##
**Numerical solution of hybrid fuzzy differential equation IVPs by a characterization theorem.**
*(English)*
Zbl 1165.65041

Hybrid fuzzy differential initial value problems are systems that evolve in continuous time like differential systems but undergo fundamental changes in their governing fuzzy equations at a sequence of discrete times. In a previous paper [Nonlinear Anal., Hybrid Syst. 2, No. 2, 626–634 (2008; Zbl 1155.93370)] the authors of this paper have studied some Runge-Kutta type methods for the numerical solution of these equations.

Now they follow an alternative approach by using a recent characterization theorem due to B. Bede [Inf. Sci. 178, No. 7, 1917–1922 (2008; Zbl 1183.65092)] for fuzzy differential equations that, under suitable conditions, permits to solve initial value problems for these equations converting them in systems of ordinary differential equations. The main result of the present paper is an extension of Bede’s characterization theorem for hybrid fuzzy IVPs so that any stable one step numerical method for ODEs can be applied piecewise to solve numerically IVPs for hybrid fuzzy differential equations.

Now they follow an alternative approach by using a recent characterization theorem due to B. Bede [Inf. Sci. 178, No. 7, 1917–1922 (2008; Zbl 1183.65092)] for fuzzy differential equations that, under suitable conditions, permits to solve initial value problems for these equations converting them in systems of ordinary differential equations. The main result of the present paper is an extension of Bede’s characterization theorem for hybrid fuzzy IVPs so that any stable one step numerical method for ODEs can be applied piecewise to solve numerically IVPs for hybrid fuzzy differential equations.

Reviewer: Manuel Calvo (Zaragoza)

### MSC:

65L05 | Numerical methods for initial value problems involving ordinary differential equations |

34A34 | Nonlinear ordinary differential equations and systems |

26E50 | Fuzzy real analysis |

65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |

### Keywords:

fuzzy differential equations; initial valus problems; Runge-Kutta methods; hybrid systems; Euler method
PDF
BibTeX
XML
Cite

\textit{S. Pederson} and \textit{M. Sambandham}, Inf. Sci. 179, No. 3, 319--328 (2009; Zbl 1165.65041)

Full Text:
DOI

### References:

[1] | Abbasbandy, S.; Allah Viranloo, T., Numerical solution of fuzzy differential equations by Taylor method, Journal of computational methods in applied mathematics, 2, 113-124, (2002) · Zbl 1019.34061 |

[2] | Abbasbandy, S.; Allah Viranloo, T., Numerical solution of fuzzy differential equation, Mathematical and computational applications, 7, 41-52, (2002) · Zbl 1013.65070 |

[3] | Abbasbandy, S.; Allah Viranloo, T., Numerical solution of fuzzy differential equation by runge – kutta method, Nonlinear studies, 11, 117-129, (2004) · Zbl 1056.65069 |

[4] | Bede, B., Note on “numerical solutions of fuzzy differential equations by predictor – corrector method, Information sciences, 178, 1917-1922, (2008) · Zbl 1183.65092 |

[5] | Buckley, J.J.; Feuring, T., Fuzzy differential equations, Fuzzy sets and systems, 110, 43-54, (2000) · Zbl 0947.34049 |

[6] | G. Debreu, Integration of correspondences, in: Proceedings of the Fifth Berkeley Symposium on Mathematics and Statistical Probability, vol. 2, Part 1, 1967, pp. 351-372. · Zbl 0211.52803 |

[7] | Dubois, D.; Prade, H., Towards fuzzy differential calculus – part 3, Fuzzy sets and systems, 8, 225-234, (1982) |

[8] | Kaleva, O., Fuzzy differential equations, Fuzzy sets and systems, 24, 301-317, (1987) · Zbl 0646.34019 |

[9] | Lakshmikantham, V.; Liu, X.Z., Impulsive hybrid systems and stability theory, International journal of nonlinear differential equations, 5, 9-17, (1999) · Zbl 0901.34018 |

[10] | Lakshmikantham, V.; Mohapatra, R.N., Theory of fuzzy differential equations and inclusions, (2003), Taylor and Francis United Kingdom · Zbl 1072.34001 |

[11] | Ma, M.; Friedman, M.; Kandel, A., Numerical solutions of fuzzy differential equations, Fuzzy sets and systems, 105, 133-138, (1999) · Zbl 0939.65086 |

[12] | Pederson, S.; Sambandham, M., Numerical solution to hybrid fuzzy systems, Mathematical and computer modelling, 45, 1133-1144, (2007) · Zbl 1123.65069 |

[13] | Pederson, S.; Sambandham, M., The runge – kutta method for hybrid fuzzy differential equations, Nonlinear analysis: hybrid systems, 2, 626-634, (2008) · Zbl 1155.93370 |

[14] | Pederson, S.; Sambandham, M., Numerical solution to hybrid stochastic differential systems, Stochastic analysis and applications, 26, 338-356, (2008) · Zbl 1143.65007 |

[15] | Puri, M.L.; Ralescu, D.A., Differentials of fuzzy functions, Journal of mathematical analysis and applications, 91, 552-558, (1983) · Zbl 0528.54009 |

[16] | Puri, M.L.; Ralescu, D.A., Fuzzy random variables, Journal of mathematical analysis and applications, 114, 409-422, (1986) · Zbl 0592.60004 |

[17] | Sambandham, M., Perturbed Lyapunov-like functions and hybrid fuzzy differential equations, International journal of hybrid systems, 2, 23-34, (2002) |

[18] | Seikkala, S., On the fuzzy initial value problem, Fuzzy sets and systems, 24, 319-330, (1987) · Zbl 0643.34005 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.