Numerical Solution for Hybrid Fuzzy Systems by Milne s Fourth Order Predictor-Corrector Method

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1 International Mathematical Forum, Vol. 9, 2014, no. 6, HIKARI Ltd, Numerical Solution for Hybrid Fuzzy Systems by Milne s Fourth Order Predictor-Corrector Method T. Jayakumar and K. Kanagarajan Department of Mathematics Sri Ramakrishna Mission Vidyalaya College of Arts and Science Coimbatore , Tamilnadu, India Copyright c 2014 T. Jayakumar and K. Kanagarajan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper we study numerical methods for hybrid fuzzy differential equations by an appllication of the Milne s fourth order predictor- corrector method for fuzzy differential equations. We prove a convergence result and give numerical examples to illustrate the theory. Keywords: Hybrid Systems; Fuzzy Differential Equations; Milne s fourth order predictor-corrector method 1. Introduction Fuzzy set theory is a powerful tool for modeling uncertainty and processing vague or subjective information in mathematical models. Its main directions of development and its applications to the very varied real-world problems have been diverse. Fuzzy differential equations are recently gaining more and more attention in the literature. The first and the most popular approach in dealing with FDEs is using the Hukuhara differentiability, or the Seikkala derivative for fuzzy-number-valued functions. Hybrid systems are devoted to modelling, design, and validation of interactive systems of computer programs and continuous systems. That is, control systems that are capable of controlling complex systems which have discrete event dynamics as well as continuous time dynamics can be modelled by hybrid systems. The differential systems containing fuzzzy valued functions and interaction with a discrete time controller are named as hybrid fuzzy differential systems.

2 274 T. Jayakumar and K. Kanagarajan In the last few years, many works have been performed by several authors in numerical solutions of fuzzy differential equations [1, 2, 3, 9, 10, 11, 12, 15, 18]. Pederson ans Sambandam [20, 21]. have investigated the numerical solution of hybrid fuzzy differential equations by using Runge-Kutta and Euler methods. Recently, the numerical solutions of fuzzy differential equations by predictorcorrector method has been studied in [4]. Allah viranloo [5] discussed numerical soultion of fuzzy differential equation by Adams-Bashforth Two-step method. In this paper we develop numerical method for hybrid fuzzy differential equation by an application of the Milne s fourth order predictor-corrector method. In Section 2 we list some basic definitions to fuzzy valued functions. Section 3 reviews hybrid fuzzy differential systems. Section 4 contains the Milne s fourth order predictor-corrector method for hybrid fuzzy differential equations and convergence theorem. Section 5, the MPC four-step algorithm is discussed. Section 6, contains numerical examples to illustrate this method. Finally, conclution is present in Section Preliminary Notes Let P k (R n ) denote the family of all nonempty compact, convex subsets of R n. If α, β R and A, B P k (R n ), then α(a + B) =αa + αb, α(βa) = (αβ)a, 1A = A and if α, β 0, then (α + β)a = αa + βa. Denote by E n the set of u : R n [0, 1] such that u satisesfies (i) (iv) mentionned below: (i) u is normal, that is, there exists an x 0 R n such that u(x 0 )=1; (ii) u is fuzzy convex, that is, for x, y R and 0 λ 1, u(λx +(1 λ)y) min[u(x),u(y)]; (iii) u is upper semicontinuous; (iv) [u] 0 = the closure of [x R : u(x) > 0] is compact. For 0 <r 1, we define [u] r =[x R n : u(x) r]. Then from (i) to (iv), it follows that r - level sets [u] r =[x p k (R n ) for 0 r 1. An example of a u E 1 is given by 4x 3, if x (0.75, 1], u(x) = 2x +3, if x (1, 1.5), (1) 0, if x / (0.75, 1.5). The r-level sets of u in (1) are given by [u] r =[ r, r]. (2) Let I be real interval. A mapping g : I E is called a fuzzy process and its r-level set is denoted by [y(t)] r =[y r (t), y r ], t I, r (0, 1].

3 Numerical solution for hybrid fuzzy systems 275 Definition 2.1. Consider the initial value problem y (t) =f(t, y(t)), y(0) = y 0, (3) where f :[a, b] R n R n. An m-step method for solving the initial-value problem is one whose difference equation for finding the approximation y(t i+1 ) at the mesh point t i+1 can be represented by the following equation : y i+1 = m 1 j=0 a m j 1yi j + h m 1 j=0 b mf(t i j+1,y i j+1 ) (4) for i = m 1,m,,N 1, such that a = t 0 t 1 t N = b, h = (b a) N = t i+1 t i and a 0,,a m 1,b 0,,b m are constants with the starting values y 0 = α 0, y 1 = α 1, y m 1 = α m 1. When b m = 0, the method is known as explicit, since Equation (4) gives y i+1 explicit in terms of previously determined values. When b m 0, the method is known as implicit, since y i+1 occurs on both sides of Equation (4) and is specfied only implicitly. A special case of multistep method is. Here, we set q y i+1 = y i 1 + h k m m f(t i,y i ), q =0, 1, 2,..., (5) m=0 where the constants ( ) n s k m =( 1) m ds 0 k are independent of f,t = t 0 + sh, f(t i,y i ) is the first backward difference of the f(t, y(t)) at the point of t = t i and higher backward difference are defined by k f(t, y i )= ( k 1 f(t i,y i )). The special case q = 3 of Milnes method are as follows: Milne explicit method: y 0 = α 0, y 1 = α 1, y 2 = α 2, y 3 = α 3, y i+1 = y i 3 + 4h 3 [2f(t i,y i ) f(t i 1,y i 1 )+2f(t i 2,y i 2 )], where i =3, 4, 5,...,N 1. Milne implicit method : y 1 = α 1, y 2 = α 2, y 3 = α 3, y i+1 = y i 1 + h 3 [f(t i 1,y i 1 )+4f(t i,y i )+f(t i+1,y i+1 )]. where i =3, 4, 5,,N 1.

4 276 T. Jayakumar and K. Kanagarajan Definition 2.2. Assosisted with the difference equation y i+1 = a m 1 y i + a m 2 y i a 0 y i+1 m + hf (t i,h,y i+1,y i,,y i+1 m ), y 0 = α 0, y 1 = α 1, y m 1 = α m 1, (6) the characteristic polynomial of the method is defined by p(λ) =λ m a m 1 λ m 1 a m 2 λ m 2 a 1 λ a 0. If λ i 1 for each i =1, 2,,m, and all roots with absolute value 1 are simple roots, then the difference method is said to satisfy the root condition. Theorem 2.1. A multistep method of the form (4) is stable if and only if it satisfies the roots condition. Notations used in this paper are as follows: A tilde is placed over a symbol to denote a fuzzy set so α 1, f(t), An arbitrary fuzzy number with an ordered pair of functions (u(r), u(r)). 0 r 1, which satisfy the following requirements is represented. 1. u(r) is a bounded left continuous non-decreasing function over [0, 1], 2. u(r) is a bounded right continuous non-increasing function over [0, 1], 3. u(r) u (r), 0 r 1. Definition 2.3. The supremum metric d on E is defined by d (U, V )=sup {d H ([U] r, [V ] r ):r I}, and (E,d ) is a complete metric space. Definition 2.4. A mapping F : T E is a Hukuhara differentiable at t 0 T R if for some h 0 > 0 the Hukuhara differences F (t 0 +Δt) h F (t 0 ), F(t 0 ) h F (t 0 Δt), exist in E for all 0 < Δt <h 0 and if there exist an F (t 0 ) E such that ( ) F lim d (t0 +Δt) h F (t 0 ) Δt 0 + =0 Δt, F (t 0 ) and ( ) F lim d (t0 ) h F (t 0 Δt) Δt 0 + =0 Δt, F (t 0 ) the fuzzy set F (t 0 ) is called the Hukuhara derivative of F at t 0. Recall that U h V = W E are defined on level sets, where [U] r h[v ] r = [W ] r for all r I. By consideration of definition of the metric d, all the level set mappings [F (.)] r are Hukuhara differentiable at t 0 with Hukuhara

5 Numerical solution for hybrid fuzzy systems 277 derivatives [F (t 0 )] r for each r I when F : T E is Hukuhara differentiable at t 0 with Hukuhara derivative F (t 0 ). Definition 2.5. The fuzzy integral b a y(t)dt, 0 a b 1, is defined by [ b r [ b ] b y(t)dt] = y r (t)dt, y r (t)dt, a a a provided the Lebesgue integrals on the right exist. Definition 2.6. If F : T E is Hukuhara differentialble and its Hukuhara derivative F is integrable over [0,1], then F (t) =F (t 0 )+ t for all values of t 0,t where 0 t 0 t 1. t 0 F (s)ds, Definition 2.7. A mapping y : I E is called a fuzzy process. We denote [y(t)] r =[y r (t), y r (t)], t I, r (0, 1]. The Seikkala derivative y (t) of a fuzzy process y is defined by [y(t)] r = [y r (t), y r (t)], t I, r (0, 1] provided the equation defines a fuzzy number y (t) E. Definition 2.8. If y : I E is Seikkala differentiable and its Seikkala derivative y is integrable over [0, 1], then y(t) =y(t 0 )+ for all values of t 0,t where t 0,t I. t t 0 y (s)ds, 3. The Hybrid Fuzzy Differential System Consider the hybrid fuzzy differential system x (t) =f(t, x(t),λ k (x k )), t [t k,t k+1 ], x(t k )=x k, (7)

6 278 T. Jayakumar and K. Kanagarajan where denotes Seikkala differentiation, 0 t 0 <t 1 <... < t k <..., t k, f C[R + E 1 E 1,E 1 ],λ k C[E 1,E 1 ]. To be specific the system would look like x (t) = x 0 (t) =f(t, x 0 (t),λ 0 (x 0 )), x 0 (t 0 )=x 0, t 0 t t 1, x 1 (t) =f(t, x 1 (t),λ 1 (x 1 )), x 1 (t 1 )=x 1, t 1 t t 2,... x k (t) =f(t, x k (t),λ k (x k )), x k (t k )=x k, t k t t 2,... Assuming that the existence and uniquness of solution of (7) hold for each [t k,t k+1 ], by the solution of (7) we mean the following function: x 0 (t), t 0 t t 1, x 1 (t), t 1 t t 2,... x (t) = x k (t), t k t t k+1..., We note that the solution of (7) are piecewise differentiable in each interval for t [t k,t k+1 ] for a fixed x k E 1 and k =0, 1, 2,... Using a representation of fuzzy numbers studied by Goestschel and Woxman [8] and Wu and Ma [23], we may represent x E 1 by a pair of fuctions (x(r), x(r)), 0 r 1, such that (i) x(r) is bounded, left continuous, and nondecreasing, (ii) x(r) is bounded, left continuons, and nonincreasing, and (iii) x(r) x(r), 0 r 1. For example, u E 1 given in (1) is represented by (u(r), u(r))=( r, r), 0 r 1, which is similar to [u] α given by (2). Therefore we may replace (7) by an equivalent system x (t) =f(t, x, λ k (x k )) F k (t, x, x), x(t k )=x k, (8) x (t) =f(t, x, λ k (x k )) G k (t, x, x), x(t k )=x k, which possesses a unique solution (x, x) which is a fuzzy function. That is for each t, the pair [x(t; r), x(t; r)] is a fuzzy number, where x(t; r), x(t; r) are respectively the solutions of the parametric form given by x (t; r) =F k [t, x(t; r), x(t; r)], x(t k ; r) =x k (r), x (t; r) =G k [t, x(t; r), x(t; r)], x(t k ; r) =x k (r), (9)

7 Numerical solution for hybrid fuzzy systems Milne s fourth order predictor-corrector method In this section for a hybrid fuzzy differential equation (7). We develop the Milne s fourth order predictor-corrector method in [3]. When f and λ k in (7) can be obtained via, the Zadeh extension principal from f [R + R R, R] and λ k [R, R]. We assume that the existance and uniqueness of solution (7) hold for each [t k,t k+1 ]. For a fixed r, to integrate the system in (7) in [t 0,t 1 ], [t 1,t 2 ],..., [t k,t k+1 ],...,we replace each interval by a set of N k + 1 discreate equally spaced grid points (including the end points) at which the exact solution (x(t; r), x(t; r)) is approximated by some (y k (t; r), y k (t; r)). For the chosen grid points on [t k,t k+1 ] at t k,n = t k + nh k,h k = t k+1 t k N k, 0 n N k. Let (Y k (t; r), Y k (t; r)) (y(t, r), y(t, r)), (Y k (t; r), Y k (t; r))and (y k (t; r), y k (t; r)) may we denoted respctively by (Y k (t; r), Y k (t; r))and (y k (t; r), y k (t; r)). The Milne s fourth order predictor-corrector method is approximation of y k (t; r), and y k (t; r) which can be written as y 0 (r) =[α 0 (r), α 0 (r)], y 2 (r) =[α 2 (r), α 2 (r)], y 1 (r) =[α 1 (r), α 1 (r)] y 3 (r) =[α 3 (r), α 3 (r)] Y p n+1 = Y n 3 (r)+ 4h 3 [2f(t n,y n (r),λ k (u k )) f(t n 1,y n 1 (r),λ k (u k )) +2f(t n 2,y n 2 (r),λ k (u k ))], Y p n+1 = Y n 3(r)+ 4h 3 [2f(t n,y n (r),λ k (u k )) f(t n 1,y n 1 (r),λ k (u k )) +2f(t n 2,y n 2 (r),λ k (u k ))], (10) Y c n+1 = Y n 1(r)+ h[f(t 3 n 1,y n 1 (r),λ k (u k ))+4f(t n,y n (r),λ k (u k )) +f(t n+1,y n+1 (r),λ k (u k ))], Y c n+1 = Y n 1 (r)+ h [f(t 3 n 1,y n 1 (r),λ k (u k ))+4f(t n,y n (r),λ k (u k )) +f(t n+1,y n+1 (r),λ k (u k ))], where Y n (r) =[y n (r), y n (r)] f(t n,y n (r),λ k (u k )) = min{f(t, u, λ k (u k )) u [y(r), y n (r)]}, f(t n,y n (r),λ k (u k )) = max{f(t, u, λ k (u k )) u [y n (r), y n (r)]}.

8 280 T. Jayakumar and K. Kanagarajan Now we define F p [t, y(r)] = [2f(t n,y n (r),λ k (u k )) f(t n 1,y n 1 (r),λ k (u k )) +2f(t n 2,y n 2 (r),λ k (u k ))], G p [t, y(r)] = [2f(t n,y n (r),λ k (u k )) f(t n 1,y n 1 (r),λ k (u k )) +2f(t n 2,y n 2 (r),λ k (u k ))], F c [t, y(r)] = [f(t n 1,y n 1 (r),λ k (u k ))+4f(t n,y n (r),λ k (u k )) (11) +f(t n+1,y n+1 (r),λ k (u k ))], G c [t, y(r)] = [f(t n 1,y n 1 (r),λ k (u k ))+4f(t n,y n (r),λ k (u k )) +f(t n+1,y n+1 (r),λ k (u k ))]. The exact and approximate solutions at t n,0 n N are denoted by [Y (t n )] r = [Y (t; r), Y (t; r)] and [y(t n )] r = [y(t; r), y(t; r)] respectively. By (10),(11) we have Y c (t n+1 ; r) Y c (t n ; r)+ h 3 F [t n; Y c (t n ; r), Y c (t n ; r)] Y c (t n+1 ; r) Y c (t n ; r)+ h 3 G[t n; Y c (t n ; r), Y c (t n ; r)] (12) we define y c (t n+1 ; r) =y c (t n ; r)+ h 3 F [t n; y c (t n ; r), y c (t n ; r)] y c (t n+1 ; r) =y c (t n ; r)+ h 3 G[t n; y c (t n ; r), y c (t n ; r)] (13) However, (12) will use y 0,0 = x 0 (r), y 0,0 = x 0 (r), and y k.0 (r) =y k 1 (r)n k 1, y k.0 (r) =y k 1 (r)n k 1. If k 1, then (12) reperesents an approximation of y k (t; r), y k (t; r) for each of intervals t 0 t t 1,t 1 t t 2,...,t k t t k+1,... Lemma 4.1. Suppose i Z +,ɛ i > 0,r [0, 1], and h i < 1 are fixed. Let {Z i,n (r)} N i n=0 be the Milne s fourth order predictor-corrector approximation with N = N i to the fuzzy IVP: { x (t) =f(t, x(t),λ i (x i )), t [t i,t i+1 ], (14) x(t i )=x i.

9 Numerical solution for hybrid fuzzy systems 281 If {y i,n (r)} N i n=0 denotes the result of (10) from some y i,0 (r), then there exist a δ i > 0 such that z i,0 (r) y i,0 (r) <δ i, z i,0 (r) y i,0 (r) <δ i implies z i,0 (r) y i,0 (r) <ɛ i, z i,0 (r) y i,0 (r) <ɛ i proof: Proof is similar to lemma 3.1 [19] Theorem 4.1. r [0, 1], Proof. See [19]. Consider the systems (9) and (11). For a fixed k Z + and lim (r) =x(t h 0,...,h k 0 k,n k+1 ; r), k (15) lim k,n h 0,...,h k 0 k (r) =x(t k+1 ; r). (16) 5. Algorithm The following algorithm is based on Milne s fourth order predictor-corrector method. To approximate the solution of following fuzzy initial value problem y (t) =f(t, y(t),λ k (u k )), t 0 t T, y α = α 0, y α = α 1 y α = α 2, y α = α 3, y α = α 4, y α = α 5 y α = α 6, y α = α 7,, positive integer N is chosen Step 1. Let h = (T t 0), N y (t) =f(t, y(t),λ k (u k )), t 0 t T, w α = α 0, w α = α 1 w α = α 2, w α = α 3, w α = α 4, w α = α 5 w α = α 6, w α = α 7, Step 2. For n =0, 1, 2, do Steps 3-4. Compute starting values using fourth order Runge - Kutta method Step 3. Let k 1 (t k,n ; y k,n (r)) = min{h k.f(t k,n,u,λ k (u k )) u [y k,n (r), y k,n (r)]}, k 1 (t k,n,y k,n (r)) = max{h.f(t k,n,u,λ k (u k )) u, [y k,n (r), y k,n (r)]}, k 2 (t k,n,y k,n (r)) = min{h.f(t k,n + h/2,u,λ k (u k ) u [z 1 (t k,n,y k,n (r)), z 1 (t k,n,y k,n (r))]}, k 2 (t k,n,y k,n (r)) = max{h.f(t k,n + h/2,u,λ k (u k )) u [z 1 (t k,n,y k,n (r)), z 1 (t k,n,y k,n (r))]},

10 282 T. Jayakumar and K. Kanagarajan k 3 (t k,n,y k,n (r)) = min{h.f(t k,n + h/2,u,λ k (u k )) u [z 2 (t k,n,y k,n (r)), z 2 (t k,n,y k,n (r))]}, k 3 ( k,n,y k,n (r)) = max{h.f(t k,n + h/2,u,λ k (u k )) u [z 2 (t k,n,y k,n (r)), z 2 (t k,n,y k,n (r))]} k 4 (t k,n,y k,n (r)) = min{h.f(t k,n + h/2,u,λ k (u k )) u [z 3 (t k,n,y(r)), z 3 (t k,n,y k,n (r))]}, k 4 (t k,n,y k,n (r)) = max{h.f(t k,n + h, u, λ k (u k )) u [z 3 (t k,n,y k,n (r)), z 3 (t k,n,y k,n (r))]}, where z 1 (t k,n,y k,n (r)) = y k,n (r)+ 1 2 k 1(t k,n,y k,n (r)), z 1 (t k,n,y k,n (r)) = y k,n (r)+ 1 2 k 1(t, y k,n (r)), z 2 (t k,n,y k,n (r)) = y k,n (r)+ 1 2 k 2(t k,n,y k,n (r)), z 2 (t k,n,y k,n (r)) = y k,n (r)+ 1 2 k 2(t k,n,y k,n (r)), z 3 (t k,n,y k,n (r)) = y k,n (r)+k 3 (t k,n,y k,n (r)), z 3 (t k,n,y k,n (r)) = y k,n (t)+k 3 (t k,n,y k,n (r)). step 4. Let y(t k,n+1,r)=y(t k,n,y k,n (r)) + 1 (k 6 1(t k,n,y k,n (r))+2k 2 (t k,n,y k,n (r)) +2k 3 (t k,n,y k,n (r)) + k 4 (t k,n,y k,n (r))), y(t k,n+1,r)=y(t n,y k,n (r)) + 1 (k 6 1(t k,n,y k,n (r))+2k 2 (t k,n,y k,n (r)) +2k 3 (t k,n,y k,n (r)) + k 4 (t k,n,y k,n (r))). Step 5. For i=3,4,...,n-1. Let y p = y + 4h[2f(t i+1 i 3 3 n,y i,λ k (u k )) f(t i 1,y i 1,λ k (u k ))+2f(t i 2,y i 2,λ k (u k )) 9f(t i 3,y i 3 )λ k (u k )], y p i+1 = y i 3 + 4h 3 [2f(t i,y i,λ k (u k )) f(t i 1,y i 1,λ k (u k ))+2f(t i 2,y i 2,λ k (u k )) 9f(t i 3,y i 3 ),λ k (u k )].

11 Numerical solution for hybrid fuzzy systems 283 step 6. Let t i+2 = t 0 +(i +2)h Step 7. Let y c i+1 = y i 1 + h 3 [f(t i 1,y i 1,λ k (u k ))+4f(t i,y i,λ k (u k )) + f(t i+1,y i+1,λ k (u k )], y c i+1 = y i 1 + h 3 [f(t i 1,y i 1,λ k (u k ))+4f(t i,y i,λ k (u k )) + f(t i+1,y i+1 ),λ k (u k )] Step 8. For i = i +1 Step 9. If i N 2 go to step 3. Step 10. Algorithm will be completed and (w α (T ), w α (T )) approximates real value of (Y α (T ), Y α (T )). 6. Numerical Examples Before illustrating the numerical solution of hybrid fuzzy IVP, first we recall the Example 3 of [18] studying the fuzzy IVP; x (t) =x(t), x(0; r) =[ r, r], 0 r 1. (17) By Milne s predictor-corrector method with N=10. Since the exact solution of (17) is x(t; r) = [( r)e t, ( r)e t ], 0 r 1, we see that x(1; r) = [( r)e, ( r)e], 0 r 1. Example 6.1. Next consider the folllowing hybrid fuzzy IVP, { x (t) =x(t)+m(t)λ k x(t k ), t [t k,t k+1 ], t k = k, k =0, 1, 2, 3,..., x(0; r) = [( r), ( r)], 0 r 1, (18) where m(t) = { 2(t(mod1)) if t(mod1) 0.5, 2(1 t(mod1) if t(mod1) > 0.5, λ k (μ) = { ˆ0, if k =0 μ, if k {1, 2,...}. The hybrid fuzzy IVP (18) is equivalent to the following systems of fuzzy IVPs: x 0(t) =x 0 (t), t [0, 1], x(0; r) = [( r), ( r)], 0 r 1, x i (t) =x i (t)+m(t)x i 1 (t),t [t i,t i+1 ],x i (t) =x i 1 (t i ),i=1, 2,...,

12 284 T. Jayakumar and K. Kanagarajan In (14) x(t)+m(t)λ k (x(t k ) is continous function of t, x and λ k (x(t k ). Therefore by Example 6.1 of Kaleva [14], for each k =0, 1, 2,..., the fuzzy IVP { x (t) =x(t)+m(t)λ k (x(t k )), t [t k,t k+1 ],t k = k, x(t k )=x tk, (19) has a unique solution on [t k,t k+1 ]. Therefore x(1; r) = [( r)e, ( r)e],x(1; r)(3 e 3). Then x(1.5; 1) is approximately and y 1 (1.5; r) is approximitly For t [1.5; 2] the exact solution satisfies Therefore x(t; r) =x(1; r)(2t 2+e t 1.5 (3 e 4). x(2.0; r) =x(1; r)(2 + 3e 4 e). Then x(2.0; r) is approximately and y(2.0; 1) is approximately The exact and approximate solution by using Milne s predictor-corrector method. With N = 10 the following results are obtained. The exact and approximate solution by fourth order Runge-Kutta method and Milne s Predictor Corrector method are compared at t = 2 see Table 1 and Figure 1, Error in fourth order Runge-Kutta method and Milne s predictor-corrector method see Table 2. Table 1. r RK-Fourth Order Predictor Corrector Exact y 1 (t i ; r) y 2 (t i ; r) y 1 (t i ; r) y 2 (t i ; r) Y 1 (t i ; r) Y 2 (t i ; r)

13 Numerical solution for hybrid fuzzy systems 285 Table 2. r Rk 4 th -Order Predictor Corrector y 1 (t i ; r) y 2 (t i ; r) y 1 (t i ; r) y 2 (t i ; r) Predictor corrector method Exact.RK order4 0 PC Method r y Figure 1: h=0.1 Example 6.2. Next consider the following hybrid fuzzy IVP, x 0(t) =x 0 (t)+m(t)λ k x(t k ), t [0, 1], x(0; r) = [( r), ( r)], 0 r 1, x i (t) =x i (t)+m(t)x i 1 (t),t [t i,t i+1 ],x i (t) =x i 1 (t i ),i=1, 2,..., where m(t) = sin(πt), k =0, 1, 2,..., λ k (μ) = { ˆ0, if k =0 μ, if k {1, 2,...}.

14 286 T. Jayakumar and K. Kanagarajan Then x(t)+m(t)λ k (x(t k )) is continous function of t, x, and λ k (x(t k )). Therefore by Example of 6.1 of Kaleva [14], for each k=0,1,2,..., the fuzzy IVP { x (t) =x(t)+m(t)λ k (x(t k )), t [t k,t k+1 ],t k = k, x(t k )=x tk, (20) has a unique solution on [t k,t k+1 ]. To numerically solve the hybrid fuzzy IVP (18) we will apply the Milne s predictor-corrector method for hybrid fuzzy differential equation with N = 10 to obtain y 1,2 (r) approximating x(2.0; r). The exact solution of (20) satisfies x(t; r) =x(1; r) πcos(πt)+sin(πt) π 2 ( + et e x(1; r) 1+ π ), π 2 +1 x(t; r) =x(1; r) πcos(πt)+sin(πt) ( + et π 2 e x(1; r) 1+ π ). π 2 +1 Therefore x(1; r) = [( r)e, ( r)e], ( ( π x(2; r) = π e 1+ π )) x(1; r). π 2 +1 Then x(2; 1) is approximately where as y 1 (2.0; 1) is approximately see in Fig.2. Table 3. r RK-Fourth Order Predictor Corrector Exact y 1 (t i ; r) y 2 (t i ; r) y 1 (t i ; r) y 2 (t i ; r) y 1 (t i ; r) y 2 (t i ; r)

15 Numerical solution for hybrid fuzzy systems 287 Table 4. r Rk 4 th -Order Predictor Corrector y 1 (t i ; r) y 2 (t i ; r) y 1 (t i ; r) y 2 (t i ; r) Predictor corrector method Exact.RK order4 0 PC Method r y Figure 2: h= Conclution In this paper, we have applied itrative solution of Milne s predictor-corrector fourth order method for finding the numerical solution of hybrid fuzzy differential equations. Comparision of solution of Example (6.1) and (6.2) shows that our proposed method gives better solution then fourth order Runge-Kutta method.

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