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Journal of Intelligent & Fuzzy Systems 27 (2014) 211–220 DOI:10.3233/IFS-130990 IOS Press
Application of hat functions to solve linear Fredholm fuzzy integral equation of the second kind Farshid Mirzaeea , Mahmoud Paripourb,∗ and Mohammad Komak Yaria a Department b Department
of Mathematics, Faculty of Science, Malayer University, Malayer, Iran of Science, Hamedan University of Technology, Hamedan, Iran
Abstract. In this paper, we present an efficient direct method for solving Fredholm fuzzy integral equation of the second kind based on hat functions. The method is based on new vector forms for representation of hat functions and its operational matrix. We have not used any integration, so all calculations can be easily implemented. Moreover, the convergence of the proposed method is proved. Finally, illustrative examples are included to show accuracy and efficiency of the proposed method. Keywords: Fuzzy number, linear Fredholm fuzzy integral equation, hat functions
1. Introduction Fuzzy integral equations and fuzzy differential equations are important for studying and solving large proportions of problems in many topics in applied mathematics, particularly in relation to physics, geography, medicine, biology, etc. In many applications, some of our parameters are represented by fuzzy numbers rather than the crisp, and hence it is important to develop mathematical models and numerical procedures which would have appropriately treated general fuzzy integral equations, fuzzy differential equations and solved them. The knowledge about differential equations is often incomplete or vague. Fuzzy differential equations were first formulated by Kaleva [26] and Seikkala [40]. The fuzzy differential equations theory is well developed [1, 27, 35, 43]. ∗ Corresponding
author. Mahmoud Paripour, Department of Science, Hamedan University of Technology, Hamedan 65156579, Iran. E-mails: m
[email protected] (Mahmoud Paripour);
[email protected] (Farshid Mirzaee); m komakyari@yahoo. com (Mohammad Komak Yari).
The concept of integration for fuzzy functions was first introduced by Dubois and Prade [16]. Alternative approaches were later suggested by Goetschel and Voxman [24], Kaleva [26], Nanda [34], Mordeson and Newman [32] and others. While Goetschel and Voxman [24] preferred a Rimann integral type approach, Kaleva [26] chose to define the integral of fuzzy function by using the Lebesgue type concept for integration. One of the first applications of fuzzy integration was given by Wu and Ma [42]. They investigated the Fuzzy Fredholm Integral equation of second kind (FFIE-2). This work which established the existence of a unique solution to (FFIE-2), was followed by other works on (FIE) [36], where a fuzzy integral equation was replaced by an original fuzzy differential equation. In [6], the authors introduced some quadrature rules for solving the integral of fuzzy-number-valued mappings. In [18], the authors gave one of the applications of fuzzy integral for solving FFIE-2. Friedman et al. [19] presented a numerical algorithm for solving FFIE-2 with arbitrary kernel. Also, they investigated numerical procedures for solving FFIE-2 by using the embedding method [21].
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