IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 10, Issue 4 Ver. V (Jul-Aug. 2014), PP 69-77 www.iosrjournals.org www.iosrjournals.org 69 | Page Homotopy perturbation and Variational iteration methods for nonlinear fractional integro-differential equations M.H.Saleh¹, S.M.Amer¹, A.S.Nagdy 1 , and M.E.Alngar 1 1 (Mathematics Department, Faculty of Science, Zagazig University, Zagazig, Egypt) Abstract: In this paper, the homotopy perturbation method (HPM) and variational iteration method (VIM) are applied to approximate solutions for nonlinear fractional integro-differential equations with boundary conditions. A comparison between these methods takes place. Numerical examples are presented to illustrate the efficiency and accuracy of the proposed methods. Keywords: Boundary value problems, Caputo fractional derivative, Fractional integro-differential equations, Homotopy perturbation method and Variational iteration method. I. Introduction In recent years, fractional differential equations have attracted much more attention of physicists and mathematicians which provides an efficient for the description of many practical dynamical arising in engineering and scientific disciplines such as, physics, biology, chemistry, economy, electrochemistry, electromagnetic, control theory and viscoelasticity, see [1 − 6]. Many mathematical formulations of physical phenomena lead to integro-differential equations such as, fluid dynamics, continuum and statistical mechanics, see 7 − 9. Integro-differential equations are usually difficult to solve analytically, so it is required to obtain an efficient approximate solution. The homotopy perturbation method and variational iteration method which are proposed by He [10 − 13] are of the methods which have received much concern. These methods have been successfully applied by many authors, such as the works in [12, 14, 15]. In this paper, we applied the HPM and VIM for approximating the solution of nonlinear fractional integro- differential equations of the second kind: − , 0 = ,0< < ,1< ≤ 2, (1) where =[()] , >1, subject to the boundary conditions 0 = , (2) = 0 , (3) where indicates the Caputo fractional derivative, and is a nonlinear continuous function, , 0 are real constants, and , are given that can be approximated by Taylor polynomials. The existence and stability of solutions for fractional integro-differential equations 16, 17. Also, in this paper we use the inverse operator of then the boundary conditions are used. There are many methods for seeking approximate solutions such as variational iteration method, homotopy perturbation method, homotopy analysis method, the fractional differential transform method and Adomian decomposition method. The outline of this paper is as follows: In section 2, we present some definitions. Section 3, contains the application of the homotopy perturbation method. Section 4, contains the application of the variational iteration method. Finally, section 5, devoted to illustrate some numerical examples on mentioned methods. II. Some Definitions And Notations Definition 2.1. A real function , >0, is said to be in the space , ∈, if there exists a real number > , such that = ₁(), where ₁() ∈[0, ∞). Definition 2.2. A real function , >0, is said to be in the space , ∈, if ∈ . Definition 2.3. denotes the fractional integral operator of order in the sense of Riemann-Liouville, defined by: = 1 − 1− 0 , > 0, (), = 0. (4) Definition 2.4. Let ∈ −1 , ∈. Then the Caputo fractional derivative of (), defined by: