Research Article Application of Sumudu Decomposition Method to Solve Nonlinear System Volterra Integrodifferential Equations Hassan Eltayeb, 1 Adem KJlJçman, 2 and Said Mesloub 1 1 Mathematics Department, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia 2 Department of Mathematics, Institute for Mathematical Research, Universiti Putra Malaysia, (UPM), 43400 Serdang, Selangor, Malaysia Correspondence should be addressed to Said Mesloub; mesloubs@yahoo.com Received 18 February 2014; Accepted 4 April 2014; Published 28 April 2014 Academic Editor: Mohamed Boussairi Jleli Copyright © 2014 Hassan Eltayeb et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We develop a method to obtain approximate solutions for nonlinear systems of Volterra integrodiferential equations with the help of Sumudu decomposition method (SDM). Te technique is based on the application of Sumudu transform to nonlinear coupled Volterra integrodiferential equations. Te nonlinear term can easily be handled with the help of Adomian polynomials. We illustrate this technique with the help of three examples and results of the present technique have close agreement with approximate solutions which were obtained with the help of Adomian decomposition method (ADM). 1. Introduction Te linear and nonlinear Volterra integral equations arise in many scientifc felds such as the population dynamics, spread of epidemics, and semiconductor devices; for more details, see [1]. Te scientists in diferent branches of science have been trying to solve this kind of problems; however, fnding an exact solution is not easy due to the nonlinear part of these type groups of equations. Diferent analytical methods have been developed and applied to fnd a solution. For example, Adomian has introduced a so-called decomposition method for solving algebraic, diferential, integrodiferential, diferential-delay, and partial diferential equations. In the nonlinear case for ordinary diferential equations and partial diferential equations, the method has the advantage of dealing directly with the problem [2, 3]. Tese equations are solved without transforming them to equivalent form which is more simple. Te method avoids linearization, perturbation, discretization, or any unrealistic assumptions; see [4, 5]. It was also suggested in [6] that the noise terms appear always for inhomogeneous equations. Tus, most recently, Wazwaz [7] established a necessary condition that is essentially needed to ensure the appearance of “noise terms” in the inhomogeneous equations. Te integral transform has been used to solve many dif- ferent types of diferential and integrodiferential equations. For similar problems, Sumudu transform was introduced and further applied to several ODEs as well as PDEs. For example, in [8], this transform was applied to the one-dimensional neutron transport equation. In [9], Sumudu transform was extended to the distributions and some of their properties were also studied in [10]. Recently, Kılıc ¸man et al. applied this transform to solve the system of diferential equations (see [11]), since there are some interesting properties that Sumudu transform satisfes such as if ()= ∞ ∑ =0     , then ()= ∞ ∑ =0 !    ; (1) see [12]. In the present paper, the intimate connection between Sumudu transform theory and decomposition method aris- ing in the solution of nonlinear Volterra integrodiferential equation is demonstrated. Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 503141, 6 pages http://dx.doi.org/10.1155/2014/503141