Abstract and Applied Analysis

Volume 2013 (2013), Article ID 735128, 10 pages

http://dx.doi.org/10.1155/2013/735128

Research Article

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

K. Ivaz,1,2 A. Khastan,³ and Juan J. Nieto4,5

¹Islamic Azad University, Shabestar Branch, Shabestar 5381637181, Iran

²Department of Applied Mathematics, University of Tabriz, Tabriz 5166616471, Iran

³Mathematics Department, Institute for Advanced Studies in Basic Sciences, Zanjan 45137-66731, Iran

⁴Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela (USC), 15782 Santiago de Compostela, Spain

⁵Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 12 June 2013; Accepted 28 August 2013

Academic Editor: Marcia Federson

Copyright © 2013 K. Ivaz et al. 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.

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.

1. 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 [6–8]. 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, [23].

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.

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  , 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 [25].

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 [11] 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 [15].

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.

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.

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 [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.

Example 15 (see [13]). 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 [13], , and Trapezoidal method, , at with , is shown in Table 1 and Figure 1.

Table 1

0
0.9636348
0.9686955
0.9636356
1.0188934
1.0138372
1.0188941

0.1
0.9677550
0.9723098
0.9677558
1.0174878
1.0129374
1.0174885

0.2
0.9718752
0.9759241
0.9718760
1.0160820
1.0120376
1.0160828

0.3
0.9759954
0.9795385
0.9759961
1.0146763
1.0111377
1.0146772

0.4
0.9801155
0.9831529
0.9801163
1.0132707
1.0102379
1.0132715

0.5
0.9842358
0.9867672
0.9842365
1.0118650
1.0093381
1.0118657

0.6
0.9883559
0.9903815
0.9883567
1.0104593
1.0084382
1.0104601

0.7
0.9924761
0.9939959
0.9924769
1.0090537
1.0075384
1.0090544

0.8
0.9965963
0.9976103
0.9965971
1.0076480
1.0066386
1.0076487

0.9
1.0007164
1.0012246
1.0007173
1.0062424
1.0057387
1.0062431

1
1.0048367
1.0048389
1.0048374
1.0048367
1.0048389
1.0048374

Figure 1: (-) Exact solution, () Trapezoidal, and (+) Midpoint approximated points.

Example 16. Let us consider the first-order fuzzy differential equation

where .

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.

Table 2

0
0.8636348
0.8686954
0.8636356
0.9188934
0.9138373
0.9188941

0.1
0.8677550
0.8723098
0.8677558
0.9174877
0.9129374
0.9174885

0.2
0.8718752
0.8759242
0.8718759
0.9160821
0.9120376
0.9160828

0.3
0.8759954
0.8795385
0.8759961
0.9146764
0.9111378
0.9146771

0.4
0.8801156
0.8831528
0.8801163
0.9132707
0.9102379
0.9132714

0.5
0.8842357
0.8867672
0.8842365
0.9118651
0.9093381
0.9118658

0.6
0.8883559
0.8903816
0.8883567
0.9104593
0.9084383
0.9104601

0.7
0.8924761
0.8939959
0.8924769
0.9090537
0.9075384
0.9090545

0.8
0.8965963
0.8976102
0.8965970
0.9076480
0.9066386
0.9076487

0.9
0.9007165
0.9012246
0.9007173
0.9062423
0.9057388
0.9062431

1
0.9048367
0.9048389
0.9048374
0.9048367
0.9048389
0.9048374

Figure 2: (-) Exact solution, () Trapezoidal, and (+) Midpoint approximated points.

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 [19].

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 .

According to Section 3, by similar computation we obtain the Trapezoidal rule for solving (60) as follows:

for , .

Next, we give the algorithm to numerically solve (55) in .

First Step. will be a numerical solution generated by (61) for as follows:

is a numerical solution of (60) over .

Second Step. For each , will be numerical solution generated by (61) for

where , is a numerical solution of (60) over for each .

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 [18] and Theorem 4.1 in [19].

Theorem 18. Consider the system of (55). Suppose for some fixed and that , where is obtained by (61). Then

Proof. See [19].

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.

Table 3

0
7.2644238
7.2370696
7.2577319
10.8966360
10.8556042
10.8865976

0.1
7.5065713
7.4783049
7.4996562
10.7755623
10.7349863
10.7656355

0.1
7.7487187
7.7195406
7.7415805
10.6544886
10.6143684
10.6446733

0.1
7.9908662
7.9607763
7.9835048
10.5334148
10.4937506
10.5237112

0.1
8.2330141
8.2020121
8.2254295
10.4123411
10.3731327
10.4027491

0.1
8.4751616
8.4432478
8.4673538
10.2912674
10.2525148
10.2817869

0.1
8.7173090
8.6844835
8.7092781
10.1701937
10.1318970
10.1608248

0.1
8.9594564
8.9257193
8.9512024
10.0491199
10.0112791
10.0398626

0.1
9.2016039
9.1669550
9.1931267
9.9280462
9.8906612
9.9189005

0.1
9.4437513
9.4081898
9.4350510
9.8069725
9.7700434
9.7979374

1
9.6858988
9.6494255
9.6769753
9.6858988
9.6494255
9.6769753

Figure 3: (-) Exact solution, () Trapezoidal, and (+) Midpoint approximated points.

5. 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.

Acknowledgments

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.

References

M. L. Puri and D. A. Ralescu, “Differentials of fuzzy functions,” Journal of Mathematical Analysis and Applications, vol. 91, no. 2, pp. 552–558, 1983.

O. Kaleva, “Fuzzy differential equations,” Fuzzy Sets and Systems, vol. 24, no. 3, pp. 301–317, 1987.

O. Kaleva, “The Cauchy problem for fuzzy differential equations,” Fuzzy Sets and Systems, vol. 35, no. 3, pp. 389–396, 1990.

S. Seikkala, “On the fuzzy initial value problem,” Fuzzy Sets and Systems, vol. 24, no. 3, pp. 319–330, 1987.

P. E. Kloeden, “Remarks on Peano-like theorems for fuzzy differential equations,” Fuzzy Sets and Systems, vol. 44, no. 1, pp. 161–163, 1991.

B. Bede, I. J. Rudas, and A. L. Bencsik, “First order linear fuzzy differential equations under generalized differentiability,” Information Sciences, vol. 177, no. 7, pp. 1648–1662, 2007.

J. J. Nieto, A. Khastan, and K. Ivaz, “Numerical solution of fuzzy differential equations under generalized differentiability,” Nonlinear Analysis: Hybrid Systems, vol. 3, no. 4, pp. 700–707, 2009.

A. Khastan, J. J. Nieto, and R. Rodríguez-López, “Variation of constant formula for first order fuzzy differential equations,” Fuzzy Sets and Systems, vol. 177, pp. 20–33, 2011.

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.

T. Allahviranloo, N. Ahmady, and E. Ahmady, “Numerical solution of fuzzy differential equations by predictor-corrector method,” Information Sciences, vol. 177, no. 7, pp. 1633–1647, 2007.

M. Friedman, M. Ma, and A. Kandel, “Numerical solutions of fuzzy differential and integral equations,” Fuzzy Sets and Systems, vol. 106, no. 1, pp. 35–48, 1999.

B. Ghazanfari and A. Shakerami, “Numerical solutions of fuzzy differential equations by extended Runge-Kutta-like formulae of order 4,” Fuzzy Sets and Systems, vol. 189, pp. 74–91, 2012.

A. Khastan and K. Ivaz, “Numerical solution of fuzzy differential equations by Nyström method,” Chaos, Solitons & Fractals, vol. 41, no. 2, pp. 859–868, 2009.

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.

M. Ma, M. Friedman, and A. Kandel, “Numerical solutions of fuzzy differential equations,” Fuzzy Sets and Systems, vol. 105, no. 1, pp. 133–138, 1999.

N. Parandin, “Numerical solution of fuzzy differential equations of nth-orderby Runge-Kutta method,” Neural Computing and Applications, vol. 21, no. 1, Supplement, pp. 347–355, 2012.

P. Prakash and V. Kalaiselvi, “Numerical solution of hybrid fuzzy differential equations by predictor-corrector method,” International Journal of Computer Mathematics, vol. 86, no. 1, pp. 121–134, 2009.

S. Pederson and M. Sambandham, “Numerical solution to hybrid fuzzy systems,” Mathematical and Computer Modelling, vol. 45, no. 9-10, pp. 1133–1144, 2007.

S. Pederson and M. Sambandham, “Numerical solution of hybrid fuzzy differential equation IVPs by a characterization theorem,” Information Sciences, vol. 179, no. 3, pp. 319–328, 2009.

J. Li, A. Zhao, and J. Yan, “The Cauchy problem of fuzzy differential equations under generalized differentiability,” Fuzzy Sets and Systems, vol. 200, pp. 1–24, 2012.

M. T. Malinowski, “Random fuzzy differential equations under generalized Lipschitz condition,” Nonlinear Analysis: Real World Applications, vol. 13, no. 2, pp. 860–881, 2012.

J. Xu, Z. Liao, and J. J. Nieto, “A class of linear differential dynamical systems with fuzzy matrices,” Journal of Mathematical Analysis and Applications, vol. 368, no. 1, pp. 54–68, 2010.

P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, John Wiley & Sons, New York, NY, USA, 1962.

P. Diamond and P. Kloeden, Metric Spaces of Fuzzy Sets, World Scientific, Singapore, 1994.

O. Kaleva, “Interpolation of fuzzy data,” Fuzzy Sets and Systems, vol. 61, no. 1, pp. 63–70, 1994.