Series Solution of the System of Fuzzy Differential Equations

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 , which provides us with a simple way to adjust and control the convergence region of solution series. Concept of -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.


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][12][13][14][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][19][20][21][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.

Preliminaries
In this section, the most basic notations used in this paper are introduced.
The set of all such fuzzy numbers is represented by E 1 .

Advances in Fuzzy Systems
Remark 2. For arbitrary u = (u(r), u(r)), v = (v(r), v(r)), and k ∈ R, we define addition and multiplication by k as Definition 3.For arbitrary fuzzy numbers u, v ∈ E 1 , we use the distance and it is shown that (E 1 , D) is a complete metric space. ( The definite integral of ( Definition 5. Let f : R 1 → E 1 be a fuzzy function and let t 0 ∈ R 1 .The derivative f (t 0 ) of f at the point t 0 is defined by provided that this limit, taken with respect to the metric D, exists.
The elements f (t 0 +h), f (t 0 ) at the right-hand side of ( 6) are observed as elements in the Banach space Clearly [ f (t 0 + h) − f (t 0 )]/h may not be a fuzzy number for all h.However, if it approaches f (t 0 ) (in B) and f (t 0 ) is also a fuzzy number (i.e., in E 1 ), this number is the fuzzy derivative of f (t) at t 0 .In this case, if f = ( f , f ), it can be easily shown that where f and f are the classic derivatives of f and f , respectively.

System of Fuzzy Differential Equations
In this section, we will review system of fuzzy differential equations of the forms where t is a scaler and A 0 (t), A where g i j (r) = g l i j (r), g u i j (r) , The superscript T denotes transpose.The component that is in rth row, kth column, 1 ≤ r, k ≤ s of matrix A p , 0 ≤ p ≤ n will be denoted by a p r,k .Then, (8) can be replaced by the following equivalent system in parametric form: where Advances in Fuzzy Systems 3 Now (11) becomes as follows:

Basic Ideas of HAM
We consider the following differential equations: where and A 0 (t), A 1 (t), . . ., A n−1 (t) are s × s matrixes and every component of them is a real function of t and A n (t) = I s , where I s denotes the s × s identity matrix.
The above system of equations can be written in following form: (16) where N i are nonlinear operators that represent the whole equations, t and r denote the independent variables, and z i (t, r), i = 1, 2, . . ., s are unknown functions, respectively.By means of generalizing the traditional homotopy method constructed the so-called zero-order deformation equations 1 − q L i φ i t, r; q − z i,0 (t, r) = qDN i φ 1 t, r; q , . . ., φ n t, r; q , i = 1, . . ., n, (17) where q ∈ [0, 1] is an embedding parameter, D is nonzero auxiliary parameter, L i , i = 1, 2, . . ., s are auxiliary linear operators, z i,0 (t, r) are initial guesses of z i (t, r), and φ i (t, r; q) are unknown functions.It is important to note that one has great freedom to choose auxiliary objects such as D and L i in HAM.Obviously, when q = 0 and q = 1, both hold.Thus, as q increases from 0 to 1, the solutions φ i (t, r; q) varies from the initial guesses z i,0 (t, r) to the solutions z i (t, r).
Expanding φ i (t, r; q) in Taylor's series with respect to q, one has where If auxiliary linear operator, initial guesses, and auxiliary parameter D are properly chosen, then the series equations ( 19) converges at q = 1 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) m time with respect to the embedding parameter q and then setting q = 0 and finally dividing them by m!, we have the so-called mth order deformation equations where Now, to simplify and solve (11), we need to know the sign of components of A i (t); 0 ≤ i ≤ n and use of Remark 2. Also we should be noted that in case that some components of A 0 (t), A 1 (t), . . ., A n (t) for some t are negative, we must divide the defined interval for t into small intervals, so that sign of each component of A 0 (t), A 1 (t), . . ., A n (t) in each small interval be unchanged.Now for each small interval we have a separate system 2s × s(n + 1).

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).
if kth equation of ( 13) On the other hand, since L i are linear operators, thus Since D / = 0, thus from the above equations for k = 1, 2, . . ., s, we can write From uniform convergency we have is the exact solution of (11) and Y = [y 1 (t, r), y 2 (t, r), . . ., y s (t, r)] T is the exact solution of ( 8) and proof is completed.

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 Y 0 = (0.1r − 0.1) + (0.088 + 0.1r)t, and the linear operator with the property where c i0 , c i1 (i = 1, 2) are integral constants.Also from (36), we can define Obviously, when q = 0 and q = 1, Therefore, when the embedding parameter q increases from 0 to 1, the homotopy solutions φ i (t, r, q) vary from z i,0 (t, r) to the solutions z i (t, r) for i = 1, 2. Now, by expanding φ i (t, r; q) in Taylor's series with respect to q, we have where Assuming that auxiliary parameter D, the initial guesses and the auxiliary linear operator are properly chosen, then the above series is convergent at q = 1, and The mth 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 D-mesh of Y and Y to get a proper interval for convergency.D-mesh is a generalization of traditional Contour plots, and for connection between D-meshes and Contour plots we plot Figures 2 and 4. Also to find the best quantity of D that lies in the convergency interval, we use the residual error of norm 2 as follows: which is a function with respect to D. Now, by minimizing the Res[Y 10 ] we obtain the best choice for auxiliary parameter to approximate of Y 10 (t, r) as follows:   and in this case absolute error for the 10th order approximation by HAM for Y n (t, r) is plotted in Figure 5.By minimizing the residual error defined by it is clear that the best choice for auxiliary parameter to approximate Y 10 (t, r) is which in this case absolute error for the 10th order approximation by HAM for Y n (t, r) is plotted in Figure 6.

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 D. The concept of traditional h-curves has been generalized to D-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.
provided that this limit exists in the metric D. If the fuzzy function f (t) is continuous in the metric D, its definite integral exists and also,
1 (t), . . ., A n−1 (t) are s × s matrixes and every component of them is a real function of t and A n (t) = I s denotes the s × s identity matrix.Y , G i , and F are fuzzy s-dimensional vectors.The rth component of Y ∈ E s will be denoted by y r , so that we may write Y = y 1 (t, r), y 2 (t, r), . . ., y s (t, r) T , = Dr − e 2t Dr + Dr 2 − e 2t Dr 2 + 2Drt + 2Dr 2 t, z 1,2 (t, r) = Dr − e 2t Dr + 21D 2 r 16 3+ 6 e 2t − 2r + 2 .