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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.

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
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