# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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, $r\in \left[0,1\right]$, are given by ${x}^{\text{'}}\left(t\right)\in {\left[f\left(t,x\left(t\right)\right)\right]}_{r},\phantom{\rule{0.277778em}{0ex}}x\left(0\right)\in {\left[{x}_{0}\right]}_{r}$ where the $r$-level set of $u$ is ${\left[u\right]}_{0}=$ supp $\left(u\right)$ and ${\left[u\right]}_{r}=\left\{x\in {ℝ}^{n};u\left(x\right)\ge r\right\}$ for $r\in \left(0,1\right]$ and ${\left[f\left(·,·\right)\right]}_{r}:\left[0,T\right]×{ℝ}^{n}\to {\kappa }_{c}^{n}$ with ${\kappa }_{c}^{n}$ the space of nonempty convex compact subsets of ${ℝ}^{n}$.

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 $r\in \left[0,1\right]$, comparing with those obtained with explicit Euler’s method.

##### MSC:
 65L05 Initial value problems for ODE (numerical methods) 26E50 Fuzzy real analysis 34A60 Differential inclusions 34A12 Initial value problems for ODE, existence, uniqueness, etc. of solutions 65L20 Stability and convergence of numerical methods for ODE 65L06 Multistep, Runge-Kutta, and extrapolation methods