IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 8, Issue 1 (Sep. - Oct. 2013), PP 18-22 www.iosrjournals.org www.iosrjournals.org 18 | Page Numerical Experiment of Systems of Non-Linear Volterra’s Integro-Differential Equations Using New Variational Homotopy Perturbation Method O. E. Abolarin Department of Industrial Mathematics Landmark University, Omu-Aran, Nigeria. Abstract: This research paper deals with the s o l u t i o n o f systems of Non-Linear Volterra’s Integro- Differential Equations using the New V ariational Homotopy Perturbation Method. The New Method does not require discritization, linearization or any restrictive assumption of any form in providing analytical or approximate solutions to linear and nonlinear equation. Theses virtues make it to be reliable and its efficiency is demonstrated with numerical examples. Keywords: systems of Non-Linear Volterra’s Integro-Differential equations; Variational Iteration Method, Homotopy Perturbation Method, N e w Variational Homotopy Perturbation Method. I. Intro duction One of the interesting topics among researchers is solving integro-differential equations. In fact, integro-differential equations arise in many physical processes, such as glass-forming process [1], nanohydrodynamics [2], drop wise condensation [3], and wind ripple in the desert [4]. There are various numerical and analytical methods to solve such problems, for example, the Homotopy perturbation method [5], the Adomian decomposition method [6], but each method limits to a special class of integro-differential equations. J.H. He used the variational iteration method for solving some integro-differential equations [7]. This Chinese mathematician chooses [7] initial approximate solution in the form of exact solution with unknown constants. M. Ghasemi et al solved the nonlinear Volterra's integro-differential equations [8] by using homotopy perturbation method. In [9], the variational iteration method was applied to solve the system of linear integro- differential equations. Also, J. Biazar et al solved systems of integro-differential equations by He's homotopy perturbation method [10]. S. Abbasbandy and E. Shivanian solved system of nonlinear volterra’s integro- differential equations using Variational Iteration Method[11].The aim of this paper is to extend the analysis of the variational Homotopy Perturbation method to solve the system of nonlinear Volterra's integro-differential equations, as demonstrated by M. Matinfar et al [12], O. A. Taiwo and O. E. Abolarin [13]. II. New Variational Homotopy Perturbation Method To demonstrate the new variational homotopy perturbation method, we consider the first order integro-differential equation given by dt t u t u t x f dx du x )) ( ), ( , ( ) ( 0      (1) where ) ( x f is the source term and ) ( x u is the unknown which is to be determined via the new variational Homotopy perturbation method. The correction functional according to variational iteration method can be constructed as follows:         s s n x n n ds dt t u t u t s f u s x u x u 0 0 1 ] )) ( ) ( , ( ) ( ) ( )[ ( ) ( ) (   (2) where n u is considered as restricted variations, which means 0  n u . To find the optimal ) ( s  , we proceed as follows:         s s n x n x n ds dt t u t u t s f u s x u u 0 0 ) ( 1 ] )) ( ) ( , ( ) ( ) ( )[ ( ) (      (3) And consequently, ds u s x u u s n x n x n ) ( ) ( ) ( 0 ) ( 1         (4)