%0 Journal Article
%T Application of Homotopy Perturbation Method for Fuzzy Linear Systems and Comparison with Adomian＊s Decomposition Method
%A H. Saberi Najafi
%A S. A. Edalatpanah
%A A. H. Refahi Sheikhani
%J Chinese Journal of Mathematics
%D 2013
%R 10.1155/2013/584240
%X We present an efficient numerical algorithm for solution of the fuzzy linear systems (FLS) based on He＊s homotopy perturbation method (HPM). Moreover, the convergence properties of the proposed method have been analyzed and also comparisons are made between Adomian＊s decomposition method (ADM) and the proposed method. The results reveal that our method is effective and simple. 1. Introduction Consider the following linear system: where denotes a vector in a finite-dimensional space and . Linear systems are of fundamental importance in various fields of science and engineering and there are numerous methods to find a solution for them; see [1每8] and the references therein. When coefficients of the system are imprecise, accessing the solution with no ambiguity will be difficult. Therefore, fuzzy logic was proposed by Zadeh (1965) and encountered seriously over Aristotle classic logic. Professor Zadeh discussed fuzzy set theory, fuzzy number concept, and arithmetical operation with these numbers in [9每12] and following that many articles and books were published in fuzzy systems. We refer the reader to [13] for more information on fuzzy numbers and fuzzy arithmetic. Fuzzy systems are used to study a variety of problems including fuzzy metric spaces [14], fuzzy differential equations [15], particle physics [16, 17], game theory [18], optimization [19], and especially fuzzy linear systems [20每24]. Fuzzy number arithmetic is widely applied and useful in computation of linear system whose parameters are all or partially represented by fuzzy numbers. Friedman et al. [20] introduced a general model for solving a fuzzy linear system whose coefficient matrix is crisp and the right-hand side column is an arbitrary fuzzy number vector. They used the parametric form of fuzzy numbers and replaced the original fuzzy linear system by a crisp linear system. Furthermore, some iterative methods have been presented for solving such fuzzy linear systems [25每32]. In this paper, we solve fuzzy linear systems (FLS) via an analytical method called homotopy perturbation method (HPM). This method was first proposed by the Chinese mathematician He in 1999 [33] and was further developed and improved by him [34每36]. This author presented a homotopy perturbation technique based on the introduction of homotopy in topology coupled with the traditional perturbation method for the solution of algebraic equations. This technique provides a summation of an infinite series with easily computable terms, which converges rapidly to the solution of the problem. In the literature, various authors
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