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
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
