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, Volume 92, Issue 2, pp 181–197 | Cite as

A new method for solving fuzzy linear differential equations

  • T. Allahviranloo
  • S. Abbasbandy
  • S. Salahshour
  • A. Hakimzadeh
Article

Abstract

In this paper, a novel operator method is proposed for solving fuzzy linear differential equations under the assumption of strongly generalized differentiability. To this end, the equivalent integral form of the original problem is obtained then by using its lower and upper functions the solutions in the parametric forms are determined. The proposed method is illustrated with numerical examples.

Keywords

Fuzzy linear differential equations Strongly generalized differentiability Equivalent integral form 

Mathematics Subject Classification (2000)

34L02 47E05 

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References

  1. 1.
    Abbasbandy S, Allahviranloo T, Lopez-Pouso O, Nieto JJ (2004) Numerical methods for fuzzy differential inclusions. Comput Math Appl 48: 1633–1641MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Allahviranloo T, Ahmady N, Ahmady E (2007) Numerical solution of fuzzy differential equations by predictor-corrector method. Inf Sci 177: 1633–1647MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Allahviranloo T, Barkhordari Ahmadi M (2010) Fuzzy Laplace transforms. Soft Comput 14: 235–243MATHCrossRefGoogle Scholar
  4. 4.
    Allahviranloo T, Kiani NA, Barkhordari M (2009) Toward the existence and uniqueness of solutions of second-order fuzzy differental equations. Inf Sci 179: 1207–1215MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Allahviranloo T, Kiani NA, Motamedi N (2009) Solving fuzzy differential equations by differential transformation method. Inf Sci 179: 956–966MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Allahviranloo T, Salahshour S (2010) Euler method for solving hybrid fuzzy differential equation. Soft Comput. doi: 10.1007/s00500-010-0659-y
  7. 7.
    Agarwal RP, Lakshmikantham V, Nieto JJ (2010) On the concept of solution for fractional differential equations with uncertainty. Nonlinear Anal 72: 2859–2862MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Bede B, Gal SG (2005) Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations. Fuzzy Sets Syst 151: 581–599MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Bede B, Rudas IJ, Bencsik AL (2006) First order linear fuzzy differential equations under generalized differentiability. Inf Sci 177: 1648–1662MathSciNetCrossRefGoogle Scholar
  10. 10.
    Chalco-Cano Y, Roman-Flores H (2006) On new solutions of fuzzy differential equations. Chaos Solitons Fractals 38: 112–119MathSciNetCrossRefGoogle Scholar
  11. 11.
    Diamond P, Kloeden P (1994) Metric spaces of fuzzy sets. World Scientific, SingaporeMATHGoogle Scholar
  12. 12.
    Georgiou DN, Nieto JJ, Rodriguez R (2005) Initial value problems for higher-order fuzzy differential equations. Nonlinear Anal 63: 587–600MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Khastan A, Bahrami F, Ivaz K (2010) New results on multiple solutions for Nth-order fuzzy differential equation under generalized differentiability. Boundary Value Problem (Hindawi Publishing Corporation). doi: 10.1155/2009/395714
  14. 14.
    Khastan A, Nieto JJ (2010) A boundary value problem for second order fuzzy differential equations. Nonlinear Anal 72: 3583–3593MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Nieto JJ, Rodriguez-Lopez R (2006) Hybrid metric dynamical systems with impulses. Nonlinear Anal 64: 368–380MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Nieto JJ, Rodriguez-Lopez R, Franco D (2006) Linear first-order fuzzy differential equations. Int J Uncertain Fuzziness Knowledge-Based Syst 14: 687–709MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Nieto JJ, Rodriguez-Lopez R, Georgiou DN (2008) Fuzzy differential systems under generalized metric spaces approach. Dyn Syst Appl 17: 1–24MathSciNetMATHGoogle Scholar
  18. 18.
    Puri ML, Ralescu D (1986) Fuzzy random variables. J Math Anal Appl 114: 409–422MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Puri ML, Ralescu D (1983) Differential for fuzzy function. J Math Anal Appl 91: 552–558MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Wu HC (1999) The improper fuzzy Riemann integral and its numerical integration. Inf Sci 111: 109–137CrossRefGoogle Scholar
  21. 21.
    Wu HC (2000) The fuzzy Riemann integral and its numerical integration. Fuzzy Set Syst 110: 1–25MATHCrossRefGoogle Scholar
  22. 22.
    Xu J, Liao Z, Nieto JJ (2010) A class of linear differential dynamical systems with fuzzy matrices. J Math Anal Appl 368: 54–68MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Zimmermann HJ (1991) Fuzzy set theory and its applications. Kluwer, DordrechtMATHGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of Mathematics, Science and Research BranchIslamic Azad UniversityTehranIran
  2. 2.Department of Mathematics, Mobarakeh BranchIslamic Azad UniversityMobarakehIran

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