THIRD ORDER RUNGE-KUTTA METHOD FOR SOLVING DIFFERENTIAL EQUATION IN FUZZY ENVIRONMENT. S. Narayanamoorthy 1, T.L. Yookesh 2

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1 International Journal of Pure Applied Mathematics Volume 101 No , ISSN: (printed version); ISSN: (on-line version) url: PAijpam.eu THIRD ORDER RUNGE-KUTTA METHOD FOR SOLVING DIFFERENTIAL EQUATION IN FUZZY ENVIRONMENT S. Narayanamoorthy 1, T.L. Yookesh 2 1,2 Department of Applied Mathematics Bharathiar University Coimbatore, , INDIA Abstract: In this research paper, we introduce a numerical algorithm for solving fuzzy initial value problem using Seikkala s derivative. The linear fuzzy differential equation is solved by Runge-Kutta method. The numerical values are calculated by choosing different step size. This approach gives a complete error analysis. AMS Subject Classification: 34A07, 65L06 Key Words: ordinary differential equation, fuzzy initial value problem, Runge-Kutta method 1. Introduction Fuzzy differential equation (FDE)has been rapidly growing in the recent years. Fuzzy differential equation forms a suitable setting for mathematical modeling of real world problems in which uncertainties or vagueness pervade. The concept of a fuzzy derivative was defined Chang Zadeh in It was followed by Dubois Prade. The term Fuzzy Differential Equation was introduced in 1987 by Kel Byatt. The most popular approach is using the Hukuhara differentablity for the fuzzy value functions, which has a draw back. Hence, the fuzzy solution behaves quite differently from the crisp so- Received: March 12, 2015 c 2015 Academic Publications, Ltd. url:

2 796 S. Narayanamoorthy, T.L. Yookesh lutions. Later, Seikkala[8] introduces the notations of fuzzy derivatives as an extension of Hukuhara derivative fuzzy integral. General formulation of fuzzy first order initial value problem is given by Buckley Feuring. Under appropriate conditions, the fuzzy initial value problem (FIVP) considered under interpretation has locally two solutions. Numerical solution of an FDE is obtained now in a natural way, by extending the existing classical methods to the fuzzy case[5]. Here we used the method called Runge-Kutta third order method to solve the fuzzy linear differential equation. 2. Preliminaries Definition 1. LetXbeaset. AfuzzysetAonXisdefinedtobeafunction A : X [0,1](orµ A : X [0,1]). Equivalently, a fuzzy set A is defined to be the class of object having the following representation A = {(x,µ A (x)) : x X}. Where, µ A : X [0,1] is a function called the membership characteristic function of A. Definition 2. The α-cut (or) α- level set of a fuzzy set A is a crisp set defined by A α = {x A(x) α} the strong α-cut defined similarly A α = {x A(x) > α}. Definition 3. A Fuzzy set Ā is the triangular Fuzzy number with peak (or center) a, left width α > 0 right β > 0 if its membership function has the following form Ā(t) = 1 a t α 1 t a β if a α < t < a, if a < t < a+β, 0, Otherwise. Lemma 4. In the sequence of non-negative numbers {W n } N n=0 satisfy W n+1 = A W n +B, 0 n N 1, W n = A n W 0 +B An 1 A 1, 0 n N, Lemma 5. Let F(t,u,v) G(t,u,v) belong to c (R f ) the partial derivatives of F G be bounded over R f. Then for arbitrarily fixed r,

3 THIRD ORDER RUNGE-KUTTA METHOD FOR SOLVING r 1, D(y(t n+1 ),y (0) (t n+1 )) h 2 L(1+2C) where L is a bound of partial derivatives of F,G C = max [ G[tn,y(t n ;r),y(t n ;r)] r [0,1] ] < α. Theorem 6. Let F(t,u,v) G(t,u,v) belong to c (R f ) the partial derivatives of F G be bounded over R f. Then for arbitrarily fixed r, 0 r 1, the numerical solutions of y(t n+1 ;r) y(t n+1 ;r) converge to the exact solution Y(t;r) Y(t;r) uniformly in t. Theorem 7. Let F(t,u,v) G(t,u,v) belong to c (R f ) the partial derivatives of F G be bounded over R f 2Lh < 1. Then for arbitrarily fixed r, 0 r 1, the iterative numerical solutions of y (j) (t n+1 ;r) y (j) (t n+1 ;r) converge to the numerical solution Y(t n ;r) Y(t n ;r) in t 0 t n t N, when j. 3. Fuzzy Initial Value Problem Consider the first order fuzzy initial value differential equation [8] is given by { y (x) = f(x,y(x)), t [x 0,X] y(x 0 ) = y 0 where y is a fuzzy function of x,f(x,y) is a fuzzy function of the crisp variable x the fuzzy derivative of y y(x 0 ) = y 0 is a triangular shaped fuzzy number. We denote the fuzzy function y by y = [y,y]. The α-level set of y(x) for x [x 0,X] is defined as [y(x)] r = [ y(x;α),y(x;α) ], (1) [y(x 0 )] r = [ y(x 0 ;α),y(x 0 ;α) ], α (0,1] Wewrite[f(x,y)] = [f(x,y),f(x,y)]f(x,y) = F[x,y,y], f(x,y) = G[x,y,y], because of y = f(x,y) we have f(x,y(x);α) = F[x,y(x;α),y(x;α)],f(x,y(x);α) = G[x,y(x;α),y(x;α)]. By using the extension principle, we have the membership function f(t,y(x))(s) = sup{y(x)(τ)/s = f(x,τ)}

4 798 S. Narayanamoorthy, T.L. Yookesh so fuzzy number f(x,y(x)). From this it follows that [f(t,y(x)] r = [f(x,y(x);α),f(x,y(x);α)],α (0,1]. Here f(x,y(x);α) = min{f(x,u) u [y(x)] r } f(x,y(x);α) = max{f(x,u) u [y(x)] r }. Theorem Ȳ(t)[α] = Ω(a) for all α [0,1],t I. 2. Ȳ(t) is a fuzzy number for all t I. 4. Third Order Runge-Kutta Method for Solving Fuzzy Differential Equations The General Third Order Fuzzy Runge-Kutta Method formula is y n+1 = y n + h 6 [k 1 +4k 2 +k 3 ], where k 1 = f(x n,y n ), k 2 = f(x n + h 2,y n + h 2 k 1), k 3 = f(x n +h,y n hk 1 +2hk 2 ). Then,letY = [Y 1,Y 2 ]beexactsolutiony = [y 1,y 2 ]betheapproximated solution of the Fuzzy initial value problem [Y(t)] r = [Y 1 (t,r),y 2 (t,r)] r = [y 1 (t,r),y 2 (t,r)] r. Keeping argument, the value of r is fixed then the exact approximate solution of t n are represented by [Y(t n )] r = [Y 1 (t n ;r),y 2 (t n ;r)][y(t n )] r = [y 1 (t n ;r),y 2 (t n ;r)],{0 n N}. The grid points at which the solution is calculated h = T t 0N t 1 = t 0 +ih, 0 i N.,

5 THIRD ORDER RUNGE-KUTTA METHOD FOR SOLVING Then we obtain Y 1 (t n+1 ;r) = Y 1 (t n ;r)+ h 6 [k 1 +4k 2 +k 3 ], where k 1 = f(t n,[y 1 (t n ;r),y 2 (t n ;r)]), k 2 = f(t n + h 2, Y 1(t n ;r)+ h 2 k 1, Y 2 (t n ;r)+ h 2 k 1) k 3 = f(t n +h,y 1 (t n ;r) hk 1 +2hk 2,Y 2 (t n ;r) hk 1 +2hk 2 ) Y 2 (t n+1 ;r) = Y 1 (t n ;r)+ h 6 [k 1 +4k 2 +k 3 ], where k 1 = g(t n,[y 1 (t n ;r),y 2 (t n ;r)]) k 2 = g(t n + h 2,Y 1(t n ;r)+ h 2 k 1,,Y 2 (t n ;r)+ h 2 k 1) k 3 = g(t n +h,y 1 (t n ;r) hk 1 +2hk 2,Y 2 (t n ;r) hk 1 +2hk 2 ). Thus we have y 1 (t n+1 ;r) = y 1 (t n ;r)+ h 6 [k 1 +4k 2 +k 3 ], where k 1 = f(t n,[y 1 (t n ;r),y 2 (t n ;r)]) k 2 = f(t n + h 2,y 1(t n ;r)+ h 2 k 1,y 2 (t n ;r)+ h 2 k 1) k 3 = f(t n +h,y 1 (t n ;r) hk 1 +2hk 2,y 2 (t n ;r) hk 1 +2hk 2 ) y 2 (t n+1 ;r) = y 1 (t n ;r)+ h 6 [k 1 +4k 2 +k 3 ]. Here k 1 = g(t n,[y 1 (t n ;r),y 2 (t n ;r)]) k 2 = g(t n + h 2,y 1(t n ;r)+ h 2 k 1,y 2 (t n ;r)+ h 2 k 1) k 3 = g(t n +h,y 1 (t n ;r) hk 1 +2hk 2,y 2 (t n ;r) hk 1 +2hk 2 ) Clearly y 1 (t n ;r) y 2 (t n ;r) converges to Y 1 (t n ;r) Y 2 (t n ;r) respectively whenever h Illustrative Example Consider the first order linear differential equation, y (t) = x 2 y, t > 0, y(0) = ( r, r)

6 800 S. Narayanamoorthy, T.L. Yookesh The exact solution is given by y(t,r) = [( r)e t3 3,( r)e t3 3 ] at t = 1 y(1,r) = [( r)e 1 3,( r)e 1 3], 0 r 1 Using the Runge-Kutta third order method approximation we denote y (0) 1 = r,y2 0 = r y (0) 1 (t n+1;r) = y 1 (t;r)+hy 1 (t,r) y (0) 2 (t n+1;r) = y 2 (t;r)+hy 2 (t,r) where I = 0,1,...N 1 h = 1/N. Now using these equation as an initial guess for the following iterative solutions respectively y1 i(t n+1;r) = y 2 (t n ;r)+ h 6 [k 1 +4k 2 +k 3 ] where, k 1 = f([y 1 (t n ;r)]) k 2 = f(y 1 (t n ;r)+ h 2 k 1) k 3 = f(y 1 (t n ;r)+ 3 4 k 2) y j 2 (t n+1;r) = y 2 (t n ;r)+ h 6 [k 1 +4k 2 +k 3 ] where, k 1 = f([y 2 (t n ;r)]) k 2 = f(y 2 (t n ;r)+ h 2 k 1) k 3 = f(y 2 (t n ;r)+ 3 4 k 2) j = 1,2,3. Thus we have y 1 (1,r) = y (3) 1 (1;r) y 2(1,r) = y (3) 2 (1;r) for i = 1,2,...,N. therefore, y 1 (1,r) = y (3) 1 (1;r) y 2(1,r) = y (3) 2 (1;r) are obtained. The solution of exact approximate solutions were tabulated compare as, r Table 1 : Exact Soln at x = 1 r Table 2 : Approx soln at x = 1

7 THIRD ORDER RUNGE-KUTTA METHOD FOR SOLVING (a) At h=0.1 (b) At h=0.01 Figure 1: Graphical representation r/h Table 3 : Error at x = 1 6. Conclusion The numerical solution for fuzzy ordinary differential equation is obtained. A scheme based on third order Runge-Kutta method to approximate the solution of fuzzy initial value problem has been formulated. Numerical example shows the efficiency of implemented numerical method it increase the accuracy. References [1] T.Allahviranloo, Numerical methods for fuzzy system of linear equation, Appl.math.comput., 155 (2004), [2] S.Abbasby, T.Allahviranloo, O.Lopez-pouso, J.J.Nieto, Numerical methods for fuzzy differential inclusions, Journal of computer mathematics with applications., 48(2004),

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