Australian Journal of Basic and Applied Sciences, 7(7): 128-139, 2013 ISSN 1991-8178 Corresponding Author: S.H. Behiry, General Required Courses Department, Jeddah Community College, King Abdulaziz University, Jeddah 21589, Kingdom of Saudi Arabia. E-mail: salah_behiry@hotmail.com 128 Nonlinear integro-differential equations by differential transform method with Adomian polynomials S. H. Behiry General Required Courses Department, Jeddah Community College, King Abdulaziz University, Jeddah 21589, Kingdom of Saudi Arabia. Abstract: A modification of differential transformation method is applied to nonlinear integro- differential equations. In this technique, the nonlinear term is replaced by its Adomian polynomials for the index k, and hence the dependent variable components are replaced in the recurrence relation by their corresponding differential transform components of the same index. Thus the nonlinear integro- differential equation can be easily solved with less computational work for any analytic nonlinearity due to the properties and available algorithms of the Adomian polynomials. New theorems for products and integrals with nonlinear functions are introduced. Several illustrative examples with different types of nonlinearities are considered to indicate the effectiveness of the present technique. Key words: Differential transform method; nonlinear integro-differential equations; Adomian polynomials. INTRODUCTION Integral and integro-differential equations play an important role in characterizing many social, biological, physical and engineering problems; for more details see (Kythe and Puri, 1992; Wazwaz, 2006; Rashed, 2004) and references cited therein. Nonlinear integral and integro-differential equations are usually hard to solve analytically and exact solutions are rather difficult to be obtained. In literature there exist many numerical methods have been studied such as Haar wavelets method (Aziz and Islam, 2013), the rationalized Haar functions method (Maleknejed and Mirzaee, 2006; Reihani and Abadi, 2007), the linearization method (Darania et al., 2006), the finite difference method (Zhao and Corless, 2006), the Tau method (Abbasbandy and Taati, 2009; Ebadi et al., 2007), the hybrid Legendre polynomials and block-pulse functions (Maleknejad et al., 2011), the Adomian decomposition method (Wazwaz, 2010; Araghi and Behzadi, 2009), the Taylor polynomial method (Darania and Ivaz, 2008; Maleknejad and Mohmoudi, 2003; Yalcinbas, 2012) and the differential transform method (Borhanifar and Abazari, 2012). The differential transform method (DTM) has been proved to be efficient for handling nonlinear problems, but the nonlinear functions used in these studies are restricted to polynomials and products with derivatives (Borhanifar and Abazari, 2012; Arikoglu and Ozkol, 2005; Arikoglu and Ozkol, 2008; Odibat, 2008; Biazar and Eslami, 2011). For other types of nonlinearities, the usual way to calculate their transformed functions as introduced by (Zhou, 1986) is to expand the nonlinear function in an infinite power series then take the differential transform of this series. The problem with this approach is that the massive computational difficulties will arise in determining the differential transform of nonlinear function while working with this infinite series. Another approach for obtaining the differential transform of nonlinear terms is the algorithm in (Chang and Chang, 2008). It is based on using the properties of differential transform and calculus to develop a canonical equation. Then this equation is solved for the required differential transform of nonlinear term. But, as seen in the simple examples in section 3 in (Chang and Chang, 2008), the algorithm requires a sequence of differentiation, algebraic manipulations and computations of differential transform for other functions which is more difficult for the case of composite nonlinearities. In this work, we introduce a comprehensive and more efficient approach for using the DTM to solve nonlinear integro-differential equations; the idea is based on the methodology in (Elsaid, 2012). The nonlinear function is replaced by its Adomian polynomials and then the dependent variable components are replaced by their corresponding differential transform component of the same index. This technique benefits the properties of the Adomian polynomials and the efficient algorithm to generate them quickly as in the work (Duan, 2010, 2011). Numerical simulations of integro-differential equations with different types of nonlinearity are treated and the proposed technique has provided good results. 2. Differential Transform Method: The basic definition and the fundamental theorems of the differential transformation and its applicability for various kinds of differential and integral equations are given in (Arikoglu and Ozkol, 2005; Arikoglu and Ozkol, 2008; Odibat, 2008; Biazar and Eslami, 2011; Zhou, 1986). For convenience of the reader, a review of