International Journal of Computer Applications (0975 – 8887) Volume 64– No.6, February 2013 5 Numerical Solution of n-th Order Fuzzy Linear Differential Equations by Homotopy Perturbation Method Smita Tapaswini Department of Mathematics, National Institute of Technology, Rourkela Odisha - 769 008, India S. Chakraverty Department of Mathematics, National Institute of Technology, Rourkela Odisha - 769 008, India ABSTRACT This paper targets to investigate the numerical solution of n - th order fuzzy differential equations with fuzzy environment using Homotopy Perturbation Method (HPM). Triangular fuzzy convex normalized sets are used for the fuzzy parameter and variables. Obtained results are compared with the existing solution depicted in term of plots to show the efficiency of the applied method. Keywords n -th order fuzzy linear differential equations, Fuzzy Number, Triangular Fuzzy Number, Homotopy Perturbation Method (HPM). 1. INTRODUCTION Theory of fuzzy differential equations plays an important role in modeling of science and engineering problems because this theory represents a natural way to model dynamical systems under uncertainty. There exist a large number of papers dealing with fuzzy differential equations and its applications in the open literatures. Some of are reviewed and cited here for better understanding of the present analysis. Chang and Zadeh [15] first introduced the concept of a fuzzy derivative, followed by Dubois and Prade [16] who defined and used the extension principle in their approach. The fuzzy differential equation and fuzzy initial value problems are studied by Kaleva [28, 29] and Seikkala [36]. Various numerical methods for solving fuzzy differential equations are introduced in [1, 2, 6, 31, 32, 37]. Very recently Tapaswini and Chakraverty [37] have proposed a new method to solve fuzzy initial value problem. Bede [11] described the exact solutions of fuzzy differential equations in his note in an excellent way. Buckley and Feurin [14] applied two analytical methods for solving n -th order linear differential equations with fuzzy initial conditions. Similarly many authors studied various other methods to solve n -th order fuzzy differential equations in [3, 4, 5, 7, 26]. Based on the idea of collocation method Allahviranloo et al. [5] investigated the numerical solution of n -th order fuzzy differential equations. Abbasbandy et al. [3] applied Runge-Kutta method for the numerical solution of n -th order fuzzy differential equations. The analytical method (eigenvalue-eigenvector method) for n -th order fuzzy differential equations with fuzzy initial value is also discussed by Allahviranloo et al. [7]. Abbasbandy et al. [4] and Jafari et al. [26] used variational iteration method for solving n -th order fuzzy differential equations recently. Besides the above approaches Homotopy Perturbation Method (HPM) is also found to be a powerful tool for solving the fuzzy differential equations. The HPM was first developed by He [20, 21] and many authors applied this method to solve various linear and non-linear differential equations of scientific and engineering problems. The solution is considered as the sum of infinite series, which converges rapidly to accurate solutions. In the homotopy technique (in topology), a homotopy is constructed with an embedding parameter which is considered as a "small parameter". Very recently HPM has been applied to a wide class of physical problems [10, 12, 13, 17, 22, 23, 24, 25, 30, 33, 34, 39, 40, 41, 42, 43]. In these papers the parameters and variables are considered as crisp (exact). Few researchers have also investigated the solution of fuzzy differential equations using HPM [8, 9, 18, 38]. Allahviranloo et al. [8, 9] applied homotopy perturbation method (HPM) to solve fuzzy Fredholm integral equations and fuzzy Volterra integral equations. Numerical solution of fuzzy initial value problems under generalized differentiability by HPM is studied by Ghanbari [18]. The example problems solved in [18] only consider the positive coefficients of the fuzzy differential equations. Also, the author did not described how to tackle the n -th order fuzzy differential equations by using HPM. Recently, Tapaswini and Chakraverty [38] used HPM for solving fuzzy quadratic Riccati differential equations. As regards in the present analysis, HPM is used to handle the numerical solution of n -th order fuzzy differential equations with fuzzy initial conditions respectively. Here the exact solutions of the respective systems are also found by the authors for the comparison. In the following sections preliminaries are first given. Next, numerical implementation of HPM for n -th order fuzzy differential equations with fuzzy initial conditions is discussed. Lastly numerical examples and conclusions are given. 2. PRELIMINARIES In this section, we present some notations, definitions and preliminaries which are used further in this paper [19, 27, 35, 44]. Definition 2.1. Fuzzy number A fuzzy number U ~ is convex normalised fuzzy set U ~ of the real line R such that } ], 1 , 0 [ : ) ( { ~ R x R x U     where, U ~  is called the membership function of the fuzzy set and it is piecewise continuous. Definition 2.2. Triangular fuzzy number A triangular fuzzy number U ~ is a convex normalized fuzzy set U ~ of the real line R such that i. There exists exactly one R x  0 with 1 ) ( 0 ~  x U  ( 0 x is called the mean value of U ~ ), where U ~  is called the membership function of the fuzzy set. ii. ) ( ~ x U  is piecewise continuous. Let us consider an arbitrary triangular fuzzy number ) , , ( ~ c b a U  . The membership function U ~  of U ~ will be define as follows