Simulink Approach to Solve Fuzzy Differential Equation under Generalized Differentiability N. Kumaresan , J. Kavikumar, and Kuru Ratnavelu Abstract—In this paper, solution of fuzzy differential equation under general differentiability is obtained by simulink. The simulink solution is equivalent or very close to the exact solution of the problem. Accuracy of the simulink solution to this problem is qualitatively better. An illustrative numerical example is presented for the proposed method. Keywords—Fuzzy differential equation, Generalized differentiabil- ity, H-difference and Simulink. I. I NTRODUCTION F UZZY set theory is a powerful tool for modelling uncer- tainty and for processing vague or subjective information in mathematical models. The main directions of development of this subject have been diverse with applications to variety of real problems like the golden mean [9], quantum optics, gravity [11], synchronize hyperchaotic systems [24], chaotic system, medicine [2], [4], and engineering problems [15]. Particularly, fuzzy differential equation is an important topic from the theoretical point of view (see [1], [12], [17], [18]) as well as its applications like in population models [13], [14], civil engineering and hydraulics. Differentiable fuzzy valued mappings were initially studied by Puri and Ralescu [19]. They generalized and extended the concept of Hukuhara differentiability (H-derivative) for set valued mappings to the class of fuzzy mappings. Subsequently, using H-derivative, Kaleva [16] started to develop a theory for fuzzy differential equations. In the last few years, many works have been done by several authors in theoretical and applied fields for fuzzy differential equations with H-derivative (see [20], [21], [22], [23]). Now, in some cases this approach suffers certain disadvantages since the diameter diam(x(t)) of the solution is unbounded as time t increases [10]. This problem demonstrates that in some case this interpretation is not a good generalization of the associated crisp case. The generalized differentiability was introduced and studied in [5], [6], [7], [8]. This concept allows us to resolve the above mentioned shortcoming. Indeed, the generalized derivative is defined for a larger class of fuzzy number valued functions than Hukuhara derivative. Hence, this differentiability concept is used in the present paper. Under appropriate conditions, the * Corresponding author. N. Kumaresan and Kuru Ratnavelu are with the Institute of Mathematical Sciences, Faculty of Science, University of Malaya, Kuala Lumpur 50603, Malaysia. e-mail: drnk2008@gmail.com, drnk2008@um.edu.my. Tel: +6 03 7967 4126, Fax:+6 03 7967 4143. J. Kavikumar is with the Department of Mathematics and Statistics, Faculty of Science, Technology and Human Development, Universiti Tun Hussein Onn Malaysia, 86400 Parit Raja, Batu Pahat, Johor, Malaysia. fuzzy initial value problem considered under this interpretation has locally two solutions. In this paper, simulink approach is used to compute the solution of fuzzy differential equation. Simulink is a MATLAB add-on package that many pro- fessional engineers use to model dynamical processes in control systems. Simulink allows to create a block diagram representation of a system and run simulations very easily. Simulink is really translating block diagram into a system of ordinary differential equations. Simulink is the tool of choice for control system design, digital signal processing (DSP) design, communication system design and other simulation applications [3]. This paper focuses upon the implementation of simulink approach for solving fuzzy differential equation. This paper is organized as follows. In section 2, the ba- sic concepts and fuzzy differential equation are described. In section 3, simulink method is presented. In section 4, numerical example is discussed. The final conclusion section demonstrates the efficiency of the method. II. BASIC CONCEPTS AND FUZZY DIFFERENTIAL EQUATION Let X be a nonempty set. A fuzzy set u in X is character- ized by its membership function u : X → [0,1]. Then u(x) is interpreted as the degree of membership of a element x in the fuzzy set u for each x ∈ X. Definition 2.1: Let F n be the space of all compact and convex fuzzy sets on R n . Let u, v ∈F n . If there exists w ∈F n such that u = v ⊕w, then w is called the H-difference of u and v and it is denoted by u ⊖ v. Definition 2.2: Let F : T →F n and t 0 ∈ T . The function F is said to be differentiable at t 0 if (I) an element F ′ (t 0 ) ∈F n exist such that, for all h> 0 sufficiently near 0, there are F (t 0 +h) ⊖F (t 0 ),F (t 0 ) ⊖F (t 0 − h) and the limits lim h→0 + F (t 0 + h) ⊖ F (t 0 ) h = lim h→0 + F (t 0 ) ⊖ F (t 0 − h) h are equal to F ′ (t 0 ). (or) (II) there is an element F ′ (t 0 ) ∈F n exist such that, for all h< 0 sufficiently near 0, there are F (t 0 + h) ⊖ F (t 0 ),F (t 0 ) ⊖ F (t 0 − h) and the limits lim h→0 - F (t 0 + h) ⊖ F (t 0 ) h = lim h→0 - F (t 0 ) ⊖ F (t 0 − h) h are equal to F ′ (t 0 ) Note that if F is differentiable in the first form (I), then it is not differentiable in the second form (II) and viceversa. World Academy of Science, Engineering and Technology Vol:6 2012-04-23 810 International Science Index Vol:6, No:4, 2012 waset.org/Publication/14396