World Applied Sciences Journal 19 (12): 1731-1745, 2012 ISSN 1818? 4952 © IDOSI Publications, 2012 DOI: 10.5829/idosi.wasj.2012.19.12.111 Corresponding Author: Dr. S. Abbasbandy, Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran 1731 A Solution of Arbitrary Fully Fuzzy Linear Systems S. Abbasbandy, T. Allahviranloo and S. Moloudzadeh Department of Mathematics, Science and Researh Branch, Islamic Azad University, Tehran, Iran Abstract: In this paper, fully fuzzy linear systems in the form   % A X=b (FFLS) will be discussed, where  nn A × is a fuzzy matrix, x and b are (n × 1) fuzzy vector. Transforming fully fuzzy linear system in to two crisp linear systems and using the Jacobi iterative and Adomian Decomposition Methods (ADM) a FFLS will be solved.We will show that to find a solution for a (FFLS) our method needs lees iterations in comparison with the Jacobi iterative and Adomian decomposition to the Dehghan methods. For a FFLS it is shown that the ADM is equivalent to the Jacobi iterative method with the less iteration. Key words: LR fuzzy number • fuzzy matrix • fuzzy arithmetic • Fully Fuzzy Linear System (FFLS) • Jacobi iterative method • Adomian decomposition method INTRODUCTION Systems of linear equations are used to solve many problems in various areas such as structural mechanic applications in civil and mechanical structures, heat transport, fluid flow, electromagnetism and etc. In many applications, at least one of the system's parameters and measurements are vague or imprecise and we can present them with fuzzy numbers rather than crisp numbers. Hence, it is important to develop mathematical models and numerical procedure that would appropriately treat general fuzzy systems and solve them. Fuzzy linear system Ax = b, where A is a crisp matrix and b is a fuzzy number vector have been solved by Friedman and his colleagues. Using the embedding approach Friedman et al . proposed a general model to solving such a fuzzy linear systems [5, 18, 19], indeed in this way whenever a solution is obtained, the original n× n fuzzy linear system is replaced by a 2n × 2n crisp linear system. They also derived conditions for the existence of a unique fuzzy solution for (FSLE) and designed a numerical procedure to calculating the solution. So far many works have been done to find the solution of the 2n × 2n crisp linear systems [2-5, 10, 16, 24, 25]. Of course a lot of works with different methods have been done in this regard, among these some works have been done in which all parameters in the fuzzy linear system are fuzzy numbers and we call it Fully Fuzzy Linear System (FFLS) [6, 12-15, 22, 23]. Here we will solve Ax = b, where A is a fuzzy matrix and x and b are fuzzy vectors. We will use fuzzy matrix defined in [17]. This class of fuzzy matrices consist of applicable matrices, which uncertain aspects can be modeled by them and working on them are too limited. Buckley and Qu [12] took the α-levels of the equations and obtained linear systems of interval equations. The exact solution of FFLS can be found by solving these interval systems. Muzzioli and Reynaerts in [22] studied FFLS of the form A 1 x + b 1 = A 2 x + b 2 . The link between interval linear systems and fuzzy linear systems is have been clarified by them. Their approach contains solving of 2 n(n+1) crisp systems for all α [0,1]. However, the classical solution to an FFLS often fails to exist [23]. Dehghan et al . [13-15] has studied some methods for solving FFLS. They have represented fuzzy numbers in LR form and applied approximate operators between fuzzy numbers to find positive solutions of FFLS, so calculating the solutions of FFLS is transformed to calculation the solutions of three crisp systems. To finding a non-zero solution for the FFLS, Allahviranloo et al . [6] introduced an algorithm. In this algorithm at first n× n fully fuzzy linear system Ax = b is transformed in to a 2n × 4n fuzzy linear system, then again it is transformed in to a 2n × 2n parametric system. Allahviranloo et al . [8, 9] proposed a new method to obtain symmetric solutions (bounded and symmetric solutions) of a fully fuzzy linear systems (FFLS) based on a 1-cut expansion. Liu proposed [21] a solution of FFLS by developing them to a block of HPM to find an approximation of the solution. The numerical results display that the block HPM converges to the exact solution rapidly than the direct method and the iterative Jacobi and Gauss-seidel methods in solving the FFLS.