Applied Mthemticl Sciences, Vol. 6, 2012, no. 26, 1267-1273 A Modified ADM for Solving Systems of Liner Fredholm Integrl Equtions of the Second Kind A. R. Vhidi nd T. Dmercheli Deprtment of Mthemtics, Shhr-e-Rey Brnch Islmic Azd University, Tehrn, Irn lrevhidi@yhoo.com, tsdmerchli@yhoo.com Abstrct In this pper, modifiction of Adomin decomposition method (MADM) for solving system of liner Fredholm integrl equtions of the second kind hs been introduced. This new method is resulted from Adomin decomposition method (ADM) by simple modifiction. To illustrte the MADM, one exmple is presented. Comprison of the result of pplying the MADM nd the ADM reveling the new technique is very effective nd convenient. Keywords: Adomin decomposition method; System of liner Fredholm integrl equtions of the second kind 1 Introduction The Adomin decomposition method ws first proposed by Adomin nd used to solve wide clss of nonliner nd prtil differentil equtions. The method is very powerful in finding the solutions for vrious physicl problems. The dvntge of the method is its fst convergence of the solution. This method hs been pplied to wide clss of deterministic nd stochstic problems of mthemticl nd physicl sciences. The method provides the solution in some rpidly convergent series with components tht re elegntly computed [1, 2]. Since mny physicl problems re modeled by system of liner Fredholm integrl equtions of the second kind, the numericl solutions of such system of liner Fredholm integrl equtions of the second kind hve been highly studied by mny uthors. In this pper, simple modifiction on the ADM will be studied nd will be pplied to solve system of liner Fredholm integrl equtions of the second kind. A system of liner Fredholm integrl equtions of the second kind cn be considered s U(t) =F (t)+ b K(s, t)u(s)ds, (1)
1268 A. R. Vhidi nd T. Dmercheli where U(t) =(u 1 (t),...,u n (t)) t, F (t) =(f 1 (t),...,f n (t)) t, K(s, t) =[k ij (s, t)], i=1,...,n, j =1,...,n. In Eq. (1), the functions K nd F re given, nd U is the vector function of the solution tht will be determined [8]. Also, we suppose tht the system (1) hs unique solution. However, the necessry nd sufficient conditions for existence nd uniqueness of the solution of the system (1) could be found in [8]. The rest of this pper is orgnized s follows: Section 2 is ssigned to brief introduction the ADM for the system of liner Fredholm integrl equtions of the second kind. In Section 3, proposed the modified technique. To illustrte nd show the efficiency of the method one exmple is presented in Section 4. And conclusions will pper in Section 5. 2 ADM for the system (1) In this section, we pply the ADM for the system of liner Fredholm integrl equtions of the second kind. To this end, consider the system of (1) s b n u i (t) =f i (t)+ k ij (s, t)u j (s)ds, i =1,...,n. (2) j=1 To solve (1) by the ADM [10], let u i (t) = u im (t), i =1,...,n. (3) Substituting (3) into (2), we obtin b n u im (t) =f i (t)+ k ij (s, t)( u jm (s))ds, i =1,...,n. (4) j=1 Bsed on the recursion scheme of the ADM, we define u i,0 (t) =f i (t), i =1,...,n, (5) nd then b n u i,k+1 (t) = k ij (s, t)u jk (s)ds, i =1,...,n, k =0, 1,... (6) j=1
A modified ADM for solving systems 1269 It is cler tht better pproximtions cn be obtined by evluting more components of the decomposition series solution u i (t). We pproximte u i (t) by ϕ ik (t) = k 1 u im (t), where lim k ϕ ik (t) =u i (t). (7) We note here tht the convergence question of this technique hs been formlly proved nd justified by [9]. In the next section, we propose simple modifiction such tht increse the rte of the convergence. 3 Description of the modified technique In this section, we propose simple modifiction to ccelerte convergence rte of the ADM pplied to systems of liner Fredholm integrl equtions of the second kind. In the ADM, thr first terms of u i (t)) is defined the known function f i (t). In modifiction technique, we define the first terms of (3) s follows u 1,0 (t) =f 1 (t), u i,0 (t) =f i (t)+ b i 1 j=1 k ij (s, t)u j,0 (s)ds, i =2,...,n. In other words, the vlues of u i,0 (t), for i =2,...,n re modified by previous vlues nd the other terms of (3) re defined s follows (8) u 1,k+1 = b nj=1 k i,j (s, t)u jk (s)ds, u i,k+1 = b ( i 1 j=1 k i,j(s, t)u j,k+1 (s)+ n j=i k ij (s, t)u j,k (s))ds, i =2,...,n, k =0, 1,... (9) in order to clculte ech u i,j, for i =2,...,n nd j =1, 2,... it s previous vlues in the sme itertion re used. For further illustrtion we now use this method with n = 2 for solving second kind Fredholm integrl eqution system. Consider the following second kind Fredholm integrl eqution system u 1 (t) =f 1 (t)+ 1 0 (k 11(s, t)u 1 (s)+k 12 (s, t)u 2 (s))ds, u 2 (t) =f 2 (t)+ 1 0 (k 21(s, t)u 1 (s)+k 22 (s, t)u 2 (s))ds, (10) To solve (10) by the ADM, let
1270 A. R. Vhidi nd T. Dmercheli u 1 (t) = u 1m (t), u 2 (t) = u 2m (t), (11) Substituting (11) into (10), we obtin u 1m (t) =f 1 (t)+ 1 0 (k 11(s, t) u 1m (s)+k 12 (s, t) u 2m (s))ds, u 2m (t) =f 2 (t)+ 1 0 (k 21(s, t) u 1m (s)+k 22 (s, t) u 2m (s))ds, (12) Bsed on the recursion scheme of the MADM, we define u 10 (t) =f 1 (t), u 20 (t) =f 2 (t)+ 1 0 k 21(s, t)u 10 (s)ds, (13) nd u 11 (t) = 1 0 (k 11(s, t)u 10 (s)+k 12 (s, t)u 20 (s))ds, u 21 (t) = 1 0 (k 21(s, t)u 11 (s)+k 22 (s, t)u 2,0 (s))ds, (14) in order to, the vlues of u 20 (t) nd u 21 (t) re modified by previous vlues nd the other terms of (10) re defined s follows u 1,k+1 = 1 0 (k 11(s, t)u 1,k (s)+k 12 (s, t)u 2,k (s))ds, u 2,k+1 = 1 0 (k 21(s, t)u 1,k+1 (s)+k 22 (s, t)u 2,k (s))ds, k =1, 2,... (15) tht is, to clculte ech u 2,k+1, for k =1, 2,... it s previous vlues in the sme itertion re used. 4 Numericl Exmple To give cler overview of the modified method, we present the following exmple. We pply the ADM nd the MADM for system of liner Fredholm integrl equtions nd compre the obtin results. The computtions ssocited with exmple were performed using the mthemtic 7.
A modified ADM for solving systems 1271 Exmple. Consider the following system of liner Fredholm integrl equtions [10] u 1 (t) =f 1 (t)+ 1 0 s+t 3 (u 1(s)+u 2 (s))ds, u 2 (t) =f 2 (t)+ 1 0 st(u 1(s)+u 2 (s))ds, where f 1 (t) = t + 17 nd f 18 36 2(t) =t 2 19t +1. The exct solutions re u 12 1(t) = t + 1 nd u 2 (t) =t 2 +1. i) Adomin decomposition method Using the ADM, we would hve the following procedure u 1,0 (t) = t 18 + 17 36, u 2,0 (t) =t 2 19 12 t +1, nd u 1,m+1 (t) = 1 0 s+t 3 (u 1m(s)+u 2m (s))ds, u 2,m+1 (t) = 1 0 st(u 1m(s)+u 2m (s))ds, m =0, 1,... Eleven terms pproximtions to the solutions re derived s [9] ϕ 1,11 (t) =u 1,0 (t)+u 1,1 (t)+...+ u 1,10 (t) =0.9813t +0.9885, ϕ 2,11 (t) =u 2,0 (t)+u 2,1 (t)+...+ u 2,10 (t) =t 2 0.03450t +1. ii) Modif y Adomin decomposition method Using the modified recursive reltion (7) nd (8), we obtin u 1,0 (t) = t 18 + 17 36, u 2,0 (t) =t 2 19 12 t +1+ 1 0 stu 10(s)ds, nd
1272 A. R. Vhidi nd T. Dmercheli u 1,m+1 (t) = 1 0 s+t 3 (u 1m(s)+u 2m (s))ds, u 2,m+1 (t) = 1 0 st(u 1,m+1(s)+u 2m (s))ds, m =0, 1,... The pproximted solution with eleven terms re ϕ 1,11 (t) =u 1,0 (t)+u 1,1 (t)+...+ u 1,10 (t) =0.993773t +0.99619, ϕ 2,11 (t) =u 2,0 (t)+u 2,1 (t)+...+ u 2,10 (t) =t 2 0.00882848t +1. Absolute errors of the ADM nd the MADM for some vlues of t re presented in Tble 1. Tble 1: Numericl results t e ADM (ϕ 1,11 (t)) e MADM (ϕ 1,11 (t)) e ADM (ϕ 2,11 (t)) e MADM (ϕ 2,11 (t)) 0 1.150E 2 3.803E 3 0 0 0.1 1.336E 2 4.425E 3 3.450E 3 8.828E 4 0.2 1.523E 2 5.047E 3 6.901E 3 1.765E 3 0.3 1.710E 2 5.670E 3 1.035E 2 2.648E 3 0.4 1.896E 2 6.293E 3 1.380E 2 3.531E 3 0.5 2.083E 2 6.915E 3 1.725E 2 4.414E 3 0.6 2.269E 2 7.538E 3 2.070E 2 5.297E 3 0.7 2.456E 2 8.161E 3 2.415E 2 6.179E 3 0.8 2.643E 2 8.783E 3 2.760E 2 7.062E 3 0.9 2.829E 2 9.406E 3 3.105E 2 7.945E 3 1 3.016E 2 1.002E 2 3.450E 2 8.829E 3 5 Conclusion In this pper, modified form of ADM, for solving the systems of liner Fredholm integrl equtions of the second kind, is studied successfully. The modified method is better thn ADM in the sense of ccurcy nd pplicbility. Illustrtive exmple presented clerly support this clim. The results hve been pproved the efficiency of this method for solving these problems.
A modified ADM for solving systems 1273 References [1] G. Adomin, Nonliner Stochstic Systems Theory nd Applictions to Physics, Kluwer, Dordrecht, (1989). [2] G. Adomin, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer, Dordrecht, (1994). [3] Yee, Eugono, Appliction of the decomposition method to the solution of the rectionconvection-diffusion eqution, Appl. Mth. Comput. 56 (1993) 1-27. [4] S. M. El-Syed, The modified decomposition method for solving non liner lgebric equtions, Appl. Mth. Comput. 132 (2002) 589-597. [5] E. Bbolin nd J. Bizr, Solving the Problem of Biologicl Species Living Together by Adomin Decomposition Method, Appl. Mth. Comput. 129 (2002) 339-343. [6] E. Bbolin nd J. Bizr, Solving Concrete Exmples by Adomin Method, Appl. Mth. Comput. 135 (2003) 161-167. [7] E. Bbolin, A.R. Vhidi nd Gh. Asdi Cordshooli, Solving differentil equtions by decomposition Appl. Mth. Comput. 167 (2005) 1150-1155. [8] Delves L. M. nd J. L. Mohmed, Computtionl methods for integrl eqution, Cmbridge University press, (1985). [9] E. Bbolin, J. Bizr, Solution of system of liner Volterr equtions by Adomin decomposition method, Fr Est J. Mth. Sci. 2 (2002) 935-945. [10] A. R. Vhidi nd M. Mokhtri, On the decomposition method for system of liner Fredholm integrl equtions of the second kind, Appl. Mth. Sci. 2 (2008) 57-62. Received: August, 2011