Open Journal of Applied Sciences, 2013, 3, 202-207 doi:10.4236/ojapps.2013.32027 Published Online June 2013 (http://www.scirp.org/journal/ojapps) Modified Adomian Techniques Applied to Non-Linear Volterra Integral Equations Haifa H. Ali, Fawzi Abdelwahid Department of Mathematics, Faculty of Science, University of Benghazi, Benghazi, Libya Email: fawziabd@hotmail.com Received January 28, 2013; revised March 1, 2013; accepted March 9, 2013 Copyright © 2013 Haifa H. Ali, Fawzi Abdelwahid. This is an open access article distributed under the Creative Commons Attribu- tion License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT In this work, we studied the performance of modified techniques of Adomian method applied to non-linear Volterra integral equations of the second kind. This study shows that the modified techniques are reliable, efficient and easy to use through recursive relations that involve simple integrals. Furthermore, we found that the right choice and the proper implementation of the modified techniques reduce the computational difficulties and increase the speed of convergent, comparing with the standard Adomian method. Keywords: Adomian Decomposition Method; Volterra Integral Equations 1. Introduction In recent years, many works have been focusing on the developing and applying of advanced and efficient meth- ods for integral equations such as implicitly linear collo- cation methods [1], product integration method [2], Her- mite-type collocation method [3] and analytical (semi- analytical) techniques such as Adomian decomposition method [4,5]. In this work, we investigate the perform- ance of modified techniques of Adomian decomposition method applied to non-linear Volterra integral equations of the second kind. This type of integral equations has the following form          0 , x ux f x kxtFut t    d . (1.1) Equation (1.1) represents a nonlinear Volterra integral equation of second kind with unknown function   ux and   F u is a non-linear function of   ux , and we assumed that, the kernel   , kxt and the function   f x are analytical functions on and , respec- tively. Hence, Equation (1.1) classifies as a linear Volterra integral equation of second kind if 2 R R   F u is a linear function of the unknown function .   ux 2. Standard Adomian Method The standard technique for the non-linear integral Equa- tion (1.1), starts by decomposing into , and assuming that   ux 0 1 2 , , , u u u      0 lim n n i i u x ux           . (2.1) For the non-linear function   F u , we set   0 n n F u     A (2.2) In (2.2), n A ,   0 n  are special polynomials known as Adomian polynomials. In ref. [6], close formulas of these polynomials, for any non-linear function   F u , introduced in the terms of the Kronecker delta , nm  . With   0 0 A Fu  , these formulas for read 1, 2, n     0 1 0 d , d n n v v F u A H u      (2.3) where 1 2 1 2 1 2 1 1 , , , , 1 ! v v v n v v ni i i i i i i i i H uu u v            Now Equation (1.1) becomes,         0 1 0 0 0 , , , , x n n n n u x f x kxt A u u u t                 d n , (2.4) Now     0 u x f x  and the can be completely determined by using the recurrent formula     ; 1, 2, i u x i   Copyright © 2013 SciRes. OJAppS