Advances in Fuzzy Systems

Volume 2012 (2012), Article ID 407647, 16 pages

doi:10.1155/2012/407647

Research Article

Series Solution of the System of Fuzzy Differential Equations

M. S. Hashemi,1,2 J. Malekinagad,² and H. R. Marasi²

¹Dipartimento di Mathematica e Informatica, Università degli Stueli di Perugia, 06123, Italy

²Department of Mathematics and Computer Science, Engineering and Technical Faculty, University of Bonab, Bonab 55517, Iran

Received 9 April 2012; Accepted 28 June 2012

Academic Editor: Rustom M. Mamlook

Copyright © 2012 M. S. Hashemi 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.

The homotopy analysis method (HAM) is proposed to obtain a semianalytical solution of the system of fuzzy differential equations (SFDE). The HAM contains the auxiliary parameter ħ, which provides us with a simple way to adjust and control the convergence region of solution series. Concept of ħ-meshes and contour plots firstly are introduced in this paper which are the generations of traditional h-curves. Convergency of this method for the SFDE has been considered and some examples are given to illustrate the efficiency and power of HAM.

1. Introduction

In many cases of the modeling real world phenomena, information about the behavior of a dynamical system is uncertain. In order to obtain a more realistic model, we have to take into account these uncertainties. Since 1965, when Zadeh published his pioneering paper [1], hundreds of examples have been supplied where the nature of uncertainty in the behavior of a given system processes is fuzzy rather than stochastic nature. The concept of fuzzy derivative was first introduced by Chang and Zadeh in [2]. It was followed up by Dubois and Prade in [3], who defined and used the extension principle. Other methods have been discussed by Puri and Ralescu in [4] and Goetschel and Voxman in [5]. The initial value problem for fuzzy differential equation (FIVP) has been studied by Kaleva in [6, 7] and by Seikkala in [8].

The purpose of this paper is to find the approximate solution of fuzzy differential equations system with the homotopy analysis method (HAM) introduced first by Liao in 1992 [9, 10], that is, analytic approach to get series solutions of various types of linear and nonlinear equations.

Some of numerical methods have been applied to obtain the solution of fuzzy differential equations [11–15]. Sami Bataineh et al. have applied the HAM for systems of ODEs and PDEs in [16, 17]. Also, recently many types of nonlinear problems solved with HAM by others [18–22].

In Section 2, some basic definitions which will be used later in the paper are provided. In Sections 3 and 4, system of fuzzy differential equations and then basic ideas of HAM applied to these types of equations have been reviewed, respectively. Convergency of HAM for SFDE that shows its reliability has been considered in Section 5. The proposed method is illustrated by solving several examples in Section 6, and finally the conclusion is drawn in Section 7.

2. Preliminaries

In this section, the most basic notations used in this paper are introduced.

Definition 1 (See [23]). An arbitrary fuzzy number in parametric form is an ordered pair of functions , ; , which satisfies the following requirements:

(1) is a bounded left-continuous nondecreasing function over ,(2) is a bounded left-continuous nonincreasing function over ,(3), .

The set of all such fuzzy numbers is represented by .

Remark 2. For arbitrary ,  , and , we define addition and multiplication by as

Definition 3. For arbitrary fuzzy numbers , , we use the distance

and it is shown that is a complete metric space.

Definition 4. Let , for each partition of and for arbitrary , suppose

The definite integral of over is

provided that this limit exists in the metric . If the fuzzy function is continuous in the metric , its definite integral exists and also,

Definition 5. Let be a fuzzy function and let . The derivative of at the point is defined by

provided that this limit, taken with respect to the metric , exists.

The elements , at the right-hand side of (6) are observed as elements in the Banach space . Thus, if and , the difference is simply .

Clearly may not be a fuzzy number for all . However, if it approaches (in ) and is also a fuzzy number (i.e., in ), this number is the fuzzy derivative of at . In this case, if , it can be easily shown that

where and are the classic derivatives of and , respectively.

3. System of Fuzzy Differential Equations

In this section, we will review system of fuzzy differential equations of the forms

where is a scaler and are matrixes and every component of them is a real function of and denotes the identity matrix. , , and are fuzzy -dimensional vectors. The th component of will be denoted by , so that we may write

where

The superscript denotes transpose. The component that is in th row, th column, ,  of matrix , will be denoted by . Then, (8) can be replaced by the following equivalent system in parametric form:

where .

We define functions and ; , ,  , in the following form:

Now (11) becomes as follows:

4. Basic Ideas of HAM

We consider the following differential equations:

where

and are matrixes and every component of them is a real function of and , where denotes the identity matrix.

The above system of equations can be written in following form:

where are nonlinear operators that represent the whole equations, and denote the independent variables, and ,   are unknown functions, respectively. By means of generalizing the traditional homotopy method constructed the so-called zero-order deformation equations

where is an embedding parameter, is nonzero auxiliary parameter, ,   are auxiliary linear operators, are initial guesses of , and are unknown functions. It is important to note that one has great freedom to choose auxiliary objects such as and in HAM. Obviously, when and , both

hold. Thus, as increases from 0 to 1, the solutions varies from the initial guesses to the solutions . Expanding in Taylor’s series with respect to , one has

where

If auxiliary linear operator, initial guesses, and auxiliary parameter are properly chosen, then the series equations (19) converges at and

which must be solution of the original nonlinear equations.

According to (20), the governing equations can be deduced from the zero-order deformation equations (17). Define the vectors

Differentiating (17) time with respect to the embedding parameter and then setting and finally dividing them by , we have the so-called th order deformation equations

where

Now, to simplify and solve (11), we need to know the sign of components of ;   and use of Remark 2. Also we should be noted that in case that some components of for some are negative, we must divide the defined interval for into small intervals, so that sign of each component of in each small interval be unchanged. Now for each small interval we have a separate system .

5. Convergency of HAM for System of Fuzzy Differential Equations

In this section, we prove a theorem which shows the convergency of approximate HAM solution applied for (8).

First we define , ; , in the following form:

if th equation of (13) is inclusive ;  , , and otherwise

Theorem 6. Let the series be uniformly converge to ; . Then , is exact solution of (13).

Proof. Since is convergent, we must have

On the other hand, since are linear operators, thus

then from (27) we can write

hence,

Since , thus from the above equations for , we can write

From uniform convergency we have

thus

Therefore, is the exact solution of (11) and   is the exact solution of (8) and proof is completed.

6. Illustrative Examples

Example 7. Consider the following second-order fuzzy linear differential equation:

The exact solution is as follows:

According to (11), we may replace (34) by the following equivalent system:

We first construct the zero-order deformation equations

subject to the initial conditions

and the linear operator

with the property

where , are integral constants. Also from (36), we can define

Obviously, when and ,

Therefore, when the embedding parameter increases from 0 to 1, the homotopy solutions vary from to the solutions for . Now, by expanding in Taylor’s series with respect to , we have

where

Assuming that auxiliary parameter , the initial guesses and the auxiliary linear operator are properly chosen, then the above series is convergent at , and

The th order deformation equations are

with the initial conditions

where

Therefore, we recursively obtain

Then the solutions obtained by HAM are as follows:

Figures 1 and 3 show the -mesh of and to get a proper interval for convergency. -mesh is a generalization of traditional Contour plots, and for connection between -meshes and Contour plots we plot Figures 2 and 4. Also to find the best quantity of that lies in the convergency interval, we use the residual error of norm 2 as follows:

which is a function with respect to . Now, by minimizing the we obtain the best choice for auxiliary parameter to approximate of as follows:

and in this case absolute error for the 10th order approximation by HAM for is plotted in Figure 5.

By minimizing the residual error defined by

it is clear that the best choice for auxiliary parameter to approximate is

which in this case absolute error for the 10th order approximation by HAM for is plotted in Figure 6.

Figure 1: -mesh of .

Figure 2: Contour plot of .

Figure 3: -mesh of .

Figure 4: Contour plot of .

Figure 5: Absolute error of for .

Figure 6: Absolute error of for .

Example 8. Consider the system of fuzzy differential equation

The exact solution is as follows:

We may replace (55) by the equivalent system

We select the following initial guesses:

and the auxiliary linear operators

with the property

where and are integral constants. Also the nonlinear operators are

Then th order deformation equations are

with the initial conditions

Therefore, we recursively obtain

Then the solutions obtained by HAM are as follows:

and Figures 7, 8, 9, and 10 show the -meshes of ,  ,  , and respectively. Also, Figures 11, 12, 13, and 14 exhibit their related Contour plots.

Figure 7: -mesh of .

Figure 8: -mesh of .

Figure 9: -mesh of .

Figure 10: -mesh of .

Figure 11: Contour plot of .

Figure 12: Contour plot of .

Figure 13: Contour plot of .

Figure 14: Contour plot of .

By minimizing the residual error, we find that the best choices for auxiliary parameter to approximate of , , , and are

respectively. Absolute errors for the 15th order approximation by HAM for and , , are plotted in Figures 15, 16, 17, and 18, respectively.

Figure 15: Absolute error of for .

Figure 16: Absolute error of for .

Figure 17: Absolute error of for .

Figure 18: Absolute error of for .

Example 9. Let us consider the first-order system of fuzzy differential equation

The exact solution is as follows:

We may replace (67) by the equivalent system

We choose the initial guesses of system (69) as follows:

and the auxiliary linear operators

with the properties

where are integral constants, and the nonlinear operators,

The th order deformation equations are

with the initial conditions

where

Therefore, we recursively obtain

Then obtained solutions are as follows:

Figures 19, 20, 21, and 22 show the -meshes of , , and , respectively. Also, Figures 23, 24, 25, and 26 show the related Contour plots.

Figure 19: -mesh of .

Figure 20: -mesh of .

Figure 21: -mesh of .

Figure 22: -mesh of .

Figure 23: Contour plot of .

Figure 24: Contour plot of .

Figure 25: Contour plot of .

Figure 26: Contour plot of .

Best choices for auxiliary parameter to approximate , and from minimizing the residual errors are

respectively. Finally, absolute errors for the 15th order approximation by HAM for and , are plotted in Figures 27, 28, 29, and 30.

Figure 27: Absolute error of for .

Figure 28: Absolute error of for .

Figure 29: Absolute error of for .

Figure 30: Absolute error of for .

7. Conclusion

In this paper, homotopy analysis method has been implemented to derive approximate analytical solutions for the system of fuzzy differential equations. Obtained results show that we can control the convergence region of HAM series solution by the auxiliary parameter . The concept of traditional -curves has been generalized to -meshes and then contour plot firstly has been introduced in HAM. Convergence theorem and given illustrative examples show the efficiency and accuracy of the HAM.

Acknowledgments

The respected anonymous referees have carefully reviewed this paper. As a result of their careful analysis, our paper has been improved. The authors would like to express their thankfulness to them for their helpful constructive comments.

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