Adams Predictor-Corrector Systems for Solving Fuzzy Differential Equations

Fuzzy differential equations (FDEs), which are utilized for the purpose of the modeling problems in science and engineering, have been studied by many researchers. Most of the practical problems require the solutions of fuzzy differential equations (FDEs) which are satisfied with fuzzy initial conditions; therefore a fuzzy initial problem occurs and should be solved. However, for the vast majority of fuzzy initial value problems, their exact solutions are difficult to be obtained. Thus it is necessary to consider their numerical methods. The concept of a fuzzy derivative was first introduced by Chang and Zadeh [1]; it was followed up byDubois and Prade [2] who used the extension principle in their approach. Other fuzzy derivative concepts have been proposed by Puri and Ralescu [3] and Goetschel Jr. and Voxman [4] as an extension of the Hukuhara derivative of multivalued functions. In the past decades, many works have been appeared on the aspects of theories and applications on fuzzy differential equations; see [5–12]. The notation of fuzzy differential equation was initially introduced by Kandel and Byatt [13, 14] and later they applied the concept of fuzzy differential equation to the analysis of fuzzy dynamical problems [7, 15]. A thorough theoretical research of fuzzy Cauchy problems was given by Kaleva [16, 17], Wu and Song [18], Ouyang and Wu [19], Kim and Sakthivel [20], and M. D. Wu [21]. A generalization of fuzzy differential equation was given by Aubin [22, 23], Băıdosov [6], Kloeden [24], and Colombo and Křivan [25]. For a fuzzy Cauchy problem


Introduction
Fuzzy differential equations (FDEs), which are utilized for the purpose of the modeling problems in science and engineering, have been studied by many researchers.Most of the practical problems require the solutions of fuzzy differential equations (FDEs) which are satisfied with fuzzy initial conditions; therefore a fuzzy initial problem occurs and should be solved.However, for the vast majority of fuzzy initial value problems, their exact solutions are difficult to be obtained.Thus it is necessary to consider their numerical methods.
The concept of a fuzzy derivative was first introduced by Chang and Zadeh [1]; it was followed up by Dubois and Prade [2] who used the extension principle in their approach.Other fuzzy derivative concepts have been proposed by Puri and Ralescu [3] and Goetschel Jr. and Voxman [4] as an extension of the Hukuhara derivative of multivalued functions.In the past decades, many works have been appeared on the aspects of theories and applications on fuzzy differential equations; see [5][6][7][8][9][10][11][12].The notation of fuzzy differential equation was initially introduced by Kandel and Byatt [13,14] and later they applied the concept of fuzzy differential equation to the analysis of fuzzy dynamical problems [7,15].A thorough theoretical research of fuzzy Cauchy problems was given by Kaleva [16,17], Wu and Song [18], Ouyang and Wu [19], Kim and Sakthivel [20], and M. D. Wu [21].A generalization of fuzzy differential equation was given by Aubin [22,23], Baȋdosov [6], Kloeden [24], and Colombo and Křivan [25].
For a fuzzy Cauchy problem   () =  (, ) ,  0 ≤  ≤ ,  ( 0 ) =  0 , in 1999, Friedman et al. [26] firstly treated it and obtained its numerical solution by Euler method.In recent years, some researchers such as Abbasbandy and Allahviranloo applied the Taylor series method, the Runge-Kutta method, and the linear multistep method to solve fuzzy differential equations [5,20,[27][28][29][30][31][32].They proposed some numerical methods and discussed the convergence and stability of their methods under the fuzzy numbers background.However, their methods always have some of low convergence order.In this paper, based on Adams-Bashforth four-step method and Adams-Moulton three-step method, two Adams predictor-corrector algorithms are proposed to solve fuzzy initial problems.The convergence of the proposed methods is also presented in detail.Finally, an example is given to illustrate our methods.The structure of this paper is organized as follows.
Let  be a real interval.A mapping  :  →  1 is called a fuzzy process and its -level set is denoted by Definition 3. A triangular fuzzy number is a fuzzy set  in  1 that is characterized by an ordered triple (  ,   ,   ) ∈  3 with The -level set of a triangular fuzzy number  is given by for any  ∈ .

Definition 7.
A mapping  :  →  1 is called a fuzzy process.
We designate The Seikkala derivative   () of a fuzzy process  is defined by provided that this equation in fact defines a fuzzy number   () ∈  1 .

A Fuzzy Cauchy
Problem.Consider the first-order fuzzy differential equation   = (, ), where  is a fuzzy function of , (, ) is a a fuzzy function of the crisp variable  and the fuzzy variable , and   is the Hukuhara or Seikkala fuzzy derivative of .Given an initial value ( 0 ) =  0 , we can define a first-order fuzzy Cauchy problem as follows: The existence theorem is obtained for the Cauchy problem (11). Let Theorem 9 (see [33]).Let one consider the FCP (11) where  : Then the FCP (11) and the system of ODEs (12) are equivalent.

Interpolation for Fuzzy Numbers.
The problem of interpolation for fuzzy sets is as follows.
Suppose that at various time instant  information () is presented as fuzzy set.The aim is to approximate the function (), for all  in the domain of .Let  0 <  1 < ⋅ ⋅ ⋅ <   be  + 1 distinct points in  and let  0 ,  1 , . . .,   be  + 1 fuzzy sets in  1 .A fuzzy polynomial interpolation of the data is a fuzzy-value function  :  →  1 satisfying the following conditions: (1) (  ) =   , for all  = 0, 1, . . ., ; (2)  is continuous; (3) if the data is crisp, then the interpolation  is a crisp polynomial.

Adams Method
which are triangular fuzzy numbers and are shown by also By fuzzy interpolation of f( for therefore the following results will be obtained: From ( 3) and ( 19) it follows that where If (22) are situated in (24), we have The following results will be obtained by integration: Thus Therefore the four-step Adams-Bashforth method for solving fuzzy initial problems is obtained as follows: 3.2.Adams-Moulton Method.From [30], the Adams-Moulton three-step method to solve fuzzy initial problem   () = (, ) is as follows:

Adams Predictor-Corrector Method.
The following algorithm is based on Adams-Bashforth four-step method as a predictor and also an iteration of Adams-Moulton three-step method as a corrector.
Algorithm 11 (predictor-corrector four-step method).To approximate the solution of the following fuzzy initial value problem: positive integer  is chosen.

Improved Adams
Thus there exists the error estimations Based on above results, we improve Adams predictorcorrector four-step method into the following iterative computation algorithm.
Algorithm 12 (improved predictor-corrector systems).To approximate the solution of the following fuzzy initial value problem: positive integer  is chosen.

Numerical Examples
Example 1 (see [33]).Consider the following fuzzy differential equation: The exact solution of equation is By using the Adams predictor-corrector four-step method with  = 10 for some  ∈ [0, 1], the results shown in Tables 1, 2, and 3 are obtained.
And by using the improved Adams predictor-corrector systems with  = 10 for some  ∈ [0, 1], the results shown in Tables 4, 5, and 6 are obtained.
The results of Example 1 are shown by Figures 1, 2, and 3.

Conclusion
In this paper two numerical methods with higher order of convergence and not much amounts of computation for     solving fuzzy differential equations were discussed in detail.The proposed algorithms were generated by updating the Adams-Bashforth four-step method and Adams-Moulton three-step method.An example showed that the proposed methods is more efficient and practical than some methods appeared in the literature before.

Figure 1 :
Figure 1: Comparisons between the exact solution  and the numerical solution .

Figure 2 :
Figure 2: Comparisons between the exact solution  1 and the numerical solution  1 .

Figure 3 :
Figure 3: Comparisons between the exact solution  and the numerical solution .

Table 1 :
Comparisons between the exact solution and the numerical solution.

Table 2 :
Comparisons between the exact solution and the numerical solution.

Table 3 :
Comparisons between the exact solution and the numerical solution.

Table 4 :
Comparisons between the exact solution and the numerical solution.

Table 5 :
Comparisons between the exact solution and the numerical solution.

Table 6 :
Comparisons between the exact solution and the numerical solution.