ISSN: 2319-5967 ISO 9001:2008 Certified International Journal of Engineering Science and Innovative Technology (IJESIT) Volume 4, Issue 3, May 2015 90 Abstract—In this paper a numerical solution for the first order fuzzy differential equations has been determined by runge- kutta method of order three with some parameters. The formula is involved with Harmonic mean of three quantitie k i ‘s. The accuracy and efficiency of the proposed method is illustrated by solving a fuzzy initial value problem. Keywords —Fuzzy Differential Equations, Runge-kutta method of order three, Trapezoidal Fuzzy Number. I. INTRODUCTION Fuzzy differential equations are a natural way to model dynamical systems under uncertainty. First order linear fuzzy differential equations are one of the simplest fuzzy differential equations, which appear in many applications. The concept of fuzzy derivative was first introduced by S.L.Chang and L.A.Zadeh in [6].D.Dubois and Prade [7] discussed differentiation with fuzzy features.M.L.puri and D.A.Ralesec [17] and R.Goetschel and W.Voxman [10] contributed towards the differential of fuzzy functions. The fuzzy differential equation and initial value problems were extensively studied by O.Kaleva[11,12] and by S.Seikkala [18].Recently many research papers are focused on numerical solution of fuzzy initial value problems (FIVPS).Numerical Solution of fuzzy differential equations has been introduced by M.Ma, M. Friedman, A. Kandel [14] through Euler method and by S.Abbasbandy and T.Allahviranloo [1] by Taylor method.Runge – Kutta methods have also been studied by authors [2,16].Numerical solutions of fuzzy differential equations by Runge-kutta method of order three with some parameters has been studied by C.Duraisamy,and B.Usha[8]. This paper is organized as follows: In section 2 some basic results of fuzzy numbers and definitions of fuzzy derivative are given. In section 3 the fuzzy initial value problem is discussed .Section 4 contains the Runge-kutta method of order three. In section 5 the third order Runge-kutta method with some parameters based on harmonic mean is discussed. The method is illustrated by a solved numerical example in section 6 and the conclusion is in the last section. II. PRELIMINARIES A trapezoidal fuzzy number u is defined by four real numbers k < < m < n, where the base of the trapezoidal is the interval [k, n] and its vertices at x = , x = m. Trapezoidal fuzzy number will be written as u = ( k, , m , n). The membership function for the trapezoidal fuzzy number u = ( k , , m , n) is defined as the following : u We have : (1) u > 0 if k > 0; (2) u > 0 if > 0; (3) u > 0 if m > 0;and A Method for Solving Fuzzy Differential Equations Using Runge-Kutta Method with Harmonic Mean of Three Quantities D.Paul Dhayabaran 1 Associate Professor & Principal, PG and Research Department of Mathematics, Bishop Heber College (Autonomous), Trichirappalli -620 017 J.Christy kingston 2 Assistant Professor , PG and Research Department of Mathematics, Bishop Heber College (Autonomous),Trichirappalli -620 017