Life Science Journal 2013; 10(4) http://www.lifesciencesite.com http://www.lifesciencesite.com 550 lifesciencej@gmail.com On Adomian’s Decomposition Method for solving nonlocal perturbed stochastic fractional integro-differential equations Mahmoud M. El-Borai 1 , M.A.Abdou 2 , Mohamed Ibrahim M. Youssef 2 1 Department of Mathematics, Faculty of Science, Alexandria University, Alexandria, Egypt 2 Department of Mathematics, Faculty of Education, Alexandria University, Alexandria, Egypt Abstract: Adomian decomposition method (ADM) is applied to approximately solve stochastic fractional integro- differential equations involving nonlocal initial condition. The convergence of the ADM for the considered problem is proved. The mean square error between approximate solution and accurate solution is also given. [Mahmoud M. El-Borai, M.A.Abdou, Mohamed Ibrahim M. Youssef. On Adomian’s Decomposition Method for solving nonlocal perturbed stochastic fractional integro-differential equations. Life Sci J 2013;10(4):550-555]. (ISSN: 1097-8135). http://wwwlifesciencesite.com. 71 Keywords: Fractional integral; stochastic integro-differential equations; Adomian decomposition method (ADM); Adomian polynomials; Mean square error 1. Introduction Recently a great deal of interest has been focused on the application of ADM for the solution of many different problems. For example in [1-7] boundary value problems, algebraic equations, nonlinear differential equations, partial differential equations, stochastic nonlinear oscillator and nonlinear Sturm-Liouville problems are considered. The ADM, which accurately computes the series solution, is of great interest to applied sciences. This method provides the solution in a rapidly convergent series with components that are elegantly computed. The main advantage of the method is that it can be applied directly for all types of differential and integral equations, linear or nonlinear, homogeneous or inhomogeneous, with constant coefficients or with variable coefficients as the method does not need linearization, weak nonlinearity assumptions or perturbation theory. Another important advantage is that the method is capable of greatly reducing the size of computation work while still maintaining high accuracy of the numerical solution. In this paper, we investigate the applicability of ADM to the following nonlocal perturbed random fractional integro- differential equations. with the nonlocal condition where the fractional derivative is provided by the Caputo derivative and (i) the supporting set of a probability measure space (ii) is the unknown stochastic process for (iii) is called the stochastic perturbing term and it is a scalar function of and scalar (iv) is a stochastic kernel defined for and satisfying and (v) is a scalar function of scalar and will be specified later. The purpose of this paper is to apply the ADM to obtain an approximate random solution to the nonlocal Cauchy problem . We shall utilize the new formula for Adomian’s polynomials developed by El-Kalla in [6] to prove the convergence of ADM and to estimate for the error between a truncated n+1-terms and accurate solution. The studied problem may be considered a generalization to the work of G. Adomain in [8]. The nonlocal Cauchy problem has applications in many fields such as viscoelasticity, fluid mechanics and electromagnetic theory, see for example [9-13]. 2. Preliminaries Let denote a probability measure space, that is is a nonempty set known as the sample space, is a sigma-algebra of subsets of , and is a complete probability measure on . We let , denote a stochastic process whose index set is . Let be the space of all random variables which have a second moment (or square- summable) with respect to -measure for each . That is: The norm of in is