IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 10, Issue 6 Ver. I (Nov - Dec. 2014), PP 01-05 www.iosrjournals.org www.iosrjournals.org 1 | Page Adomian Decomposition Method for Solving Delay Differential Equations of Fractional Order Osama H. Mohammed and Abbas I. Khlaif Al-Nahrain University, College of Science, Department of Mathematics and Computer Applications Abstract: In this paper, we implement Adomian decomposition method for solving numerically non-linear delay differential equations of fractional order. The fractional derivative will be in the Caputo sense. In this approach, the solutions are found in the form of a convergent power series with easily computed components. Some numerical examples are presented to illustrate the accuracy and ability of the proposed method. Keywords: Adomian decomposition method, delay differential equations, fractional calculus, fractional delay differential equations. I. Introduction The subject of fractional calculus (that is calculus of integrals and derivatives of any arbitrary real or complex order) has gained considerable popularity and importance during the past three decades or so, due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering. It does indeed provide several potentially useful tools for solving differential and integral equations, and various other problem involving special functions of mathematical physics as well as their extensions and generalizations in one and more variables [1]. In real world systems, delays can be recognized everywhere and there has been widespread interest in the study of delay differential equations for many years. However, fractional delay differential equations(FDDE S ) are a very recent topic.Although it seems natural to model certain processes and systems in engineering and other science (with memory and heritage properties) with this kind of equations, only in the last few years has the attention of the scientific community been devoted to them [2]. Concerning the existence of solutions of (FDDE S ) we refer [3-5]. In [3] Lakshmikantham provides sufficient conditions for the existence of solutions to initial value problems to single term nonlinear delay fractional differential equations, with the fractional derivative defined in the Riemann-Liouville sense. In [4],Yeetal. investigate the existence of positive solutions for a class of single term delay fractional differential equations.Later in [5], for the same class of equations, sufficient condition for the uniqueness of the solution are reported [2]. For the stability issues of the (FDDE S ) we refer the references [6-9]. In this paper we shall use the Adomian decomposition method to find the approximate solution of the (FDDE S ) with variable delays. At the beginning of 1980 Adomian proposed new method to solve some functional equations [10,11].The Adomian decomposition method has the advantage of converging to the exact solution, and can easily handle a wide class of both linear and nonlinear differential and integral equations. It decomposes the solution into the series with easily computed components which converges rapidly to the exact solution.The theoretical treatment of the convergence of the Adomian decomposition method has been considered in [12,13,14,15]. The structure of this paper is organized as follows: In section 2, we recall the definitions of fractional derivatives and fractional integration in section 3 the basic concept of the Adomian decomposition method will be given in section 4 we present our approach to solve the delay differential equation of fractional order in section 5 numerical examples are given followed by conclusions in section 6. II. Fractional Derivative and Integration In this section we shall review the basic definitions and properties of fractional integral and derivatives, which are used further in this paper[16]. Definition(1):The Riemann-Liouville fractional integral operator of order ˃0 is defined as: