Indonesian Journal of Electrical Engineering and Computer Science Vol. 11, No. 3, September 2018, pp.857∼867 ISSN: 2502-4752, DOI:10.11591/ijeecs.v11.i3.pp857-867  857 Homotopy Analysis Method for the First Order Fuzzy Volterra-Fredholm Integro-Differential Equations Ahmed A. Hamoud 1 and Kirtiwant P. Ghadle 2 1,2 Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad-431004 (M.S.) India. 1 Department of Mathematics, Taiz University, Taiz, Yemen. Article Info Article history: Received April 28, 2018 Revised May 27, 2018 Accepted May 27, 2018 Keyword: Homotopy analysis method Fuzzy Volterra-Fredholm integro-differential equation Existence and uniqueness results. ABSTRACT A fuzzy Volterra-Fredholm integro-differential equation (FVFIDE) in a parametric case is converted to its related crisp case. We use homotopy analysis method to find the approximate solution of this system and hence obtain an approximation for the fuzzy solution of the FVFIDE. This paper discusses existence and uniqueness results and convergence of the proposed method. Copyright c  2018 Institute of Advanced Engineering and Science. All rights reserved. Corresponding Author: Ahmed A. Hamoud Department of Mathematics, Taiz University, Taiz, Yemen. Email: drahmed985@yahoo.com 1. INTRODUCTION In recent years, the topics of fuzzy integral equations which attracted increasing interest, in particular in relation to fuzzy control, have been rapidly developed. The concept of fuzzy numbers and arithmetic operations firstly introduced by Zadeh [1], and then by Dubois and Prade (1978). Also, in [2] have introduced the concept of integration of fuzzy functions. The fuzzy mapping function was introduced by Cheng and Zadeh [1]. Moreover, Dubois and Prade [3] presented an elementary fuzzy calculus based on the extension principle. The fuzzy integro-differential equations are a natural way to model uncertainty of dynamical systems. Kaleva [4] chose to define the integral of the fuzzy function, using the Lebesgue-type concept for integration. Recently, Hence various other methods for solving them such as using homotopy perturbation method (Narayanamoorthy and Sathiyapriya 2016) [5], expansion method (Allahviranloo et al. 2014) [6], Laplace transformation method (Das and Talukdar 2014) [7], homotopy analysis method (Hussain and Ali 2013) [8], differential transform method (Behiry and Mohamed 2012) [9], fixed point theorems (Rahimi et al. 2011) [10], variational iteration method (Hashemi and Abbasbandy 2011) [11]. Also, some mathematicians have studied fuzzy integral and integro-differential equation by numerical techniques [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 26]. As we know the fuzzy integral and integro-differential equations are one of the important parts of the fuzzy analysis theory that play a main role in the numerical analysis. In this work, we will examine HAM to approximate the solution of the fuzzy Volterra-Fredholm integro-differential equation of the second kind. The structure of this paper is organized as follows: In Section 2, we state some known notations and definitions and also some theorems which are used throughout this paper. In Section 3, the fuzzy Volterra-Fredholm integro-differential equation of the second kind is briefly presented. In Section 4, we convert a fuzzy Volterra-Fredholm integro-differential equation of the second kind to the system of Volterra-Fredholm integro- differential equation of the second kind in a crisp case and approximate with HAM. In Section 5, the existence and uniqueness results and convergence of the proposed method is proved. In Section 6, the analytical example is presented illustrate the accuracy of this method. Finally, we will give a report on our paper and a brief conclusion in Section 7. Journal Homepage: http://iaesjournal.com/online/index.php/IJECE