Abstract
In this paper, we consider the linear system of Fredholm integral equation of the second kind. Three methods are used to solve this system, successive approximation method, Aitken's method depending on successive approximation method and a new procedure which is Aitken's method depending on Adomian decomposition method. A comparison between approximate and exact results for two numerical examples depending on the least-square error, are given to show the accuracy of the results obtained by using these methods.
Highlights
(Saeed, 2006) used iteration methods for solving linear system of Volterra integral and integro differential equations.(Babolian et al, 2004), used the Adomian decomposition method for solving linear system of Fredholm integral equations of the second kinds (LSFIEs,2nd)
LSFIEs,2nd can be written in the following form (Babolian, et al 2004): nb ui (x) fi (x) kij (x,t)u j (t)dt, x [a,b], j 1 a where fi (x), kij (x,t), i, j=1, 2, ..., n are known continuous functions and ui (x), i=1, 2, ..., n are unknown functions
This paper presents the use of successive approximation method, Aitken’s method depending on successive approximation method, and Aitken’s method depending on Adomian decomposition method for solving linear system of Fredholm integral equation of the second kind
Summary
(Saeed, 2006) used iteration methods for solving linear system of Volterra integral and integro differential equations.(Babolian et al, 2004), used the Adomian decomposition method for solving linear system of Fredholm integral equations of the second kinds (LSFIEs,2nd). We apply Aitken’s method depending on successive approximation method for solving equation (1) as follows: In equations (2.a0), (2.a1) and (2.a2) find ui0(x), ui1(x) and ui2(x) respectively and substitute these values in Aitken’s formula we get ui (x) ui0 (x)ui (x) ui21(x) ui (x) 2ui1(x) ui0 (x). For the first time Aitken’s method has been used successfully on Adomian decomposition method as extrapolated formula to find approximate (sometime exact, depending on the number of iterations) solution of equation (1) where four successive approximated values of the unknown function ui(x) are found to activate this method by using Adomian decomposition method as follows: Step 1: Recall equation (1): ui (x) fi (x) kij (x,t)u j (t)dt, j 1 a.
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