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Numerical Methods for Fuzzy Initial Value Problems under Different Types of Interpretation: A Comparison Study

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Book cover Informatics Engineering and Information Science (ICIEIS 2011)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 252))

Abstract

In this paper, we present the solution of fuzzy initial value problem of the form X ′(t) = f(t,X(t)), X(t 0) = X 0, where X 0 is a symmetric triangular fuzzy interval. In the first part, we review the existing analytical methods for solving such problem under different types of interpretation. We then provide detailed numerical procedures based on the different interpretations given in the first part. Finally, we give a property and establish some results on relationship between the existing numerical methods under different types of interpretation.

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References

  1. Abbasbandy, S., Allahviranloo, T.: Numerical solutions of fuzzy differential equations by Taylor method. Computational Methods in Applied Mathematics 2, 113–124 (2002)

    Article  MathSciNet  Google Scholar 

  2. Abbasbandy, S., Allahviranloo, T.: Numerical solution of fuzzy differential equation by Runge-Kutta method. Nonlinear Studies 11, 117–129 (2004)

    MathSciNet  MATH  Google Scholar 

  3. Agarwal, R.P., O’Regan, D., Lakshmikantham, V.: Viability theory and fuzzy differential equations. Fuzzy Sets and Systems 151, 563–580 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ahmad, M.Z., De Baets, B.: A predator-prey model with fuzzy initial populations. In: IFSA/EUSFLAT Conf. 2009, pp. 1311–1314 (2009)

    Google Scholar 

  5. Ahmad, M.Z., Hasan, M.K.: A new approach to incorporate uncertainty into Euler method. Applied Mathematical Sciences 4, 2509–2520 (2010)

    MathSciNet  MATH  Google Scholar 

  6. Ahmad, M.Z., Hasan, M.K.: Incorporating optimisation technique into Zadeh’s extension principle for computing non-monotone functions with fuzzy variable. Sains Malaysiana 40, 643–650 (2011)

    MATH  Google Scholar 

  7. Ahmad, M.Z., Hasan, M.K.: A new fuzzy version of Euler’s method for solving differential equations with fuzzy initial values. Sains Malaysiana 40, 651–657 (2011)

    MATH  Google Scholar 

  8. Ahmad, M.Z., Hasan, M.K., De Baets, B.: A new method for computing continuous functions with fuzzy variable. Journal of Applied Sciences 11, 1143–1149 (2011)

    Article  Google Scholar 

  9. Bede, B., Gal, S.G.: Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations. Fuzzy Sets and Systems 151, 581–599 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Buckley, J.J., Feuring, T.: Fuzzy differential equations. Fuzzy Sets and Systems 110, 43–54 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chalco-Cano, Y., Román-Flores, H.: On new solutions of fuzzy differential equations. Chaos, Solitons & Fractals 38, 112–119 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chalco-Cano, Y., Román-Flores, H.: Comparation between some approaches to solve fuzzy differential equations. Fuzzy Sets and Systems 160, 1517–1527 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Diamond, P.: Stability and periodicity in fuzzy differential equations. IEEE Transactions on Fuzzy Systems 8, 583–590 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ding, Z., Ma, M., Kandel, A.: Existence of the solutions of fuzzy differential equations with parameters. Information Sciences 99, 205–217 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  15. Dubois, D., Prade, H.: Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York (1980)

    MATH  Google Scholar 

  16. Goetschel, R., Voxman, W.: Elementary fuzzy calculus. Fuzzy Sets and Systems 18, 31–43 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hüllermeier, E.: An approach to modelling and simulation of uncertain dynamical systems. Int. J. Uncertainty Fuzziness Knowledge-Based Systems 5, 117–137 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hüllermeier, E.: Numerical methods for fuzzy initial value problems. Int. J. Uncertainty Fuzziness Knowledge-Based Systems 7, 439–461 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kaleva, O.: Fuzzy differential equations. Fuzzy Sets and Systems 24, 301–317 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kaleva, O.: A note on fuzzy differential equations. Nonlinear Analysis 64, 895–900 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ma, M., Friedman, M., Kandel, A.: Numerical solution of fuzzy differential equations. Fuzzy Sets and Systems 105, 133–138 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mizukoshi, M.T., Barros, L.C., Chalco-Cano, Y., Romn-Flores, H., Bassanezi, R.C.: Fuzzy differential equations and the extension principle. Information Sciences 177, 3627–3635 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Nieto, J.J., Khastan, A., Ivaz, K.: Numerical solution of fuzzy differential equations under generalized differentiability. Nonlinear Analysis: Hybrid Systems 3, 700–707 (2009)

    MathSciNet  MATH  Google Scholar 

  24. Palligkinis, S.C., Papageorgiou, G., Famelis, I.T.: Runge-Kutta methods for fuzzy differential equations. Applied Mathematics and Computation 209, 97–105 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Puri, M.L., Ralescu, D.A.: Differentials of fuzzy functions. J. Math. Anal. Appl. 91, 552–558 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  26. Román-Flores, H., Barros, L., Bassanezi, R.: A note on Zadeh’s extension principle. Fuzzy Sets and Systems 117, 327–331 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  27. Song, S., Wu, C.: Existence and uniqueness of solutions to the Cauchy problem of fuzzy differential equations. Fuzzy Sets and Systems 110, 55–67 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  28. Vorobiev, D., Seikkala, S.: Towards the theory of fuzzy differential equations. Fuzzy Sets and Systems 125, 231–237 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wu, C., Song, S.: Existence theorem to the Cauchy problem of fuzzy differential equations under compactness-type conditions. Information Sciences 108, 123–134 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wu, C., Song, S., Lee, E.S.: Approximate solutions, existence and uniqueness of the Cauchy problem of fuzzy differential equations. Journal of Mathematical Analysis and Applications 202, 629–644 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  31. Zadeh, L.A.: Fuzzy sets. Information and Control 8, 338–353 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zimmermann, H.J.: Fuzzy Set Theory and Its Applications. Kluwer Academic Publishers, Dordrecht (1991)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Engineering Mathematics, Universiti Malaysia Perlis, 02000, Kuala Perlis, Perlis, Malaysia

    Muhammad Zaini Ahmad

  2. School of Information Technology, Faculty of Information Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM, Bangi, Selangor, Malaysia

    Mohammad Khatim Hasan

Authors
  1. Muhammad Zaini Ahmad
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  2. Mohammad Khatim Hasan
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Editor information

Editors and Affiliations

  1. Advanced Informatics School (UTM AIS), UTM International Campus, 54100, Kuala Lumpur, Malaysia

    Azizah Abd Manaf , Akram Zeki , Mazdak Zamani  & Suriayati Chuprat , ,  & 

  2. Information Systems Department, King Saud University, Riyadh, Saudi Arabia

    Eyas El-Qawasmeh

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© 2011 Springer-Verlag Berlin Heidelberg

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Ahmad, M.Z., Hasan, M.K. (2011). Numerical Methods for Fuzzy Initial Value Problems under Different Types of Interpretation: A Comparison Study. In: Abd Manaf, A., Zeki, A., Zamani, M., Chuprat, S., El-Qawasmeh, E. (eds) Informatics Engineering and Information Science. ICIEIS 2011. Communications in Computer and Information Science, vol 252. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25453-6_25

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  • DOI: https://doi.org/10.1007/978-3-642-25453-6_25

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-25452-9

  • Online ISBN: 978-3-642-25453-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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