Numerical solution of hybrid fuzzy differential equation IVPs by a characterization theorem.

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

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