Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 312328, 12 pages

http://dx.doi.org/10.1155/2013/312328

Research Article

Adams Predictor-Corrector Systems for Solving Fuzzy Differential Equations

Dequan Shang¹ and Xiaobin Guo²

¹Department of Public Courses, Gansu College of Traditional Chinese Medicine, Lanzhou 730000, China

²College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

Received 20 March 2013; Revised 31 May 2013; Accepted 2 June 2013

Academic Editor: Valentina E. Balas

Copyright © 2013 Dequan Shang and Xiaobin Guo. 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.

A predictor-corrector algorithm and an improved predictor-corrector (IPC) algorithm based on Adams method are proposed to solve first-order differential equations with fuzzy initial condition. These algorithms are generated by updating the Adams predictor-corrector method and their convergence is also analyzed. Finally, the proposed methods are illustrated by solving an example.

1. 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 [5–12]. 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], Baĭdosov [6], Kloeden [24], and Colombo and Křivan [25].

For a fuzzy Cauchy problem

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, 27–32]. 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.

In Section 2, some basic definitions and results are recalled. An explicit Adams-Bashforth method and an implicit Adams-Moulton method for solving FDEs are mentioned in Section 3. The predictor-corrector method and the improved predictor-corrector (IPC) systems algorithm are introduced in Section 4. The convergence of the proposed methods is discussed in Section 5. An illustrating example is given in Section 6 and the conclusion is drawn in Section 7.

2. Preliminaries

2.1. Fuzzy Numbers

Definition 1 (see [1]). A fuzzy number is a fuzzy set like which satisfies the following:

(1)is upper semicontinuous;(2) is fuzzy convex; that is, , for all , , ;(3) is normal; that is, there exists such that ;(4) is the support of the , and its closure cl(supp ) is compact.

Let be the set of all fuzzy numbers on .

Definition 2 (see [2]). A fuzzy number in parametric form is a pair of functions , , , which satisfies the following requirements:

(1) is a bounded monotonic increasing left continuous function;(2) is a bounded monotonic decreasing left continuous function;(3), .

Let be a real interval. A mapping is called a fuzzy process and its -level set is denoted by

Definition 3. A triangular fuzzy number is a fuzzy set in that is characterized by an ordered triple with such that and .

The -level set of a triangular fuzzy number is given by

for any .

Definition 4 (see [9]). The supremum metric on is defined by

With the spermium metric, the space is a complete metric space.

Definition 5 (see [9]). A mapping is Hukuhara differentiable at if for some the Hukuhara difference

exists in , for all , and if there exists an such that

The fuzzy set is called the Hukuhara derivative of at .

Recall that is defined on -level sets, where , for all . By the definition of the metric , all the -level set mappings are Hukuhara differentiable at with Hukuhara derivative for each when is Hukuhara differentiable at with Hukuhara derivative .

Remark 6. If is Hukuhara differentiable and its Hukuhara derivative is integrable over , then

for all .

Definition 7. A mapping 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 .

Remark 8. If is Seikkala differentiable and its Seikkala derivative is integrable over , then

for all .

2.2. 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 , we can define a first-order fuzzy Cauchy problem as follows:

The existence theorem is obtained for the Cauchy problem (11).

Let ; if is Hukuhara differentiable then . So (11) translates into the following system of ODEs:

Theorem 9 (see [33]). Let one consider the FCP (11) where is such that

(i);(ii) and are equicontinuous and uniformly bounded on any bounded set;(iii) and satisfy the Lipschitz conditions.

Then the FCP (11) and the system of ODEs (12) are equivalent.

2.3. 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 be distinct points in and let be fuzzy sets in . A fuzzy polynomial interpolation of the data is a fuzzy-value function satisfying the following conditions:

(1), for all ;(2) is continuous;(3)if the data is crisp, then the interpolation is a crisp polynomial.

A function fulfilling these conditions may be constructed as follows. Let , for any , . For each , the unique polynomial of degree is denoted by such that

Finally, for all , are defined by such that

The interpolation polynomial can be written level setwise as

for .

Theorem 10 (see [34]). Let , be the observed data and suppose that each of the is an element of . Then for each , ,

where , .

3. Adams Method

3.1. Adams-Bashforth Method

Now we are going to solve fuzzy initial problem by Adams-Bashforth four-step method. Let the fuzzy initial values be , , , , that is,

which are triangular fuzzy numbers and are shown by

also

By fuzzy interpolation of , , , , we have

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:

4. Predictor-Corrector Method

4.1. 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.

Step 1. Let ,

Step 2. Let .

Step 3. Let

Step 4. Let .

Step 5. Let

Step 6. .

Step 7. If goto Step 3.

Step 8. Algorithm is completed and approximates real value of to the original differential equations .

4.2. Improved Adams Predictor-Corrector Method

In the above section, the predicted values and the corrected values have the local truncation errors as follows:

Thus there exists the error estimations

Based on above results, we improve Adams predictor-corrector 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.

Step 1. Let ,

Step 2. Let .

Step 3. Let

Step 4. Let , .

Step 5. Let

Step 6. Let

Step 7. .

Step 8. If goto Step 3.

Step 9. Algorithm is completed and approximates real value of to the original differential equations .

5. Convergence

To integrate the system given in (11) from to a prefixed , the interval is replaced by a set of discrete equally spaced grid points , and the exact solution is approximated by some . The exact and approximate solutions at , are denoted by and , respectively. The grid points at which the solution is calculated are , , .

From (30), the polygon curves

are the implicit three-step approximation to and , respectively, over the interval . The following lemma will be applied to show the convergence of these approximations; that is,

Lemma 13. Let a sequence of numbers satisfy

for some given positive and , . Then

when is odd and

when is even, where , , , , are constants for all , , , , and .

Theorem 14. For any arbitrary fixed , the implicit three-step Simpson approximations of (30) converge to the exact solutions , for , .

Proof. It is sufficient to show

By Taylor’s theorem, we have

where , . Consequently

Denote , . Then

where and .

Set

then

are obtained, where , so by Lemma 13, since and we have

If then , which concludes the proof.

Theorem 15. For any arbitrary fixed , the explicit four-step Milne approximations of (29) converge to the exact solutions , for .

Proof. Like Theorem 14, the conclusion can be obtained easily.

6. 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 for some , the results shown in Tables 1, 2, and 3 are obtained.

And by using the improved Adams predictor-corrector systems with for some , the results shown in Tables 4, 5, and 6 are obtained.

The results of Example 16 are shown by Figures 1, 2, and 3.

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

Error

0.1
0.963635583813
0.963082998354
0.000552585459

0.2
0.975914722827
0.975304021448
0.000610701379

0.3
0.995959477182
0.995284547778
0.000674929403

0.4
1.022969627904
1.022223715555
0.000745912348

0.5
1.056214668049
1.055390307414
0.000824360635

0.6
1.095026491542
1.094115432142
0.000911059400

0.7
1.138792706547
1.137785830194
0.001006876353

0.8
1.186950506443
1.185837735978
0.001112770464

0.9
1.238981037065
1.237751235509
0.001229801555

1
1.294404203842
1.293045062928
0.001359140914

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

Error

0.1
1.004837418035
1.006647092872
0.001809674836

0.2
1.018730753077
1.020368214584
0.001637461506

0.3
1.040818220681
1.042299857123
0.001481636441

0.4
1.070320046035
1.071660686127
0.001340640092

0.5
1.106530659712
1.107743721032
0.001213061319

0.6
1.148811636094
1.149909259366
0.001097623272

0.7
1.196585303791
1.197578474398
0.000993170607

0.8
1.249328964117
1.250227622045
0.000898657928

0.9
1.306569659740
1.307382799060
0.000813139319

1
1.367879441171
1.368615200053
0.000735758882

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

Error

0.1
1.018894129717
1.019446715176
0.000552585459

0.2
1.036984860735
1.037595562114
0.000610701379

0.3
1.063452417560
1.064127346964
0.000674929403

0.4
1.097560862786
1.098306775134
0.000745912348

0.5
1.138650731584
1.139475092219
0.000824360635

0.6
1.186132431562
1.187043490962
0.000911059400

0.7
1.239480341921
1.240487218275
0.001006876353

0.8
1.298227552867
1.299340323332
0.001112770464

0.9
1.361961192623
1.363190994178
0.001229801555

1
1.430318295265
1.431677436179
0.001359140914

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

Error

0.1
0.963635583813
0.963525066721

0.2
0.975914722827
0.975792582551

0.3
0.995959477182
0.995824491301

0.4
1.022969627904
1.022820445434

0.5
1.056214668049
1.056049795922

0.6
1.095026491542
1.094844279662

0.7
1.138792706547
1.138591331277

0.8
1.186950506443
1.186727952350

0.9
1.238981037065
1.238735076754

1
1.294404203842
1.294132375659

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

Error

0.1
1.004837418035
1.005742255454

0.2
1.018730753077
1.019549483831

0.3
1.040818220681
1.041559038902

0.4
1.070320046035
1.070990366081

0.5
1.106530659712
1.107137190372

0.6
1.148811636094
1.149360447730

0.7
1.196585303791
1.197081889095

0.8
1.249328964117
1.249778293081

0.9
1.306569659740
1.306976229400

1
1.367879441171
1.368247320612

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

Error

0.1
1.018894129717
1.019004646809

0.2
1.036984860735
1.037107001011

0.3
1.063452417560
1.063587403441

0.4
1.097560862786
1.097710045255

0.5
1.138650731584
1.138815603711

0.6
1.186132431562
1.186314643442

0.7
1.239480341921
1.239681717192

0.8
1.298227552867
1.298450106960

0.9
1.361961192623
1.362207152934

1
1.430318295265
1.430590123448

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

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

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

7. 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.

Acknowledgments

The work is supported by the Natural Scientific Funds of PR China (no. 71061013) and the Youth Research Ability Project of Northwest Normal University (NWNU-LKQN-1120).

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