Journal of Interpolation and Approximation in Scientific Computing 2013 (2013) 1-13 Available online at www.ispacs.com/jiasc Volume 2013, Year 2013 Article ID jiasc-00016, 13 Pages doi:10.5899/2013/jiasc-00016 Research Article A new computational method for solving the first order linear fuzzy Fredholm integro-differential equations Mojtaba Ghanbari 1 ∗ , Rahele Nuraei 2 (1)Department of Mathematics, Aliabad Katoul Branch, Islamic Azad University, Aliabad Katoul, Iran. (2)Department of Mathematics, South Tehran Branch, Islamic Azad University, Tehran, Iran. Copyright 2013 c ⃝ Mojtaba Ghanbari and Rahele Nuraei. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Recently, fuzzy integro-differential equations (FIDEs) have attracted some interest. In this paper, we focus on linear fuzzy Fredholm integro-differential equation and propose a new method for solving it. In fact, using parametric form of fuzzy numbers we convert a linear fuzzy Fredholm integro-differential equation to a linear system of Fredholm integro-differential equations in crisp case. We use variational iteration method (VIM) to obtain solution of this sys- tem and hence obtain a fuzzy solution of the linear fuzzy Fredholm integro-differential equation. Finally, using the proposed method, we give some illustrative examples. Keywords: Fuzzy functions; Fuzzy integro-differential equations; Fuzzy numbers; System of linear Fredholm integro-differential equations; Variational iteration method 1 Introduction Prior to discussing fuzzy integro-differential equations and their solving, it is necessary to present an appropriate brief introduction to preliminary topics such as fuzzy numbers and fuzzy calculus. The concept of fuzzy sets, was originally introduced by Zadeh [26], led to the definition of fuzzy numbers and its implementation in fuzzy control [3] and approximate reasoning problems [27]. The basic arithmetic structure for fuzzy numbers was later developed by Mizumoto and Tanaka [16], Nahmias [17], Dubois and Prade [4], all of them observed fuzzy numbers as a collection of α -levels, 0  α  1. Goetschel and Voxman [8] suggested a new approach. They represented fuzzy number as a parameterized triple (see Section 2) and then embedded the set of fuzzy numbers into a topological vector space. This enabled them to design the basics of a fuzzy calculus. The subject of embedding fuzzy numbers in either a topological or a Banach space was investigated also by Puri and Ralescu [21, 22], Kaleva [12] and Ouyang [20]. Solving integro-differential equations requires appropriate and applicable definitions of fuzzy function, the fuzzy derivative and the fuzzy integral of a fuzzy function. The fuzzy mapping function was introduced by Chang and Zadeh [3]. Later, Dubois and Prade [5] presented an elementary fuzzy calculus based on the extension principle [26]. Puri and Ralescu [21] suggested two definitions for fuzzy derivative of fuzzy functions. The first method was based on the H-difference notation and was further investigated by Kaleva [12]. The second method was derived from the embedding technique and was followed by Goetchel and Voxman [8] who gave it a more applicable representation. The concept of integration ∗ Corresponding author. Email address: Mojtaba.Ghanbari@gmail.com, Tel:+989113222331.