IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.23 (2) 2010 Solutins of Systems for the Linear Fredholm-Volterra Integral Equations of the Second Kind H. H. Omran Department of Mathematics, College of Education, Ibn Al-Haitham,University of Baghdad. Abstract In this paper, we present some numerical methods for solving systems of linear Fredholm- Volterra integral equations of the second kind. These methods namely are the Repeated Trapezoidal Method (RTM) and the Repeated Simpson's 1/3 Method (RSM). Also some numerical examples are presented to show the efficiency and the accuracy of the presented work. 1- Introduction Integral equations have received a considerable interest in the mathematical literature, because of their many filed of applications in different areas of sciences (see, for example [1]-[4]). Many authors gave some numerical solutions for different types of Fredholm integral equations and Volterra integral equations (see, for example [3]-[9]). In this paper, we show how the numerical methods which are based on the Repeated Trapezoidal Method (RTM) and Repeated Simpson’s 1/3 Method (RSM) can be used to solve the following system of linear Fredholm-Volterra integral 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 the 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 integral equations. Also, if ij  0 for each i,j=1,2,…,m then equation (1.1) is called system of linear Volterra integral equations. The solution exists for these special types of equation (1.1), (see, [10]-[15]). 2- The Repeated Trapezoidal Method: Consider the system 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 point of subdivision is denoted by xr, such that rx a rh,  r  0, 1, …, n. If we approximate the integrals that appeared in equation (1.1) by the (RTM ) which will yield the following system 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. SCI. 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 system given by equation (2.2) which consists of m(n+1) equations and m(n+1) unknowns, the approximated solution of (1.1) is obtained. 3- The Repeated Simpson’s 1/3 Method: Consider the system of the linear Fredholm-Volterra integral 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 approximate the integrals that appeared in equations (3.1) - (3.2) one can get the following system 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,2r,2s 1 ij,2r,2s 1 j,2s 1 s 1 4 L k u           r 1 ij,2r,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,2r,2s j,2s ij,2r,2n j,2n 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,2s 5 2L K u 2         n n 1 ij,2r 1,2r 1 ij,2r 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 1,2r 1 2j 1L u   ,r 0, 1, …, n  1, i 1,2,...,m , IHJPAS IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.23 (2) 2010   m i, 2n i,2n 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,2n,2s j,2s s 1 2 L K u       ij,2n,2n ij,2n,2n j,2nL K u , i 1,2,...,m. (3.3) By solving the linear system 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 present some examples and their absolute errors to show the high accuracy of the solution obtained by (RTM) and (RSM) for solving systems of linear Fredholm-Volterra integral equations of the second kinds. The results for these examples, are computed by using Matlab Version 2007. Example (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 5y)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)=x3 . We solved this system with (RTM) and (RSM). Tables 1 and 2 shows the a selection absolute error of approximated solutions for example (1) for h  0.1, 0.02, 0.01 and figure (1) shows that the comparison 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) 2sin(1))x 2e x 2xcos(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. SCI. 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 listed in table (3) and (4). Figure (2) shows comparison between the exact solutions and the numerical solutions of that example (2) for h = 0.1. 5- Conclusions and Recommendations: The systems of linear Fredholm-Volterra integral equations are usually difficult to solve analytically. In many cases, it is required to obtain the numarical solutions, for this purpose the presented methods can be proposed. From numerical examples it can be seen that the proposed numerical methods are efficient and accurate to estimate the solution of these equations, Also, we show that when the values of h decreases, the absolute errors decrease to small values and the (RSM) then more accurate results than (RTM). 6-References 1. Jerri, A. J. (1985). “Introduction to Integral Equations with Applications", 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. Appl. Math. 4. Polyanin, A. D. and Manzhirov, A.V. (1998). "Handbook of integral equations", CRC Press LLC. 5. Atkinson, K. E. (1997). “The Numerical Solution of Integral Equations of the Second Kind", Cambridge University Press. 6. Brunner, H. (2004). "Collocation Methods for Volterra Integral and Related Functional Differential Equations", Cambridge University Press. 7. Golberg, M. A. (1979). "Solution Methods for Integral Equations Theory and Applications", Plenum Press, New York and London. 8. Kyte, P. K. and Puri, P. (2002). “Computational Methods for Linear Integral Equations", Birkhauser, Boston. 9. Majeed, S. J. and Omran, H. H. (2009). "Numerical Metods for Solving Linear Fredholm-Volterra Integral Equations", J. Al-Nahrain University, 11, No. 3, pp 131-134. 10. Maleknejad, K. and Shahrezaee, M. (2004)." Using Runge–Kutta method for numerical. solution of the system of Volterra integral equation", J. Appl. Math. and Comupt., 149, No. 2, pp 399-410. 11. Maleknejad, K., Aghazadeh, N. and Rabbani, M. (2006). " Numerical solution of second kind Fredholm integral equations system by using a Taylor-series expansion method", J. Appl. Math. and Comupt., 175, No. 2, pp 1229-1234. 12. Rabbani, M., Maleknejad, K. and N. Aghazadeh, (2007). "Numerical computational solution of the Volterra integral equations system of the second kind by using an expansion method", J. Appl. Math. and Comupt., 187, No. 2, pp 1143-1146. 13. Babolian, E., Biazar, J. and Vahidi, A. R. (2004)."On the decomposition method for system of linear equations and system of linear Volterra integral equations", J. Appl. Math. and Comupt., 147, No. 1, pp 19-27. 14. Vahidi, A. R. and Mokhtari, M. (2008)."On the decomposition method for system of linear Fredholm integral equations of the Second Kind", J. Appl. Math. Scie., 2, No. 2, pp 57-62. 15. Majeed, S. J. (2009), “Modified Trapezoidal Method for Solving System 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. SCI. VOL.23 (2) 2010 Table (1): The Absolute Errors at Some Mesh Points of Example (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 2.06630×10 -5 5.18322×10 -5 5.16887×10 -6 0.1 4.70150×10-3 8.77017×10-4 1.92551×10-4 3.56507×10-5 4.84789×10-5 8.91723×10-6 0.2 4.47285×10-3 1.44803×10-3 1.83361×10-4 5.87847×10-5 4.60420×10-5 1.47031×10-5 0.3 4.38597×10-3 2.21049×10-3 1.79978×10-4 8.96918×10-5 4.50450×10-5 2.24331×10-5 0.4 4.46029×10 -3 3.16647×10 -3 1.83197×10 -4 1.28468×10 -4 4.58372×10 -5 3.21313×10 -5 0.5 4.73251×10 -3 4.33054×10 -3 1.94515×10 -4 1.75718×10 -4 4.86701×10 -5 4.39494×10 -5 0.6 5.26697×10-3 5.73390×10-3 2.16546×10-4 2.32730×10-4 5.41830×10-5 5.82094×10-5 0.7 6.17244×10-3 7.43297×10-3 2.53692×10-4 3.01822×10-4 6.34768×10-5 7.54914×10-5 0.8 7.63063×10-3 9.52479×10-3 3.13269×10-4 3.86967×10-4 7.83812×10-5 9.67892×10-5 0.9 9.94575×10 -3 1.21739×10 -2 4.07468×10 -4 4.94878×10 -4 1.01943×10 -4 1.23782×10 -4 1 1.36332×10 -2 1.56602×10 -2 5.56811×10 -4 6.36931×10 -4 1.39294×10 -4 1.59316×10 -4 Table( 2): The Absolute Errors Some Mesh Points of Example (1) byUsing (RSM). 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 2.01117×10 -6 1.01950×10 -6 2.51490×10 -7 0.1 7.35843×10 -4 3.81895×10 -5 5.98137×10 -6 1.41866×10 -7 9.65248×10 -7 2.24867×10 -7 0.2 9.02691×10-4 1.85301×10-4 7.48408×10-6 1.45430×10-6 9.40272×10-7 1.81698×10-7 0.3 5.89914×10-4 1.20343×10-4 4.80683×10-6 1.10028×10-6 9.43165×10-7 1.22708×10-7 0.4 9.30264×10-4 5.57167×10-5 7.74846×10-6 3.83134×10-7 9.74053×10-7 4.72401×10-8 0.5 4.91389×10 -4 2.62787×10 -4 3.97669×10 -6 2.19672×10 -6 1.03426×10 -6 4.65516×10 -8 0.6 1.07408×10 -3 1.44862×10 -4 8.95561×10 -6 1.27950×10 -6 1.12600×10 -6 1.61511×10 -7 0.7 4.09887×10-4 3.36511×10-4 3.22345×10-6 2.69014×10-6 1.25191×10-6 3.01138×10-7 0.8 1.35720×10-3 4.42686×10-4 1.12569×10-5 3.73015×10-6 1.41443×10-6 4.68890×10-7 0.9 3.29348×10 -4 2.63176×10 -4 2.33478×10 -6 1.88579×10 -6 1.61460×10 -6 6.66812×10 -7 1 1.80589×10 -3 8.64267×10 -4 1.47532×10 -5 7.12125×10 -6 1.84972×10 -6 8.92877×10 -7 Table( 3): The Absolute Errors at Some Mesh Points of Example (2) by Using (RTM). xr E1r E2r E1r E2r E1r E2r h=0.1 h=0.02 h=0.01 0 0 4.59400×10 -2 0 1.68901×10 -3 0 4.21188×10 -4 0.1 1.39964×10-3 3.94492×10-2 5.28576×10-5 1.45109×10-3 1.31920×10-5 3.61864×10-4 0.2 4.82389×10-3 3.31494×10-2 1.80143×10-4 1.22046×10-3 4.49441×10-5 3.04359×10-4 0.3 1.01980×10 -2 2.71552×10 -2 3.79220×10 -4 1.00132×10 -3 9.45993×10 -5 2.49722×10 -4 0.4 1.76153×10-2 2.16013×10-2 6.53606×10-4 7.98625×10-4 1.63036×10-4 1.99188×10-4 0.5 2.73715×10-2 1.66865×10-2 1.01421×10-3 6.19689×10-4 2.52974×10-4 1.54580×10-4 0.6 4.00359×10 -2 1.27385×10 -2 1.48187×10 -3 4.76515×10 -4 3.69612×10 -4 1.18892×10 -4 0.7 5.65854×10 -2 1.03185×10 -2 2.09218×10 -3 3.89560×10 -4 5.21819×10 -4 9.72234×10 -5 0.8 7.86479×10-2 1.04044×10-2 2.90402×10-3 3.94267×10-4 7.24273×10-4 9.84091×10-5 0.9 1.08946×10-1 1.47285×10-2 4.01514×10-3 5.52950×10-4 1.00133×10-3 1.37977×10-4 1 1.52124×10 -1 2.64217×10 -2 5.59065×10 -3 9.77200×10 -4 1.39412×10 -3 2.43730×10 -4 IHJPAS IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.23 (2) 2010 Table( 4): The Absolute Errors Some Mesh Points of Example (2) by Using (RSM). xr E1r E2r E1r E2r E1r E2r h=0.1 h=0.02 h=0.01 0 0 1.72067×10 -3 0 1.42600×10 -5 0 1.79214×10 -6 0.1 3.94896×10-4 1.48330×10-3 3.24894×10-6 1.22918×10-5 4.22744×10-8 1.50903×10-6 0.2 1.57813×10-4 1.18168×10-3 1.30728×10-6 9.79618×10-6 1.64235×10-7 1.23117×10-6 0.3 7.70606×10 -4 1.04351×10 -3 6.35692×10 -6 8.64332×10 -6 3.63302×10 -7 9.63412×10 -7 0.4 6.19239×10 -4 6.82325×10 -4 5.12401×10 -6 5.66094×10 -6 6.43696×10 -7 7.11472×10 -7 0.5 1.47546×10-3 6.89838×10-4 1.22051×10-5 5.71102×10-6 1.01797×10-6 4.83757×10-7 0.6 1.45303×10-3 2.81265×10-4 1.20207×10-5 2.33981×10-6 1.51008×10-6 2.94049×10-7 0.7 2.66905×10 -3 5.22096×10 -4 2.21131×10 -5 4.32653×10 -6 2.16111×10 -6 1.65889×10 -7 0.8 2.92364×10-3 1.33308×10-4 2.41924×10-5 1.11631×10-6 3.03940×10-6 1.40236×10-7 0.9 4.81510×10-3 8.24729×10-4 3.99044×10-5 6.84114×10-6 4.25899×10-6 2.89457×10-7 1 5.78126×10 -3 7.12361×10 -4 4.78547×10 -5 5.91702×10 -6 6.01328×10 -6 7.43744×10 -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): Comparison between the exact solution and numerical solution via (RTM) and (RSM) of example(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): Comparison between the exact solution and numerical solution via (RTM) and (RSM) of example(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 . فولتیرا التكاملیة الخطیة من النوع الثاني - لحل انظمة معادالت فریدهولم ةق العددیائلطر قدمنا بعض ا اعطیت لتبیان ةالعددی ةاالمثل. المتكررة 3/ 1ق هي طریقة شبه المنحرف المتكررة وطریقة سمبسون ائهذه الطر .ه ودقة العمل المقدمیكفا IHJPAS