On the Decomposition Method for System of Linear Fredholm Integral Equations of the Second Kind

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1 Applied Mthemticl Sciences, Vol. 2, 28, no. 2, On the Decomposition Method for System of Liner Fredholm Integrl Equtions of the Second Kind A. R. Vhidi 1 nd M. Mokhtri Deprtment of Mthemtics, Shhr-e-Rey Brnch Islmic Azd University, Irn lrevhidi@yhoo.com m-mokhtri@iusr.c.ir Abstrct In this pper, system of liner Fredholm integrl equtions of the second kind is hndled by pplying the decomposition method. For system of liner equtions we show tht the Adomin decomposition method is equivlent to the clssicl successive pproximtions method, so clled Picrd s method. Finlly, numericl exmples re prepred to illustrte these considertions. Keywords: Adomin decomposition method, System of liner Fredholm integrl equtions 1 Introduction The topic of the Adomin decomposition method hs been rpidly growing in recent yers. The concept of this method ws first introduced by G. Adomin in the beginning of 198 s [1, 2]. In this method the solution of functionl equtions is considered s the sum of n infinite series usully converging to the solution [3]. The Adomin decomposition method for solving liner nd nonliner integrl equtions is known s subject of extensive nlyticl nd numericl studies [4, 5]. Our im here is to compre the decomposition method with the clssicl successive pproximtions method [6] for solving system of liner Fredholm integrl equtions. Consider the following system of liner Fredholm integrl equtions: 1 Corresponding uthor F (t) =G(t)+ b K(t, s)f (s)ds, t [, b] (1)

2 58 A. R. Vhidi nd M. Mokhtri where F (t) =(f 1 (t),..., f n (t)) t, G(t) =(g 1 (t),..., g n (t)) t, K(t, s) =[k i,j (t, s)] i =1,..., n, j =1,..., n. We suppose tht system (1) hs unique solution. 2 The Decomposition Method Applied to (1) Consider the i-th eqution of (1): f i (t) =g i (t)+ k ij (t, s)f j (s)ds. (2) From (2), we obtin cnonicl f orm of Adomin s eqution by writing f i (t) =g i (t)+n i (t) (3) where N i (t) = k ij (t, s)f j (s)ds. (4) To solve (3) by Adomin s method, let f i (t) = m= f im (t), nd N i (t) = m= A im where A im,m=, 1,..., re polynomils depending on f 1,..., f 1m,..., f n,..., f nm nd they re clled Adomin polynomils. Hence, (3) cn be rewritten s: f im (t) =g i (t)+ A im (f 1,..., f 1m,..., f n,..., f nm ). (5) m= m= From (4) we define: f i (t) =g i (t),. (6) f i,m+1 (t) =A im (f 1,..., f 1m,..., f n,..., f nm ), i =1,..., n, m =, 1, 2,... In prctice, ll terms of the series f i (t) = m= f im (t) cn not be determined nd so we use n pproximtion of the solution by the following truncted series: ϕ ik (t) = k 1 m= f im (t), with lim k ϕ ik (t) =f i (t). (7)

3 Liner Fredholm integrl equtions 59 To determine Adomin polynomils, we consider the expnsions: f iλ (t) = λ m f im (t), (8) m= N iλ (f 1,..., f n )= λ m A im, (9) m= where, λ is prmeter introduced for convenience. From (9) we obtin: A im = 1 [ ] d m m! dλ N iλ(f m 1,..., f n ), (1) λ= nd from (4), (8) nd (1) we hve: A im (f 1,..., f 1m,..., f n,..., f nm )= = b n [ 1 v ij (s, t) m! d m ] λ l f dλ m jl ds l= λ= v ij (s, t)f jm ds. (11) So, (6) for the solution of the system of liner Fredholm integrl equtions will be s follow: f i (t) =g i (t). (12) f i,m+1 (t) = v ij (s, t)f jm (t)ds, i =1,..., n, m =, 1, 2,... Considering (7), we obtin: t n ϕ ik (t) =g i (t)+ k ij (t, s)f jm (s)ds, i =1,..., n, m =, 1, 2,... (13) In fct (6) is exctly the sme s the well known successive pproximtions method for solving the system of liner Fredholm integrl equtions defining s: t n f i,m+1 (t) =g i (t)+ k ij (t, s)f jm (s)ds, i =1,..., n, m =, 1, 2,...(14) The initil pproximtions for the successive pproximtions method is usully zero function. In other words, if the initil pproximtions in this method is selected g i (t), then the Adomin decomposition method nd the successive pproximtions method re exctly the sme.

4 6 A. R. Vhidi nd M. Mokhtri 3 Numericl Exmple Exmple Consider the following system of liner Fredholm integrl equtions with the exct solutions f 1 (t) =t + 1 nd f 2 (t) =t f 1 (t) = t s + t 3 (f 1(s)+f 2 (s))ds, f 2 (t) =t t +1+ st(f 1 (s)+f 2 (s))ds. To derive the solutions by using the decomposition method, we cn use the following Adomin scheme: f 1 (t) = t t , 36 f 2 (t) =t t +1 t t +1, nd f 1,m+1 (t) = f 2,m+1 (t) = (s + t) (f 1m (s)+f 2m (s))ds, 3 st(f 1m (s)+f 2m (s))ds, m =, 1, 2,... For the first itertion, we hve: f 11 (t) = (s + t) (f 1 (s)+f 2 (s))ds = t t +.159, f 21 (t) = st(f 1 (s)+f 2 (s))ds = t.4769t. Considering (7), the pproximted solutions with two terms re: ϕ 12 (t) =f 1 (t)+f 11 (t).428t , ϕ 22 (t) =f 2 (t)+f 21 (t) t t +1. f 12 (t) = Next terms re: f 22 (t) = (s + t) (f 11 (s)+f 21 (s))ds = t st(f 11 (s)+f 21 (s))ds = t.3542t..193t ,

5 Liner Fredholm integrl equtions 61 Solutions with three terms re: ϕ 13 (t) =f 1 (t)+f 11 (t)+f 12 (t).5931t , ϕ 23 (t) =f 2 (t)+f 21 (t)+f 22 (t) t t +1. In the sme wy, the components ϕ 1k (t) nd ϕ 2k (t) cn be clculted for k =3, 4,... The solutions with eleven terms re given s: ϕ 1,11 (t) =f 1 (t)+f 11 (t) f 1,1 (t).9813t , ϕ 2,11 (t) =f 2 (t)+f 21 (t) f 2,1 (t) t 2.345t +1. Approximted solutions for some vlues of t nd the corresponding bsolute errors re presented in Tble 3.1. t f 1 (t) ϕ 1,11 (t) e(ϕ 1,11 (t)) f 2 (t) ϕ 2,11 (t) e(ϕ 2,11 (t)) Tble 3.1 ii) T he successive pproximtions method Clerly, in this method by choosing the initil pproximtion of f 1 (t) = nd f 2 (t) = nd considering (21), in the first itertion, we will get the initil pproximtion of the Adomin decomposition method tht is f 1 (t) nd f 2 (t). In the second itertion f1 1 (t) nd f2 1 (t) pproximtions for f 1 (t) nd f 2 (t) re derived. And subsequently in the third itertion, we get f1 2 (t) nd f2 2 (t) pproximtions for f 1 (t) nd f 2 (t). Hence, if the initil pproximtion for f 1 (t) nd f 2 (t) re respectively chosen s g 1 (t) nd g 2 (t), the Adomin decomposition method nd successive pproximtions method will be exctly the sme.

6 62 A. R. Vhidi nd M. Mokhtri 4 Conclusion This pper presents the use of the Adomin decomposition method, for the system of liner Fredholm integrl equtions. As it cn be seen, the Adomin decomposition method for system of liner Fredholm integrl equtions is equivlent to successive pproximtions method. Although, the Adomin decomposition method is very powerful device for solving the functionl equtions, this method for system of liner Fredholm integrl equtions of the second kind is not new method. References [1] Abboui K. nd Y. Cherruult, Convergence of Adomin s Method Applied to Differentil Equtions, Mthl. Comput. Modeling 28(5), 13-11, (1994). [2] Adomin G., Nonliner Stochstic Systems Theory nd Applictions to Physics, Kluwer, Dordrech, Hollnd, (1989). [3] Adomin G., Solving Frontier Problems of Physics: The Decomposition Method, Kluwer, (1994). [4] Cherruult Y. nd G. Sccomndi, New results for convergence of Adomin method pplied to integrl equtions, Mthl. Comput. Modelling 16 (2), 83-93, (1992). [5] Cherruult Y. nd V. Seng, The resolution of nonliner integrl equtions of the first kind using the decomposition method of Adomin, Kybernetes, 26 (2), , (1997). [6] Delves L. M. nd J. L. Mohmed, Computtionl methods for integrl equtions, Cmbridge University Press, (1985). Received: September 1, 27