Generalized Differential and Integral EquationsView this Special Issue
A Numerical Method for Fuzzy Differential Equations and Hybrid Fuzzy Differential Equations
Numerical algorithms for solving first-order fuzzy differential equations and hybrid fuzzy differential equations have been investigated. Sufficient conditions for stability and convergence of the proposed algorithms are given, and their applicability is illustrated with some examples.
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 . Under this setting, the existence and uniqueness of the solution of a fuzzy differential equation are studied by Kaleva [2, 3], Seikkala , and Kloeden . This approach has the disadvantage that it leads to solutions which have an increasing length of their support . A generalized differentiability was studied in [6–8]. 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 [9–19]. Some other approaches to study FDE and fuzzy dynamical systems have been investigated in [20–22].
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.
2. 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, .
Consider the first-order ordinary differential equation where and . A linear multistep method applied to (1) is with , , given starting values . In the case , the corresponding methods (2) are explicit and are implicit otherwise. The constant step size leads to time discretizations with respect to the grid points . The value is an approximation of the exact solution at . The special case of explicit methods, , , , , , and , corresponds to the Midpoint rule: and the especial case of implicit methods, , , , and , corresponds to the Trapezoidal rule: For an explicit method, (2) yields the current value directly in terms of , , , 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 , , . 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 , , where and , for .
According to Definition 3, Midpoint rule and Trapezoidal rule are second-order methods.
Definition 4. Let be a nonempty set. A fuzzy set in is characterized by its membership function , and is interpreted as the degree of membership of an element in fuzzy set for each .
Let us denote by the class of fuzzy subsets of the real axis, that is, satisfying the following properties: (i)is normal, that is, there exists such that ,(ii) is a convex fuzzy set (i.e., , ), (iii) is upper semicontinuous on , (iv) is compact, where denotes the closure of a subset. The space is called the space of fuzzy numbers. Obviously, . For , we denote Then from (i)–(iv), it follows that the -level set is a nonempty compact interval for all . The notation denotes explicitly the -level set of . The following remark shows when is a valid -level set.
Remark 5. The sufficient conditions for to define the parametric form of a fuzzy number are as follows: (i) is a bounded monotonic increasing (nondecreasing) left-continuous function on and right-continuous for ,(ii) is a bounded monotonic decreasing (nonincreasing) left-continuous function on and right-continuous for .(iii), .
For and , the sum and the product are defined by , , , where means the usual addition of two intervals (subsets) of , and means the usual product between a scaler and a subset of .
The metric structure is given by the Hausdorff distance by The following properties are well known: , , , , , , ,and is complete metric spaces.
Let be a real interval. A mapping is called a fuzzy process and its -level set is denoted by A triangular fuzzy number is defined by an ordered triple 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 .
Definition 6. Let . If there exists 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 . Usually we denote by , while stands for the H-difference.
Definition 7. Let be a fuzzy function. We say is Hukuhara differentiable at if there exists an element such that the limits exist and are equal to . Here the limits are taken in the metric space .
Definition 8. Let . The fuzzy integral is defined by provided the Lebesgue integrals on the right exist.
Remark 9. Let . If is Hukuhara differentiable and its Hukuhara derivative is integrable over , then for all values of , , where .
Theorem 10. Let , , be the observed data, and suppose that each of the is a triangular fuzzy number. Then for each , the fuzzy polynomial interpolation is a fuzzy-value continuous function , where ,, and such that .
Proof. See .
3. 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 is given, a fuzzy Cauchy problem of first order will be obtained as follows: By Theorem 5.2 in  we may replace (21) by equivalent system The parametric form of (22) is given by for . 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.
3.1. Trapezoidal Rule for Fuzzy Differential Equations
In the interval we consider a set of discrete equally spaced grid points . The exact and approximate solutions at , , are denoted by and , respectively. The grid points at which the solution is calculated are Let , which is triangular fuzzy number. We have By fuzzy interpolation, Theorem 10, we get where , interpolates with the interpolation data given by the value , and , .
For we have From (16) and (25) it follows that where According to (25), if (26) and (27) are situated in (31), (27) and (28) in (32), we obtain By integration we have By (16) deduce Similarly we obtain Therefore, Trapezoidal rule is obtained as follows: for .
3.2. Convergence and Stability
Suppose the exact solution is approximated by some . The exact and approximate solutions at , are denoted by and , respectively. Our next result determines the pointwise convergence of the Trapezoidal approximates to the exact solution. The following lemma will be applied to show convergence of these approximates; that is,
Lemma 11. Let a sequence of numbers satisfy for some given positive constant and . Then
Proof. See .
Let and be the functions and of (22), where and are constants and . The domain where and are defined is therefore
Theorem 12. Let and belong to , and let the partial derivatives of , be bounded over . Then for arbitrary fixed , the Trapezoidal rule approximate of (37) converges to the exact solutions , uniformly in , for .
Proof. It is sufficient to show that By using Taylor’s theorem, we get where , . Consequently, Denote and . Then where and , and is a bound for partial derivatives of and in , . Thus, If we put and , then Then by Lemma 11 and , we have If , then , which concludes the proof.
Theorem 14. Trapezoidal rule is stable.
Proof. For Trapezoidal rule exists only one characteristic polynomial , and it is clear that satisfies the root condition. Then by Theorem 2, the Trapezoidal rule is stable.
3.3. 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  have presented the Midpoint rule for numerical solution of FDEs as follows: The Midpoint rule is a second-order and stable method .
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.
Example 15 (see ). Consider the initial value problem The exact solution at for is given by A comparison between the exact solution, , and the approximate solutions by Midpoint method , , and Trapezoidal method, , at with , is shown in Table 1 and Figure 1.
Example 16. Let us consider the first-order fuzzy differential equation
The exact solution at is given by A comparison between the exact solution, , and the approximate solutions by Midpoint method, , and Trapezoidal method, , at with , is shown in Table 2 and Figure 2.
4. Hybrid Fuzzy Differential Equations
Consider the hybrid fuzzy differential equation where is strictly increasing and unbounded, denotes , is continuous, and each is a continuous function. A solution to (55) will be a function satisfying (55). For , let , where . The hybrid fuzzy differential equation in (55) can be written in expanded form as and a solution of (55) can be expressed as We note that the solution of (55) is continuous and piecewise differentiable over and differentiable on each interval for any fixed and .
Theorem 17. Suppose for that each is such that If for each there exists such that for all , then (55) and the hybrid system of ODEs are equivalent.
Proof. See .
4.1. Trapezoidal Rule for Hybrid Fuzzy Differential Equations
For each , to numerically solve (55) in , replace each interval , by a set of regularly spaced grid points (including the endpoints). The grid point on will be , , at which the exact solution will be approximated by some . We set , and , if .
Next, we give the algorithm to numerically solve (55) in .
For arbitrary fixed and , we can prove that the numerical solution of (55) converges to the exact solution; that is, 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  and Theorem 4.1 in .
Proof. See .
Example 19. Consider the following hybrid fuzzy system:
where is a triangular fuzzy number having -level sets ,
By [19, Example ], we know (66) has a unique solution and the exact solution on is given by
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 with is shown in Table 3 and Figure 3.
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.
The first author would like to thank the financial support of the Islamic Azad University, Shabestar Branch. The research of the third author has been partially supported by Ministerio de Economia y Competitividad (Spain), Project MTM2010-15314, and cofinanced by the European Community fund FEDER. This research was completed during the visit of the second author to the USC.
S. Abbasbandy, T. Allahviranloo, and P. Darabi, “Numerical soluion of nth-order fuzzy differential equations by Runge-Kutta method,” Journal of Mathematical and Computational Applications, vol. 16, no. 4, pp. 935–946, 2011.View at: Google Scholar
H. Kim and R. Sakthivel, “Numerical solution of hybrid fuzzy differential equations using improved predictor-corrector method,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 10, pp. 3788–3794, 2012.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, John Wiley & Sons, New York, NY, USA, 1962.View at: MathSciNet
P. Diamond and P. Kloeden, Metric Spaces of Fuzzy Sets, World Scientific, Singapore, 1994.