822×10 -4 6.34768×10 -5 7.5 4914×10 -5 0.8 7.63 063×10 -3 9.52 479×10 -3 3.13269×10 -4 3.8 6967×10 -4 7.83812×10 -5 9.6 7892×10 -5 0.9 9.94 575×10 -3 1.21 739×10 -2 4.07468×10 -4 4.9 4878×10 -4 1.01943×10 -4 1.2 3782×10 -4 1 1.36 332×10 -2 1.56 602×10 -2 5.56811×10 -4 6.3 6931×10 -4 1.39294×10 -4 1.5 9316×10 -4 Tabl e ( 2): The Absolu te Errors Some Me sh Points of Example (1) byUsi ng (RS M). xr E1r E2r E1r E2r E1r E2r h=0 .1 h=0 .02 h=0 .01 0 9.85011×10 -4 2.53612×10 -4 8.12018×10 -6 2.0 1117×10 -6 1.01950×10 -6 2.5 1490×10 -7 0.1 7.35 843×10 -4 3.81 895×10 -5 5.98137×10 -6 1.4 1866×10 -7 9.65248×10 -7 2.2 4867×10 -7 0.2 9.02 691×10 -4 1.85 301×10 -4 7.48408×10 -6 1.4 5430×10 -6 9.40272×10 -7 1.8 1698×10 -7 0.3 5.89 914×10 -4 1.20 343×10 -4 4.80683×10 -6 1.1 0028×10 -6 9.43165×10 -7 1.2 2708×10 -7 0.4 9.30 264×10 -4 5.57 167×10 -5 7.74846×10 -6 3.83134×10 -7 9.74053×10 -7 4.7 2401×10 -8 0.5 4.91 389×10 -4 2.62 787×10 -4 3.97669×10 -6 2.1 9672×10 -6 1.03426×10 -6 4.6 5516×10 -8 0.6 1.07 408×10 -3 1.44 862×10 -4 8.95561×10 -6 1.2 7950×10 -6 1.12600×10 -6 1.6 1511×10 -7 0.7 4.09 887×10 -4 3.36 511×10 -4 3.22345×10 -6 2.69014×10 -6 1.25191×10 -6 3.0 1138×10 -7 0.8 1.35 720×10 -3 4.42 686×10 -4 1.12569×10 -5 3.7 3015×10 -6 1.41443×10 -6 4.6 8890×10 -7 0.9 3.29 348×10 -4 2.63 176×10 -4 2.33478×10 -6 1.8 8579×10 -6 1.61460×10 -6 6.6 6812×10 -7 1 1.80 589×10 -3 8.64 267×10 -4 1.47532×10 -5 7.1 2125×10 -6 1.84972×10 -6 8.9 2877×10 -7 Table( 3): The Absol ute Errors at S ome Mesh Points of Example (2) by Usi ng (RTM). xr E1r E2r E1r E2r E1r E2r h=0 .1 h=0 .02 h=0 .01 0 0 4.59400×10 -2 0 1.6 8901×10 -3 0 4.2 1188×10 -4 0.1 1.39 964×10 -3 3.94 492×10 -2 5.28576×10 -5 1.4 5109×10 -3 1.31920×10 -5 3.6 1864×10 -4 0.2 4.82 389×10 -3 3.31 494×10 -2 1.80143×10 -4 1.2 2046×10 -3 4.49441×10 -5 3.0 4359×10 -4 0.3 1.01 980×10 -2 2.71 552×10 -2 3.79220×10 -4 1.0 0132×10 -3 9.45993×10 -5 2.4 9722×10 -4 0.4 1.76 153×10 -2 2.16 013×10 -2 6.53606×10 -4 7.9 8625×10 -4 1.63036×10 -4 1.9 9188×10 -4 0.5 2.73 715×10 -2 1.66 865×10 -2 1.01421×10 -3 6.1 9689×10 -4 2.52974×10 -4 1.5 4580×10 -4 0.6 4.00 359×10 -2 1.27 385×10 -2 1.48187×10 -3 4.7 6515×10 -4 3.69612×10 -4 1.1 8892×10 -4 0.7 5.65 854×10 -2 1.03 185×10 -2 2.09218×10 -3 3.8 9560×10 -4 5.21819×10 -4 9.7 2234×10 -5 0.8 7.86 479×10 -2 1.04 044×10 -2 2.90402×10 -3 3.9 4267×10 -4 7.24273×10 -4 9.8 4091×10 -5 0.9 1.08 946×10 -1 1.47 285×10 -2 4.01514×10 -3 5.5 2950×10 -4 1.00133×10 -3 1.3 7977×10 -4 1 1.52 124×10 -1 2.64 217×10 -2 5.59065×10 -3 9.7 7200×10 -4 1.39412×10 -3 2.4 3730×10 -4 IHJPAS IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (2) 2010 Table( 4): The Absol ute Errors S ome Mesh Points of Example (2) by Usi ng (RS M). xr E1r E2r E1r E2r E1r E2r h=0 .1 h=0 .02 h=0 .01 0 0 1.72067×10 -3 0 1.4 2600×10 -5 0 1.7 9214×10 -6 0.1 3.94 896×10 -4 1.48 330×10 -3 3.24894×10 -6 1.2 2918×10 -5 4.22744×10 -8 1.5 0903×10 -6 0.2 1.57 813×10 -4 1.18 168×10 -3 1.30728×10 -6 9.7 9618×10 -6 1.64235×10 -7 1.2 3117×10 -6 0.3 7.70 606×10 -4 1.04 351×10 -3 6.35692×10 -6 8.6 4332×10 -6 3.63302×10 -7 9.6 3412×10 -7 0.4 6.19 239×10 -4 6.82 325×10 -4 5.12401×10 -6 5.6 6094×10 -6 6.43696×10 -7 7.1 1472×10 -7 0.5 1.47 546×10 -3 6.89 838×10 -4 1.22051×10 -5 5.7 1102×10 -6 1.01797×10 -6 4.8 3757×10 -7 0.6 1.45 303×10 -3 2.81 265×10 -4 1.20207×10 -5 2.3 3981×10 -6 1.51008×10 -6 2.9 4049×10 -7 0.7 2.66 905×10 -3 5.22 096×10 -4 2.21131×10 -5 4.3 2653×10 -6 2.16111×10 -6 1.6 5889×10 -7 0.8 2.92 364×10 -3 1.33 308×10 -4 2.41924×10 -5 1.1 1631×10 -6 3.03940×10 -6 1.4 0236×10 -7 0.9 4.81 510×10 -3 8.24 729×10 -4 3.99044×10 -5 6.8 4114×10 -6 4.25899×10 -6 2.8 9457×10 -7 1 5.78 126×10 -3 7.12 361×10 -4 4.78547×10 -5 5.9 1702×10 -6 6.01328×10 -6 7.4 3744×10 -7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.5 0 0.5 1 1.5 2 x  Exact RTM RSM u1 u2 Fig (1) : C omparison be tween the exact solution an d nume ri cal solu tion via (RTM) an d (RSM) of exam ple(1) for h=0.1 . 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 2.5 3 x  Exact RTM RSM u1 u2 Fig (2): C om pari son be tween the exact solu tion an d nume ri cal solu tion via (RTM) an d (RSM) of exam ple(2) for h=0.1 . IHJPAS 2010) 2( 23مجلة ابن الھیثم للعلوم الصرفة والتطبیقیة المجلد فولتیرا التكاملیة الخطیة من النوع الثاني -حلول انظمة معادالت فریدهول هدى حمودي عمران .جامعة بغداد، كلیة التربیة ابن الهیثم ،قسم الریاضیات الخالصة ل انظمة معادالت فریدهولم ةق العددیائلطر قدمنا بعض ا . فولتیرا التكاملیة الخطیة من النوع الثاني - لح اعطیت لتبیان ةالعددی ةاالمثل. المتكررة 3/ 1ق هي طریقة شبه المنحرف المتكررة وطریقة سمبسون ائهذه الطر .ه ودقة العمل المقدمیكفا IHJPAS