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**Numerical methods for fuzzy differential inclusions.**
*(English)*
Zbl 1074.65072

A numerical method for solving initial value problems for fuzzy differential inclusions is given. The fuzzy differential inclusions at each \(r\)-level, \(r\in [0,1]\), are given by \(x'(t) \in [f(t, x(t))]_r, x(0)\in [x_0]_r \) where the \(r\)-level set of \(u\) is \([u]_0= \text{supp}(u)\) and \( [u]_r = \{x\in\mathbb R^n; u(x)\geq r \}\) for \(r\in (0,1]\) and \( [f(\cdot, \cdot)]_r :[0,T] \times\mathbb R^n \to \kappa_c^n\) with \(\kappa_c^n\) the space of nonempty convex compact subsets of \(\mathbb R^n\).

The paper is organized as follows: After introducing some basic results on fuzzy derivative and initial value problems, the authors propose a version of the two-stage Runge-Kutta method of Heun with order two for the above class of problems and prove the convergence of the approximate solution. Finally, the results of some numerical experiments are presented showing the \(r\)-level sets for a discrete set of values \( r \in [0,1]\), comparing with those obtained with explicit Euler’s method.

The paper is organized as follows: After introducing some basic results on fuzzy derivative and initial value problems, the authors propose a version of the two-stage Runge-Kutta method of Heun with order two for the above class of problems and prove the convergence of the approximate solution. Finally, the results of some numerical experiments are presented showing the \(r\)-level sets for a discrete set of values \( r \in [0,1]\), comparing with those obtained with explicit Euler’s method.

Reviewer: Manuel Calvo (Zaragoza)

### MSC:

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

26E50 | Fuzzy real analysis |

34A60 | Ordinary differential inclusions |

34A07 | Fuzzy ordinary differential equations |

65L20 | Stability and convergence of numerical methods for ordinary differential equations |

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

### Keywords:

differential inclusions; fuzzy initial value problems; comparison of methods; Heun method; Runge-Kutta method; convergence; numerical experiments; Euler’s method
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\textit{S. Abbasbandy} et al., Comput. Math. Appl. 48, No. 10--11, 1633--1641 (2004; Zbl 1074.65072)

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