A Runge-Kutta Method with Lower Function Evaluations for Solving Hybrid Fuzzy Differential Equations

  • Ali Ahmadian
  • Mohamed Suleiman
  • Fudziah Ismail
  • Soheil Salahshour
  • Ferial Ghaemi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7802)


In this paper, we apply a Runge-Kutta method for solving first order fuzzy differential equations using lower number of function evaluations in comparison with classical Runge-Kutta method. It is assumed that the user will evaluate both f and f readily instead of the evaluations of f only when solving hybrid fuzzy differential equation which enhance the order of accuracy of the solutions. Numerical example is provided which compares the new results with previous findings.


Fuzzy ordinary differential equation Hybrid system Bede’s Characterization Theorem High order Runge-Kutta method Seikkala derivative 


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  1. 1.
    Abbod, M.F., Von Keyserlingk, D.G., Linkens, D.A., Mahfouf, M.: Survey of utilisation of fuzzy technology in medicine and healthcare. Fuzzy Set Syst. 120, 331–349 (2001)CrossRefGoogle Scholar
  2. 2.
    Allahviranloo, T., Ahmady, N., Ahmady, E.: Numerical solution of fuzzy differential equations by predictor-corrector method. Information Sciences 177, 1633–1647 (2007)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Allahviranloo, T., Salahshour, S.: Euler method for solving hybrid fuzzy differential equation. Soft Comput. J. 15, 1247–1253 (2011)MATHCrossRefGoogle Scholar
  4. 4.
    Barro, S., Marn, R.: Fuzzy logic in medicine. Physica-Verlag, Heidelberg (2002)MATHCrossRefGoogle Scholar
  5. 5.
    Bede, B.: Note on ”Numerical solutions of fuzzy differential equations by predictor corrector method”. Information Sciences 178, 1917–1922 (2008)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Casasnovas, J., Rossell, F.: Averaging fuzzy biopolymers. Fuzzy Set Syst. 152, 139–158 (2005)MATHCrossRefGoogle Scholar
  7. 7.
    Chang, B.C., Halgamuge, S.K.: Protein motif extraction with neuro-fuzzy optimization. Bioinformatics 18, 1084–1090 (2002)CrossRefGoogle Scholar
  8. 8.
    Dubios, D., Prade, H.: Towards fuzzy differential calculus-part3. Fuzzy Sets and Systems 8, 225–234 (1982)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Enright, W.H.: Second derivative multi-step methods for stiff ordinary differential equations. SIAM J. Numer. Anal. 11, 321–331 (1974)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Feng, G., Chen, G.: Adaptative control of discrete-time chaotic systems: a fuzzy control approach. Chaos, Solitons & Fractals 23, 459–467 (2005)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Goeken, D., Johnson, O.: Runge-Kutta with higher derivative approximations. Appl. Numer. Math. 39, 249–257 (2000)MathSciNetGoogle Scholar
  12. 12.
    Jiang, W., Guo-Dong, Q., Bin, D.: H  ∞  variable universe adaptative fuzzy control for chaotic systems. Chaos, Solitons Fractals 24, 1075–1086 (2005)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Kima, H., Sakthivel, R.: Numerical solution of hybrid fuzzy differential equations using improved predictorcorrector method. Commun. Nonlinear Sci. Numer. Simulat. 17, 3788–3794 (2012)CrossRefGoogle Scholar
  14. 14.
    Kaleva, O.: Fuzzy differential equations. Fuzzy Sets and Systems 24, 301–317 (1987)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Kloeden, P.: Remarks on Peano-like theorems for fuzzy differential equations. Fuzzy Set Syst. 44, 161–164 (1991)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Ma, M., Friedman, M., Kandel, A.: Numerical solution of fuzzy differential equations. Fuzzy Sets Syst. 105, 133–138 (1999)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Palligkinis, S.C., Papageorgiou, G., Famelis, I.T.: Runge-Kutta methods for fuzzy differential equations. Appl. Math. Comput. 209, 97–105 (2009)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Pederson, S., Sambandham, M.: Numerical solution of hybrid fuzzy differential equation IVPs by a characterization theorem. Information Sciences 179, 319–328 (2009)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Pederson, S., Sambandham, M.: The Runge-Kutta method for hybrid fuzzy differential equations. Nonlinear Anal. Hybrid Syst. 2, 626–634 (2008)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Pederson, S., Sambandham, M.: Numerical solution to hybrid fuzzy systems. Mathematical and Computer Modelling 45, 1133–1144 (2007)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Prakash, P., Kalaiselvi, V.: Numerical solution of hybrid fuzzy differential equations by predictor-corrector method. Int. J. Comput. Math. 86, 121–134 (2009)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Puri, M.L., Ralescu, D.: Differential for fuzzy function. J. Math. Anal. Appl. 91, 552–558 (1983)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Rosenbrock, H.H.: Some general implicit processes for the numerical solution of differential equations. Comp. J. 5, 329–330 (1963)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Salahshour, S., Allahviranloo, T., Abbasbandy, S.: Solving fuzzy fractional differential equations by fuzzy Laplace transforms. Commun. Nonlinear Sci. Numer. Simulat. 17, 1372–1381 (2012)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Seikkala, S.: On the fuzzy initial value problem. Fuzzy Sets and Systems 24, 319–330 (1987)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Zhang, H., Liao, X., Yu, J.: Fuzzy modeling and synchronization of hyperchaotic systems. Chaos, Solitons & Fractals 26, 835–843 (2005)MATHCrossRefGoogle Scholar
  27. 27.
    Wu, C., Song, S., Stanley Lee, E.: Approximate solution, existence and uniqueness of the Cauchy problem of fuzzy differential equations. J. Math. Anal. Appl. 202, 629–644 (1996)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Xu, J., Liao, Z., Hu, Z.: A class of linear differential dynamical systems with fuzzy initial condition. Fuzzy Sets Syst. 158, 2339–2358 (2007)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Ali Ahmadian
    • 1
  • Mohamed Suleiman
    • 1
  • Fudziah Ismail
    • 1
  • Soheil Salahshour
    • 2
  • Ferial Ghaemi
    • 3
  1. 1.Institute for Mathematical ResearchUniversiti Putra Malaysia, UPMSerdangMalaysia
  2. 2.Department of Mathematics, Mobarakeh BranchIslamic Azad UniversityMobarakehIran
  3. 3.Institute of Advanced TechnologyUniversiti Putra Malaysia, UPMSerdangMalaysia

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