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

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.


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.
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 7.

Preliminary Notes
(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, ( The r-level sets of u in (1) are given by 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].

Definition 2.1. Consider the initial value problem
where 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 : 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 where the constants 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 . The special case q = 3 of Milnes method are as follows: Milne explicit method : where i = 3, 4, 5, . . ., N − 1.
Definition 2.2.Assosisted with the difference equation the characteristic polynomial of the method is defined by and (E, d ∞ ) is a complete metric space.

Definition 2.4. A mapping
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 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 ) .
for all values of t 0 , t where 0 ≤ t 0 ≤ t ≤ 1.
Definition 2.7.A mapping y : 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 for all values of t 0 , t where t 0 , t ∈ I.

The Hybrid Fuzzy Differential System
Consider the hybrid fuzzy differential system where denotes Seikkala differentiation, 0 To be specific the system would look like 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: 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 which is similar to [u] α given by ( 2).Therefore we may replace ( 7) by an equivalent system 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

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 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 )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 where

Algorithm
The following algorithm is based on Milne's fourth order predictor-corrector method.

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.

Definition 2 . 5 .Definition 2 . 6 .
The fuzzy integral b a y(t)dt, 0 ≤ a ≤ b ≤ 1, is defined by t)dt , provided the Lebesgue integrals on the right exist.If F : T → E is Hukuhara differentialble and its Hukuhara derivative F is integrable over [0,1], then

−6 1 .Figure 2 :
Figure 2: h=0.1 , • • • , m, and all roots with absolute value 1 are simple roots, then the difference method is said to satisfy the root condition.

Table 2 .
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 Table1and Figure1, Error in fourth order Runge-Kutta method and Milne's predictor-corrector method see Table2.