Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 406240, 12 pages
http://dx.doi.org/10.1155/2013/406240
Research Article

Approximate Solution of th-Order Fuzzy Linear Differential Equations

1College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China
2Department of Public Courses, Gansu College of Traditional Chinese Medicine, Lanzhou 730000, China

Received 29 January 2013; Accepted 19 March 2013

Academic Editor: Ker-Wei Yu

Copyright © 2013 Xiaobin Guo and Dequan Shang. 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 approximate solution of th-order fuzzy linear differential equations in which coefficient functions maintain the sign is investigated by the undetermined fuzzy coefficients method. The differential equations is converted to a crisp function system of linear equations according to the operations of fuzzy numbers. The fuzzy approximate solution of the fuzzy linear differential equation is obtained by solving the crisp linear equations. Some numerical examples are given to illustrate the proposed method. It is an extension of Allahviranloo's results.

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 practical problems require the solutions of fuzzy differential equations (FDEs) which are satisfied with fuzzy initial conditions, and therefore a fuzzy initial problem occurs and needs to be solved. However, most fuzzy initial value problems could not be solved exactly. So it is necessary to consider their approximating methods.

Prior to discussing fuzzy differential equations and their associated numerical algorithms, it is necessary to present an appropriate brief introduction to derivative of the fuzzy-valued function. The concept of a fuzzy derivative was first introduced by Chang and Zadeh [1], and 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 recent years, many works have been produced in the aspects of theories and applications on fuzzy differential equations; see [515]. The notation of fuzzy differential equation was initially introduced by Kandel and Byatt [16, 17] who later applied the concept of fuzzy differential equation to the analysis of fuzzy dynamical problems [18, 19]. A thorough theoretical research of fuzzy Cauchy problems was given by Kaleva [20, 21], Seikkala [22], Ouyang and Wu [23], Kloeden [24], and Wu [25]. A generalization of fuzzy differential equation was given by Aubin [26, 27], Baĭdosov [6], Leland [28], and Colombo and Kivan [29]. Some numerical methods for solving fuzzy differential equations were introduced in [3033].

For an th-order linear differential equation with fuzzy initial conditions where are continuous functions, Buckley and Feuring [34] presented two analytical methods for solving them. The first was to fuzzify the crisp solution and then check to see if it satisfies the fuzzy differential equations with fuzzy initial conditions. The second method was the reverse of the first one, in that they firstly solved the fuzzy initial value problem and then checked to see if it defines a fuzzy function. In 2008, Allahviranloo et al. [35] utilized the collocation method to transfer (1) and (2) into a crisp system of linear equations for three specious cases; that is, all coefficient functions are positive, negative, , are negative are positive, respectively. By computing (3), they obtained the approximate solution of (1) and (2). However, their methods will be restricted for the general case.

In this paper, the th-order linear differential equation with fuzzy initial conditions is further investigated. It shows that the result obtained in this paper is an extension of Allahviranloo's conclusions. In addition, considering that the case of the order of the fuzzy differential equation and the number of basic functions in assumed solution are not always equal, we obtain the approximate solution of the original equations (1) and (2) by calculating the minimal norm least squares solution of crisp system of linear equations. Three illustrating examples are given, and one of them is compared with Allahviranloo's work and is shown be more accurate. The structure of this paper is organized as follows.

In Section 2, we recall some basic definitions and results about fuzzy numbers and the undetermined fuzzy coefficients method. In Section 3, a class of th-order fuzzy linear differential equations is investigated by converting it to a crisp system of linear equations, and some corollaries for the special cases are given. The proposed algorithms are illustrated by solving some examples in Section 4, and the conclusion is drawn in Section 5.

2. Preliminaries

2.1. Fuzzy Number

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)Supp  is the support of , and its closure cl(supp ) is compact.
Let be the set of all fuzzy numbers on .

Definition 2 (see [36]). 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), .
For arbitrary fuzzy number , , and real number ,(1) if and only if and ;(2);(3);(4)

Definition 3 (see [23]). For arbitrary , the quantity is the distance between fuzzy numbers and .

2.2. The Undetermined Fuzzy Coefficients Method

The undetermined fuzzy coefficients method is to seek an approximate solution as where , are positive basic functions whose all differentiations are positive. We compute the fuzzy coefficients in (6) by setting the error to zero as follows:

We substitute (6) in (7) and represent them in parametric forms; then,

Lemma 4 (see [35]). Let basic functions , and all of their differentiations be positive; without loss of generality, then and .

3. Basic Results

In order to solve (1) and (2), we need to consider the system of linear equations (8). In this section, we study its general case at first then give some corollaries for some special cases.

3.1. New Models

Theorem 5. Suppose that each , in (1) is either nonnegative or negative over ; then, the th-order fuzzy linear differential equations can be extended into a system of linear equations where
By setting , and by solving (10), the fuzzy approximate solution of the original fuzzy linear differential equations is obtained as follows:

Proof. Let (5) be substituted in (1) and (2); we have
We express the pervious equations in parametric forms; then,
By setting
thus we have the corresponding systems as follows: being linear systems.
By setting , and by solving (10), we obtain the values of parameters therefore, we get the fuzzy approximate solution of the original fuzzy equation as follows:

Now, we consider another special case; that is, coefficient functions , are positive and negative alternately.

Corollary 6. Suppose that the coefficients functions are positive and , are negative, and is an odd number. Then, the th-order fuzzy linear differential equations (1) and (2) can be extended into the following linear equations: where
By setting
(19) is a system of linear equations such that where
When is an even number, then the th-order fuzzy linear differential equations (1) and (2) can be extended into a system of linear equations where
Three special cases in Allahviranloo et al.’s paper [35] are viewed as corollaries from Theorem 5.

Corollary 7. Suppose that coefficients functions , are nonnegative; then, the th-order fuzzy linear differential equations (1) and (2) can be extended into the following linear equations: where

Corollary 8. Suppose that coefficients functions , are negative; then, the th-order fuzzy linear differential equations (1) and (2) can be extended into a system of linear equations where

Corollary 9. Suppose that coefficients functions are nonnegative and are negative; then, the th-order fuzzy linear differential equations (1) and (2) can be extended into a system of linear equations where

3.2. Method to Solve Linear Equations

The previous crisp linear equations (10), (19), (24), (26), (28), and (30) are all linear systems, and they have the same form as follows:

In the process of solving (32) by setting , whether it is consistent or inconsistent, we obtain the minimal norm least squares solution [37] by using the generalized inverse of the coefficient matrix ; that is,

Thus, we get

Therefore, we obtain the fuzzy approximate solution of the original fuzzy equation as follows:

4. Numerical Examples

Example  1. Consider the following second-order fuzzy linear differential equation:

The exact solution of equation is

Let ; then,

From (10), the extended linear equation is as follows:

By setting , the parameters are obtained, and by putting them into (38), we have

Tables 1 and 2 show comparisons between the exact solution and the approximate solution at for some , and all data were calculated by MATLAB 6.x.

tab1
Table 1: Comparisons between the exact solution and the approximate solution.
tab2
Table 2: Comparisons between the exact solution and the approximate solution.

Example  2. Consider the following three-order fuzzy linear differential equation:

The exact solution of equation is

Let ; then,

From (19), the extended linear equation is as follows:

By setting , the parameters are obtained, and by putting them into (43), we have

Tables 3, 4, 5, and 6 show comparisons between the exact solution and the approximate solution at and for some .

tab3
Table 3: Comparisons between the exact solution and the approximate solution ().
tab4
Table 4: Comparisons between the exact solution and the approximate solution ().
tab5
Table 5: Comparisons between the exact solution and the approximate solution ().
tab6
Table 6: Comparisons between the exact solution and the approximate solution ().

Example  3 (see [35]). Consider the following fuzzy linear differential equation:

The exact solution of equation is

Let ; then,

From (26), the extended linear equation is as follows:

By setting , the parameters are obtained, and by putting them into (48), we have

Tables 7, 8, 9, and 10 show comparisons between the exact solution and the approximate solution at and for some .

tab7
Table 7: Comparisons between the exact solution and the approximate solution ().
tab8
Table 8: Comparisons between the exact solution and the approximate solution ().
tab9
Table 9: Comparisons between the exact solution and the approximate solution ().
tab10
Table 10: Comparisons between the exact solution and the approximate solution ().

From Tables 7, 8, 9, and 10, we can see that our results are more accurate than those of Example 3.1 [35] in Allahviranloo et al.’s work.

5. Conclusion

In this paper, an approximate method similar to the undetermined fuzzy coefficients method, based on a positive basis for solving fuzzy differential equations, was further discussed. The more general case was considered, and the model system of linear equation was set up according to different case of coefficient functions. Fuzzy approximate solution was obtained by solving the model system. Our work enriched the theory of fuzzy linear differential equations.

Acknowledgments

The work is supported by the Natural Scientific Funds of PR China (71061013) and the Youth Research Ability Project of Northwest Normal University (NWNU-LKQN-11-20).

References

  1. S. S. L. Chang and L. A. Zadeh, “On fuzzy mapping and control,” vol. SMC-2, pp. 330–340, 1972. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. D. Dubois and H. Prade, “Operations on fuzzy numbers,” International Journal of Systems Science, vol. 9, no. 6, pp. 613–626, 1978. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. M. L. Puri and D. A. Ralescu, “Differentials of fuzzy functions,” Journal of Mathematical Analysis and Applications, vol. 91, no. 2, pp. 552–558, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. R. Goetschel, Jr. and W. Voxman, “Elementary fuzzy calculus,” Fuzzy Sets and Systems, vol. 18, no. 1, pp. 31–43, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. R. P. Agarwal, D. O'Regan, and V. Lakshmikantham, “Viability theory and fuzzy differential equations,” Fuzzy Sets and Systems, vol. 151, no. 3, pp. 563–580, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. V. A. Baĭdosov, “Fuzzy differential inclusions,” Journal of Applied Mathematics and Mechanics, vol. 54, no. 1, pp. 8–13, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  7. K. Kirkiakidis and A. Tzez, “Fuzzy logic adaptive sliding control design with an application to air-handling systems,” in Proceedings of the European Congress on Intelligent Techniques and Soft Computing (EUFIT '95), pp. 954–959, 1995.
  8. M. T. Mizukoshi, L. C. Barros, Y. Chalco-Cano, H. Román-Flores, and R. C. Bassanezi, “Fuzzy differential equations and the extension principle,” Information Sciences, vol. 177, no. 17, pp. 3627–3635, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. J. J. Nieto and R. Rodríguez-López, “Bounded solutions for fuzzy differential and integral equations,” Chaos, Solitons and Fractals, vol. 27, no. 5, pp. 1376–1386, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  10. J. J. Nieto, R. Rodríguez-López, and D. Franco, “Linear first-order fuzzy differential equations,” International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, vol. 14, no. 6, pp. 687–709, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. R. Rodríguez-López and T. Feuring, “Monotone method for fuzzy differential equations,” Fuzzy Sets and Systems, vol. 159, no. 16, pp. 2047–2076, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. R. Palm and D. Driankov, “Fuzzy inputs,” Fuzzy Sets and Systems, vol. 70, no. 2-3, pp. 315–335, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  13. B. S. Moon, “A tuning algorithm for the fuzzy logic controllers,” in Proceedings of the European Congress on Intelligent Techniques and Soft Computing (EUFIT '95), pp. 620–624, 1995.
  14. C. X. Wu and S. J Shih, “Existence theorem to the Cauchy problem of fuzzy differential equations under compactness-type conditions,” Information Sciences, vol. 108, no. 1–4, pp. 123–134, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  15. C. X. Wu and G. Wang, “Convergence of sequences of fuzzy numbers and fixed point theorems for increasing fuzzy mappings and application,” Fuzzy Sets and Systems, vol. 130, no. 3, pp. 383–390, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. A. Kandel and W. J. Byatt, “Fuzzy sets, Fuzzy algebra and fuzzy statistics,” in Proceedings of the IEEE, pp. 1619–1639, 1978. View at MathSciNet
  17. A. Kandel and W. J. Byatt, “Fuzzy differential equations,” in Proceedings of the international Conference on Cybernetics and Society, pp. 1213–1216, Tokyo, Japan, 1987.
  18. A. Kandel and W. J. Byatt, “Fuzzy processes,” Fuzzy Sets and Systems, vol. 4, no. 2, pp. 117–152, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. A. Kandel, “Fuzzy dynamical systems and the nature of their solutions,” in Fuzzy Sets: Theory and Application To Policy Analysis and Information Systems, P. P. Wang and S. K. Chang, Eds., pp. 93–122, Plenum Press, New York, NY, USA, 1980. View at Google Scholar
  20. O. Kaleva, “Fuzzy differential equations,” Fuzzy Sets and Systems, vol. 24, no. 3, pp. 301–317, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. O. Kaleva, “The Cauchy problem for fuzzy differential equations,” Fuzzy Sets and Systems, vol. 35, no. 3, pp. 389–396, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. S. Seikkala, “On the fuzzy initial value problem,” Fuzzy Sets and Systems, vol. 24, no. 3, pp. 319–330, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. H. Ouyang and Y. Wu, “On fuzzy differential equations,” Fuzzy Sets and Systems, vol. 32, no. 3, pp. 321–325, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. P. E. Kloeden, “Remarks on Peano-like theorems for fuzzy differential equations,” Fuzzy Sets and Systems, vol. 44, no. 1, pp. 161–164, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. M. D. Wu, “Linear fuzzy differential equation systems on R1,” Journal of Fuzzy Mathematics, vol. 2, pp. 51–56, 1988 (Chinese). View at Google Scholar
  26. J.-P. Aubin, “Fuzzy differential inclusions,” Problems of Control and Information Theory, vol. 19, no. 1, pp. 55–57, 1990. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. J.-P. Aubin, “A survey of viability theory,” SIAM Journal on Control and Optimization, vol. 28, no. 4, pp. 749–788, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. R. P. Leland, “Fuzzy differential systems and Malliavin calculus,” Fuzzy Sets and Systems, vol. 70, no. 1, pp. 59–73, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. G. Colombo and V. Křivan, “Fuzzy differential inclusions and nonprobabilistic likelihood,” Dynamic Systems and Applications, vol. 1, no. 4, pp. 419–439, 1992. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. S. Abbasbandy and T. Allahviranloo, “Numerical solutions of fuzzy differential equations by Taylor method,” Computational Methods in Applied Mathematics, vol. 2, no. 2, pp. 113–124, 2002. View at Google Scholar · View at MathSciNet
  31. S. Abbasbandy, T. A. Viranloo, Ó. López-Pouso, and J. J. Nieto, “Numerical methods for fuzzy differential inclusions,” Computers & Mathematics with Applications, vol. 48, no. 10-11, pp. 1633–1641, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. T. Allahviranloo, N. Ahmady, and E. Ahmady, “Numerical solution of fuzzy differential equations by predictor-corrector method,” Information Sciences, vol. 177, no. 7, pp. 1633–1647, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. T. Allahviranloo, S. Abbasbandy, N. Ahmady, and E. Ahmady, “Improved predictor-corrector method for solving fuzzy initial value problems,” Information Sciencesl, vol. 179, no. 7, pp. 945–955, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. J. J. Buckley and T. Feuring, “Fuzzy initial value problem for nth-order linear differential equations,” Fuzzy Sets and Systems, vol. 121, no. 2, pp. 247–255, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. T. Allahviranloo, E. Ahmady, and N. Ahmady, “Nth-order fuzzy linear differential equations,” Information Sciences, vol. 178, no. 5, pp. 1309–1324, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. P. Diamond, “Stability and periodicity in fuzzy differential equations,” IEEE Transactions on Fuzzy Systems, vol. 8, pp. 583–590, 2000. View at Google Scholar
  37. B. R. Fang, J. D. Zhou, Y. M. Li, and Y. M., Matrix Theory, Tsinghua university and Springer press, BeiJing, China, 2004.