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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 735128, 10 pages
http://dx.doi.org/10.1155/2013/735128
Research Article

A Numerical Method for Fuzzy Differential Equations and Hybrid Fuzzy Differential Equations

1Islamic Azad University, Shabestar Branch, Shabestar 5381637181, Iran
2Department of Applied Mathematics, University of Tabriz, Tabriz 5166616471, Iran
3Mathematics Department, Institute for Advanced Studies in Basic Sciences, Zanjan 45137-66731, Iran
4Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela (USC), 15782 Santiago de Compostela, Spain
5Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 12 June 2013; Accepted 28 August 2013

Academic Editor: Marcia Federson

Copyright © 2013 K. Ivaz et al. 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.

Linked References

  1. 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
  2. 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
  3. 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
  4. 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
  5. P. E. Kloeden, “Remarks on Peano-like theorems for fuzzy differential equations,” Fuzzy Sets and Systems, vol. 44, no. 1, pp. 161–163, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. B. Bede, I. J. Rudas, and A. L. Bencsik, “First order linear fuzzy differential equations under generalized differentiability,” Information Sciences, vol. 177, no. 7, pp. 1648–1662, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. J. J. Nieto, A. Khastan, and K. Ivaz, “Numerical solution of fuzzy differential equations under generalized differentiability,” Nonlinear Analysis: Hybrid Systems, vol. 3, no. 4, pp. 700–707, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. A. Khastan, J. J. Nieto, and R. Rodríguez-López, “Variation of constant formula for first order fuzzy differential equations,” Fuzzy Sets and Systems, vol. 177, pp. 20–33, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. S. Abbasbandy, T. Allahviranloo, and P. Darabi, “Numerical soluion of nth-order fuzzy differential equations by Runge-Kutta method,” Journal of Mathematical and Computational Applications, vol. 16, no. 4, pp. 935–946, 2011.
  10. 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
  11. M. Friedman, M. Ma, and A. Kandel, “Numerical solutions of fuzzy differential and integral equations,” Fuzzy Sets and Systems, vol. 106, no. 1, pp. 35–48, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. B. Ghazanfari and A. Shakerami, “Numerical solutions of fuzzy differential equations by extended Runge-Kutta-like formulae of order 4,” Fuzzy Sets and Systems, vol. 189, pp. 74–91, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. A. Khastan and K. Ivaz, “Numerical solution of fuzzy differential equations by Nyström method,” Chaos, Solitons & Fractals, vol. 41, no. 2, pp. 859–868, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. H. Kim and R. Sakthivel, “Numerical solution of hybrid fuzzy differential equations using improved predictor-corrector method,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 10, pp. 3788–3794, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. M. Ma, M. Friedman, and A. Kandel, “Numerical solutions of fuzzy differential equations,” Fuzzy Sets and Systems, vol. 105, no. 1, pp. 133–138, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. N. Parandin, “Numerical solution of fuzzy differential equations of nth-orderby Runge-Kutta method,” Neural Computing and Applications, vol. 21, no. 1, Supplement, pp. 347–355, 2012. View at Publisher · View at Google Scholar
  17. P. Prakash and V. Kalaiselvi, “Numerical solution of hybrid fuzzy differential equations by predictor-corrector method,” International Journal of Computer Mathematics, vol. 86, no. 1, pp. 121–134, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. S. Pederson and M. Sambandham, “Numerical solution to hybrid fuzzy systems,” Mathematical and Computer Modelling, vol. 45, no. 9-10, pp. 1133–1144, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. S. Pederson and M. Sambandham, “Numerical solution of hybrid fuzzy differential equation IVPs by a characterization theorem,” Information Sciences, vol. 179, no. 3, pp. 319–328, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. J. Li, A. Zhao, and J. Yan, “The Cauchy problem of fuzzy differential equations under generalized differentiability,” Fuzzy Sets and Systems, vol. 200, pp. 1–24, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. M. T. Malinowski, “Random fuzzy differential equations under generalized Lipschitz condition,” Nonlinear Analysis: Real World Applications, vol. 13, no. 2, pp. 860–881, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. J. Xu, Z. Liao, and J. J. Nieto, “A class of linear differential dynamical systems with fuzzy matrices,” Journal of Mathematical Analysis and Applications, vol. 368, no. 1, pp. 54–68, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, John Wiley & Sons, New York, NY, USA, 1962. View at MathSciNet
  24. P. Diamond and P. Kloeden, Metric Spaces of Fuzzy Sets, World Scientific, Singapore, 1994.
  25. O. Kaleva, “Interpolation of fuzzy data,” Fuzzy Sets and Systems, vol. 61, no. 1, pp. 63–70, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet