A Numerical Method for Fuzzy Differential Equations and Hybrid Fuzzy Differential Equations

and Applied Analysis 3 For u, V ∈ R F and λ ∈ R, the sum u + V and the product λu are defined by [u + V]α = [u]α + [V]α, [λu]α = λ[u]α, ∀α ∈ [0, 1], where [u]α+[V]αmeans the usual addition of two intervals (subsets) of R, and λ[u]α means the usual product between a scaler and a subset of R. The metric structure is given by the Hausdorff distance D : R F ×R F 󳨀→ R + ∪ {0} , (13) by D (u, V) = sup α∈[0,1] max {󵄨󵄨󵄨󵄨u α − Vα󵄨󵄨󵄨󵄨 , 󵄨󵄨󵄨󵄨u α − Vα󵄨󵄨󵄨󵄨} . (14) The following properties are well known: D(u + w, V + w) = D(u, V), ∀u, V, w ∈ R F , D(ku, kV) = |k|D(u, V), ∀k ∈ R, u, V ∈ R F , D(u + V, w + e) ≤ D(u, w) + D(V, e), ∀u, V, w, e ∈ R F , and (R F , D) is complete metric spaces. Let I be a real interval. A mapping y : I → R F is called a fuzzy process and its α-level set is denoted by [y (t)] α = [y α (t) , y α (t)] , t ∈ I, α ∈ (0, 1] . (15) A triangular fuzzy numberN is defined by an ordered triple (x l , x c , x r ) ∈ R3 with xl ≤ xc ≤ x, where the graph of N(s) is a triangle with base on the interval [xl, xr] and vertex at s = x . An α-level ofN is always a closed, bounded interval. We writeN = (xl, xc, x); then [N] α = [x c − (1 − α) (x c − x l ) , x c + (1 − α) (x r − x c )] , (16) for any 0 ≤ α ≤ 1. Definition 6. Let x, y ∈ R F . If there exists z ∈ R F such that x = y + z, then z is called the H-difference of x and y, and it is denoted by x ⊖ y. In this paper the sign “⊖” stands always for H-difference, and let us remark that x ⊖ y ̸ = x + (−1)y. Usually we denote x + (−1)y by x − y, while x ⊖ y stands for the H-difference. Definition 7. Let F : I → R F be a fuzzy function. We say F is Hukuhara differentiable at t 0 ∈ I if there exists an element F 󸀠 (t 0 ) ∈ R F such that the limits lim h→0 + F (t 0 + h) ⊖ F (t 0 ) h , lim h→0 + F (t 0 ) ⊖ F (t 0 − h)


Introduction
Hybrid systems are devoted to modeling, 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 modeled by hybrid systems.The differential systems containing fuzzy valued functions and interaction with a discrete time controller are named hybrid fuzzy differential systems.
The Hukuhara derivative of a fuzzy-number-valued function was introduced in [1].Under this setting, the existence and uniqueness of the solution of a fuzzy differential equation are studied by Kaleva [2,3], Seikkala [4], and Kloeden [5].This approach has the disadvantage that it leads to solutions which have an increasing length of their support [2].A generalized differentiability was studied in [6][7][8].This concept allows us to resolve the previously mentioned shortcoming.Indeed, the generalized derivative is defined for a larger class of fuzzy-number-valued functions than the Hukuhara derivative.Some applications of numerical methods in FDE and hybrid fuzzy differential equation (HFDE) are presented in [9][10][11][12][13][14][15][16][17][18][19].Some other approaches to study FDE and fuzzy dynamical systems have been investigated in [20][21][22].
In engineering and physical problems, Trapezoidal rule is a simple and powerful method to solve numerically related ODEs.Trapezoidal rule has a higher convergence order in comparison to other one step methods, for instance, Euler method.
In this work, we concentrate on numerical procedure for solving FDEs and HFDEs, whenever these equations possess unique fuzzy solutions.
In Section 2, we briefly present the basic definitions.Trapezoidal rule for solving fuzzy differential equations is introduced in Section 3, and convergence and stability of the mentioned method are proved.The proposed algorithm is illustrated by solving two examples.In Section 4 we present Trapezoidal rule for solving hybrid fuzzy differential equations.

Preliminary Notes
In this section the most basic definition of ordinary differential equations (ODEs) and notation used in fuzzy calculus are introduced.See, for example, [23].
Consider the first-order ordinary differential equation   () =  (,  ()) , ( 0 ) =  0 , where  : [ 0 ,   ] × R  → R  and  0 ∈ R. A linear multistep method applied to (1) is with   ,   ∈ R,   ̸ = 0, given starting values  0 ,  1 , . . .,  −1 .In the case   = 0, the corresponding methods (2) are explicit and are implicit otherwise.The constant step size ℎ > 0 leads to time discretizations with respect to the grid points   :=  0 + ℎ.The value  + is an approximation of the exact solution at  + .The special case of explicit methods,  = 2,  0 = −1,  1 = 0,  2 = 1,  0 =  2 = 0, and  1 = 2, corresponds to the Midpoint rule: and the especial case of implicit methods,  = 1,  0 = −1,  1 = 1, and  0 =  1 = 1/2, corresponds to the Trapezoidal rule: For an explicit method, (2) yields the current value  + directly in terms of  + ,  + ,  = 0, 1, . . .,  − 1, which, at this stage of the computation, have already been calculated.An implicit method will call for the solution, at each stage of computation, of the the equation where  is a known function of previously calculated values  + ,  + ,  = 0, 1, . . .,  − 1.When the original differential equation in ( 1) is linear, then (5) is linear in  + , and there is no problem in solving it.When  is nonlinear, for finding solution of (1), we can use the following iteration: Definition 1. Associated with the multistep method (2), we define the first characteristic polynomial as follows: Theorem 2. A multistep method is stable if the first characteristic polynomial satisfies the root condition, that is, the roots of () lie on or within the unit circle, and further the roots on the unit circle are simple.
According to Theorem 2, we know the Midpoint rule and Trapezoidal rule are stable.

Definition 3. The difference operator
and the associated linear multistep method (2) are said to be of order  if for the following equation: we have According to Definition 3, Midpoint rule and Trapezoidal rule are second-order methods.
We now recall some general concepts of fuzzy set theory; see, for example, [2,24].Definition 4. Let  be a nonempty set.A fuzzy set  in  is characterized by its membership function  :  → [0, 1], and () is interpreted as the degree of membership of an element  in fuzzy set  for each  ∈ .
Let us denote by R  the class of fuzzy subsets of the real axis, that is, satisfying the following properties: The space R  is called the space of fuzzy numbers.Obviously, R ⊂ R  .For 0 <  ≤ 1, we denote Then from (i)-(iv), it follows that the -level set []  is a nonempty compact interval for all 0 ≤  ≤ 1.The notation denotes explicitly the -level set of .The following remark shows when [  ,   ] is a valid -level set.

Abstract and Applied Analysis 3
For , V ∈ R  and  ∈ R, the sum  + V and the product  are defined by , where []  +[V]  means the usual addition of two intervals (subsets) of R, and []  means the usual product between a scaler and a subset of R.
The metric structure is given by the Hausdorff distance by The following properties are well known: and (R  , ) is complete metric spaces.
Let  be a real interval.A mapping  :  → R  is called a fuzzy process and its -level set is denoted by A triangular fuzzy number  is defined by an ordered triple (  ,   ,   ) ∈ R 3 with   ≤   ≤   , where the graph of () is a triangle with base on the interval [  ,   ] and vertex at  =   .An -level of  is always a closed, bounded interval.We write  = (  ,   ,   ); then for any 0 ≤  ≤ 1.
Definition 6.Let ,  ∈ R  .If there exists  ∈ R  such that  =  + , then  is called the H-difference of  and , and it is denoted by  ⊖ .
In this paper the sign "⊖" stands always for H-difference, and let us remark that  ⊖  ̸ =  + (−1).Usually we denote  + (−1) by  − , while  ⊖  stands for the H-difference.Definition 7. Let  :  → R  be a fuzzy function.We say  is Hukuhara differentiable at  0 ∈  if there exists an element   ( 0 ) ∈ R  such that the limits exist and are equal to   ( 0 ).Here the limits are taken in the metric space (R  , ).
provided the Lebesgue integrals on the right exist.

Fuzzy Differential Equations
Consider the first-order fuzzy differential equation   = (, ), where  is a fuzzy function of , (, ) is a fuzzy function of crisp variable  and fuzzy variable , and   is Hukuhara fuzzy derivative of .If an initial value ( 0 ) =  0 ∈ R  is given, a fuzzy Cauchy problem of first order will be obtained as follows: By Theorem 5.2 in [11] we may replace (21) by equivalent system The parametric form of ( 22) is given by   (; ) =  (,  (; ) ,  (; )) ,  ( 0 ; ) =   0 ,   (; ) =  (,  (; ) ,  (; )) ,  ( 0 ; ) =   0 , for 0 ≤  ≤ 1.In some cases the system given by ( 23) can be solved analytically.In most cases analytical solutions may not be found, and a numerical approach must be considered.Some numerical methods such as the fuzzy Euler method, Nyström method, and predictor-corrector method presented in [7,10,11,13,15].In the following, we present a new method to numerical solution of FDE.
Let (, , V) and (, , V) be the functions  and  of (22), where  and V are constants and  ≤ V.The domain where  and  are defined is therefore ( Then by Lemma 11 and  0 = V 0 = 0, we have If ℎ → 0, then   → 0, V  → 0 which concludes the proof.
Remark 13.According to Definition 3, Trapezoidal rule is a second-order method.In fact we may consider the definition of convergence order given in Definition 3 for system of ODEs.
Proof.For Trapezoidal rule exists only one characteristic polynomial () =  − 1, and it is clear that satisfies the root condition.Then by Theorem 2, the Trapezoidal rule is stable.

Numerical Results.
In this section we apply Trapezoidal rule for numerical solution of two linear fuzzy differential equations.We compare our results with Midpoint rule.The authors in [13] have presented the Midpoint rule for numerical solution of FDEs as follows: The Midpoint rule is a second-order and stable method [13].
In the following two examples, the implicit nature of Trapezoidal rule for solving linear fuzzy differential equation is implemented by solving a linear system at each stage of computation.
The exact solution at  = 0.1 is given by A comparison between the exact solution, (; ), and the approximate solutions by Midpoint method,  Mid (; ), and Trapezoidal method, (; ), at  = 0.1 with  = 10, is shown in Table 2 and Figure 2.
For arbitrary fixed  ∈ [0, 1] and , we can prove that the numerical solution of (55) converges to the exact solution; that is, lim ℎ 0 ,...,ℎ  → 0 The Trapezoidal rule is a one-step method as the Euler method.Therefore, the proof of the convergence closely follows the idea of the proof of Theorem 3.2 in [18]    (68) To numerically solve the hybrid fuzzy initial value problem (66) we apply the Trapezoidal rule for hybrid fuzzy differential equations.A comparison between the exact solution and the approximate solutions by Midpoint method and Trapezoidal method at  = 2 with  = 10 is shown in Table 3 and Figure 3.

Conclusion
We have presented Trapezoidal rule for numerical solution of first-order fuzzy differential equations and hybrid fuzzy differential equations.Also convergence and stability of the method are studied.To illustrate the efficiency of the new method, we have compared our method with the Midpoint rule in some examples.We have shown the global error in Trapezoidal rule is much less than in Midpoint rule.
For future research, we will apply Trapezoidal rule to fuzzy differential equations and hybrid fuzzy differential equations under generalized Hukuhara differentiability.Also one can apply Trapezoidal rule and Midpoint rule as a predictor-corrector method to solve FDE and HFDE.