IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.22 (4) 2009 Approximation Solutions for System of Linear Fredhom Integral Equatio ns by Using Decomposition Method H. S. Ali,W. S. Ali*, N. M. Rahmah College of Engineering, Unive rsity of Al-Mustansriya *College of Agriculture, Unive rsity of Baghdad Abstract In this p ap er, the Decomp osition method was used to find ap p roximation solutions for a system of linear Fredholm inte gral equations of the second kind. In this method the solution of a functional equations is considered as the sum of an infinite series usually converging to the solution, and Adomian decomposition method for solving lin ear and nonlin ear inte gral equations. Finally, numerical examples are prepared to illustrate these considerations. Introduction The integral equation is an equation in which the unknown function y (x) app ears under the integral si gn. The gener al form of integr al equation is given by :[1]  dttytxkxfxyxh )(),()()()( …(1) Where )(xh , )(xf and the kernel ),( txk are known functions ; )(xy is the function to be determined. To satisfy linearity condition: ))(())(())()(( 22112211 tfLatfLatfatfaL  …(2) Where 21, aa are constants and  dttftxktfL )(),())(( . The Integral equation is called ho mogenous If 0)( xf , otherwise it is called non homogenous.[2] We can d istinguish between two typ es of integral equations which are: 1. Integral equation of the first kind when 0)( xh in equation (1).  dttytxkxf )(),()( …(3) 2. Integral equation of the second kind when 1)( xh in equation (1).  dttytxkxfxy )(),()()( …(4) Integral equations can be classified into different kinds accordin g to the limits of integr al[3]: 1. If the limits of equation (1) are constants then the equation is called Fredho lm inte gral equation. The Fredholm integr al equation of the first kind is:-  b a dttytxkxf )(),()( …(5) Where a, b are constants. 2. Fredholm inte gral equation of the second kind is:-  b a dttytxkxfxy )(),()()( …(6) IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.22 (4) 2009 The De composition Method Applied to Fredholm Inte gral Eq The top ic of Adomian decomposition method has been rapidly growin g in recent y ears. The concept of this method was first introduced by G. Adomian in the beginning of 1980’s[4]. In this subsection use a decomposition method to find the a app roximation solutions for Sy stem of linear Fredholm integral equations. Let us reconsid er the following sy st em of linear Fredholm inte gral equations of the se cond kind(4)(5)(6). ,)(),()()(  b a dssFstKtGtF ],[ bat  …(7) Where ,))(,),(),(()( 21 t n tftftftF  ,))(,),(),(()( 21 t n tgtgtgtG    n.,1,2,j ,,1,2,i ),(),( ,   nstkstK ji We supp ose that sy stem [7] has a unique solution. Consider the i-th equation of (7): .)(),()()( 1    b a n j jijii dssfstktgtf …(8) From (8), we obtain canonical form of Adom ian's equation by writing )()()( tNtgtf iii  …(9) Where .)(),()( 1    b a n j jiji dssfstktN …(10) To solve (10) by Adomian's method, let )()( 0 tftf m imi     , and     0 )( m imi AtN where ,0,1,m , imA are p olynomials depending on nmnm fffff ,,,,,,, 011110  and they are called Adom ian p olynomials. Hence, (9) can be rewritten as:       0 1011110 0 ).,,,,,,,,()()( m nmnnmimi m im ffffffAtgtf  …(11) From [10] we define:         0,1,2,m ),,,,,,,,()( n,,1,2,i ),()( 0111101, 0 nmnmimmi ii fffffAtf tgtf …(12) In p ractice, all terms of the series )()( 0 tftf m imi     can not be determin ed and so we use an approximation of the solution by the following truncated series: ),()( 1 0 tft k m imik     with ).()(lim tft iik k    …(13) To determine Adomian polynomials, we consider the exp ansions: ),()( 0 tftf m im m i      …(14) ,),,,( 0 21     m im m ni AfffN   …(15) Where,  is a parameter introduced for conven ience. From (15) we obtain: ,),,,( ! 1 0 21            nim m im fffN d d m A  …(16) IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.22 (4) 2009 And from (10), (14) and (16) we have: dsf d d m tsvffffA l jl l m mb a n j ijnmnmim 001 0110 ! 1 ),(),,,,,,(                .),( 1    b a n j jmij dsftsv …(17) So, (12) for the solution of the sy stem of linear Fredholm inte gral equations will be as follows:            0,1,2,m n,,1,2,i ,)(),()( ),()( 1 1, 0 b a n j jmijmi ii dstftsvtf tgtf … (18) Considerin g (13), we obtain:  0,1,2,m n,,1,2,i ,)(),()()( 0 1    t n j jmijiik dssfstktgt …(19) In fact (12) is exactly the same as the well known successive app roximations method for solving the sy st em of linear Fredholm integr al equations definin g as:  0,1,2,m ,1,2,i ,)(),()()( 0 1 1,     t n j jmijimi dssfstktgtf … (20) The initial app roximations for t he successive app roximations method is usually zero function. In the other words, if the initial approximations in this method is selected )(tgi , then the Adomian decomp osition method and the successive app roximations method are exactly the same. The followin g algor ithm summar izes the st ep s for finding the app roximation solutions for the second kind of sy stem Fredholm integr al equations. Algorithm (ADSFI) Input: ( nibasfstktg iii ,1,,),(),,(),(  ); Output: series solution of given equation S tep1: Put )()(0 tgtf ii  for ni ,...,2,1,0 S tep2: Compute     b a jm n j ijmi dssftxktf )(),()( 1 1, ni ,...,0 , ,...1,0m S tep3: Find the solution ),()( 1 0 tft k m imik     ni ,...,0 , ,...1,0m End Example Consider the followin g sy stem of linear Fredholm inte gral equations of the second k ind with the exact solutions 1)(1  ttf and 1)( 2 2  ttf . IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.22 (4) 2009              1 0 21 2 2 1 0 211 .))()(((1 12 19 )( ,))()(( 3 1 36 17 18 )( dssfsfsttttf dssfsf st tf To derive the solutions by using the decomp osition method, we can use the following Adomian scheme:         ,15833.11 12 19 )( ,4722.00556.0 36 17 18 )( 22 20 10 tttttf t t tf And 0,1,2,m ,))()(()( ,))()(( 3 )( )( 1 0 211,2 1 0 211,1                 dssfsfsttf dssfsf ts tf mmm mmm For t he first iteration, we have: .4769.0 216 103 ))()(()( ,1590.03472.0 648 103 72 25 ))()(( 3 )( )( 1 0 201021 1 0 201011              ttdssfsfsttf ttdssfsf ts tf Considerin g (13), the ap p roximated solutions with two terms are:      .11065.1)()()( ,6312.04028.0)()()( 2 212022 111012 tttftft ttftft   Next term are: .3542.0 48 17 ))()(()( ,1181.01903.0 144 17 972 185 ))()(( 3 )( )( 1 0 211121 1 0 211112              ttdssfsfsttf ttdssfsf ts tf Solution with three terms are:      .17523.0)()()( ,7492.05931.0)()()()( 2 22212023 12111013 ttftftft ttftftft   In the same way , the comp onents )(1 tk and )(2 tk can b e calculated for ,4,3k The solutions with seven terms and eleven terms are given as:       .11609.0)()()( ,9468.09129.0)()()()( 2 6,221207,2 6,111107,1 ttftftft ttftftft           .10345.0)()()( ,9885.09813.0)()()()( 2 10,2212011,2 10,1111011,1 ttftftft ttftftft     IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.22 (4) 2009 App roximated solutions for some values of t by using Decomp osition method and exact values 1)(1  ttf and 1)( 2 2  ttf of Example, depending on least square error ( L.S. E) ar e p resented in Table(1), Fig.1, and Fig.2. Conclusion This p ap er p resents the use of the Adomian deco mposition method, for the sy stem of linear Fredholm integr al equations. As it can be se en, the Adomian d ecomposition method for a sy stem of linear Fredho lm inte gral equations is equiv alent to successive app roximations method. Although, the Adomian decomposition method is a very p owerful device for solv ing the integral equations. From solving a numer ical examp le the followin g p oints were identified: 1- this method can be used to solve the secant kinds of linear Fredholm inte gral equation. 2- It is clear that using the decomposition method basis function to app roximate when the m order that increases the error would be decrease, as in Fig.1 and Fig.2. Re ferences 1. Abdul J. jerri, (1985) " Introduction to Integral Equation with applications" M arcel Dekker, Inc , New York. 2. Lap idus L. and Seinf eid J., (1979) "Numerical solution of Differential Equations", Wiley East ern Limited. New Delhi. 3. Diskunov ON., (1974) "Differential and Integral Calculus, "English translation. M ir Publishers, M oscow. 4. Ioan A. Rus and Egri Edith, (2006), "Numerical and App roximate methods in some mathematical models", Babes-Bolyai University of Cluj-Nap oca. 5. Delves L. M . and M ohamed, J. L. (1985), "Comp utational M ethods for Integral Equations, Cambridge University Press. 6. Ray S. Saha and Bera R. K, (2004), "Solution of an Extraordinary Differential Equatation by Adomian Decomp osition M ethod", Journal App lied M athematics 4 , 331-338 URL: http://dx.doi.org/10.1155/S1110757X04311010 .. Table (1): The results of the e xample by using (ADS FI) algorithm t )(1 tf Exact )(7,1 t )(11,1 t )(2 tf Exact )(7,2 t )(11,2 t 0 1 0.9468 0.9885 1 1 1 0.1 1.1 1.0381 1.0866 1.01 0.9939 1.0066 0.2 1.2 1.1294 1.1848 1.04 1.0078 1.0331 0.3 1.3 1.2207 1.2829 1.09 1.0417 1.0797 0.4 1.4 1.3120 1.3810 1.16 1.0956 1.1462 0.5 1.5 1.4033 1.4792 1.25 1.1695 1.2328 0.6 1.6 1.4945 1.5773 1.36 1.2634 1.3393 0.7 1.7 1.5858 1.6754 1.49 1.3773 1.4659 0.8 1.8 1.6771 1.7735 1.64 1.5112 1.6124 0.9 1.9 1.7684 1.8717 1.81 1.6652 1.7790 1 2 1.8597 1.9698 2 1.8391 1.9655 L.S .E 0.1113 0.0052 0.0997 0.0046 0 0.5 1 1.5 2 2.5 0 0. 2 0. 4 0. 6 0. 8 1 1.2 t f( t) IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.22 (4) 2009 Fig.(1 ) :Approximati ons and Exact soluti on of Fredholm integral equati ons of Example Fig.(2): Approximati ons and Exact soluti on of Fredholm integral equati ons of Example 0 0.5 1 1.5 2 2.5 0 0.2 0.4 0.6 0.8 1 1.2 t f( t) Exact =f2(t ) 2,7(t) 2,11(t) Exact =f1(t ) 1,7(t) 1,11(t) (22مجلة ابن الھیثم للعلوم الصرفة والتطبیقیة المجلد 4 (2009 طریقة عمالباستحلول تقریبیة لنظام معادالت فریدھوم التكاملیة الخطیة االنحالل ، نسرین مزھر رحمة* حلیمة سویدان علي ، ولیدة سویدان علي كلیة الھندسة ، الجامعة المستنصریة یة الزراعة ، جامعة بغدادكل * الخالصة طریقة االنحالل إلیجاد حلول تقریبیة لنظام معادالت فریدهوم التكاملیة الخطیة من النوع عمالاستفي هذا البحث د كما أن طریقة أ. متسلسلة غیر محددة تتقارب إلى الحلةوالحل في هذه الطریقة یكون دالة تمثل مجموع. الثاني دمون . والمثال المطبق یوضح هذه الطریقة. االنحاللیة هي طریقة لحل المعادالت التكاملیة الخطیة وغیر الخطیة