IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 10, Issue 5 Ver. V (Sep-Oct. 2014), PP 72-77 www.iosrjournals.org www.iosrjournals.org 72 | Page Modification Adomian Decomposition Method for solving Seventh OrderIntegro-Differential Equations Samaher M. Yassien Department of MathematicsCollege of Education for Sciences pure Baghdad University,Iraq Abstract: In this paper, a method based on modified adomian decomposition method for solving Seventh order integro-differential equations (MADM). The distinctive feature of the method is that it can be used to find the analytic solution without transformation of boundary value problems. To test the efficiency of the method presented two examples are solved by proposed method. Keyword: Adomian decomposition method; boundary-value problems; integro-differential equation I. Introduction An analytical method called the Adomian decomposition method (ADM) proposed by Adomian [1] aims to solve frontier nonlinear physical problems. It has been applied to a wide class of deterministic and stochastic problems, linear and nonlinear, in physics, biology and chemical reactions etc. For nonlinear models, the method has shown reliable results in supplying analytical approximations that converge rapidly [2].The Adomian decomposition method (ADM) [3,4] has been efficiently used to solve linear and nonlinear problems such as differential equations and integral equations.The methodprovides the solution as an infinite series in which each term can be easily determined. The rapid convergence of the series obtained by this method is horoughly discussed byCherruault et al. [5]. Recently, Wazwaz [6] proposed a reliable modified techniqueof ADM that accelerates the rapid convergence of decomposition series solution. The modified decomposition needs only a slight variation from the standard decompositionmethod. Although the modified decomposition method may provide the exact solution by using two iterations only and sometimes without using the so-called Adomian polynomials, its effectiveness is based on the assumption that the function can be divided into two parts, and thus the success of themodifiedmethod depends on the proper choice of 1 and 2.The ADM[7,8,9] is a well-known systematic method for solving linear and nonlinear equations, including ordinary differential equations,partial differential equations, integral equations and integro-differential equations. The method permits us to solve both nonlinearinitial value problems and boundary value problems. The method is well known, and several advanced progresses are conducted inthis regard. II. Description of the Modification Adomian decomposition method Since the beginning of the 1980s, theAdomian decompositionmethod has been applied to a wide class of integral equations.To illustrate the procedure, consider the following Volterraintegral equations of the second kind given by L(y ()) =()+ K(x, t) x a (R (y ()) + (y ())) d, ≠ 0, (1) where the kernel (, ) and the function () are given realvalued functions, is a parameter, R(y()) and (y()) are linear and nonlinear operators of y()[10],the differential operator L(y ())is the highest order derivative in the equation, respectively, Then, we assume that L is invertible byusing the given conditions and applying the inverse operator L -1 to both sides of (1) ,we get the following equation: y () = 0 +L -1 ()+L -1 ( K(x, t) x a (R (y ()) + (y ())) d), ≠ 0, (2) where the function 0 is arising from integrating the source term from applying the given conditions which are prescribed. And so on the Adomian decomposition method admits the decomposition of y into an infinite series of components [11] y(x)= y ∞ n=0 n (x)(3) Moreover, the Adomian decomposition method identifies the nonlinearterm(y()) by the decomposition series