# 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)
Runge-Kutta methods for fuzzy differential equations. (English) Zbl 1161.65058
Summary: Fuzzy differential equations (FDEs) generalize the concept of crisp initial value problems. In this article, we deal with the numerical solution of FDEs. The notion of convergence of a numerical method is defined and a category of problems which is more general than the one already found in the numerical analysis literature is solved. Efficient $s$-stage Runge-Kutta methods are used for the numerical solution of these problems and the convergence of the methods is proved. Several examples comparing these methods with the previously developed Euler method are displayed.
##### MSC:
 65L06 Multistep, Runge-Kutta, and extrapolation methods 65L05 Initial value problems for ODE (numerical methods) 34A34 Nonlinear ODE and systems, general 26E50 Fuzzy real analysis 65L20 Stability and convergence of numerical methods for ODE
##### References:
 [1] Abbasbandy, S.; Viranloo, T. Allah: Numerical solution of fuzzy differential equation by Runge – Kutta method, Nonlinear studies 11, No. 1, 117-129 (2004) · Zbl 1056.65069 [2] Buckley, J. J.; Feuring, T.: Fuzzy differential equations, Fuzzy sets and systems 110, 43-54 (2000) · Zbl 0947.34049 · doi:10.1016/S0165-0114(98)00141-9 [3] Butcher, J. C.: The numerical analysis of ordinary differential equations, (1987) · Zbl 0616.65072 [4] Dubois, D.; Prade, H.: Towards fuzzy differential calculus part 3: differentiation, Fuzzy sets and systems 8, 225-233 (1982) · Zbl 0499.28009 · doi:10.1016/S0165-0114(82)80001-8 [5] Goetschel, R.; Voxman, W.: Elementary fuzzy calculus, Fuzzy sets and systems 18, 31-43 (1986) · Zbl 0626.26014 · doi:10.1016/0165-0114(86)90026-6 [6] Hairer, E.; Norsett, S. P.; Wanner, G.: Solving ordinary differential equations I, (1987) [7] Kaleva, O.: Fuzzy differential equations, Fuzzy sets and systems 24, 301-317 (1987) · Zbl 0646.34019 · doi:10.1016/0165-0114(87)90029-7 [8] Kolmogorov, A. N.; Fomin, S. V.: Introductory real analysis, (1970) · Zbl 0213.07305 [9] Lambert, J. D.: Numerical methods for ordinary differential systems, (1990) [10] Ma, M.; Friedman, M.; Kandel, A.: Numerical solutions of fuzzy differential equations, Fuzzy sets and systems 105, 133-138 (1999) · Zbl 0939.65086 · doi:10.1016/S0165-0114(97)00233-9 [11] Puri, M. L.; Ralescu, D. A.: Differentials of fuzzy functions, Journal of mathematical analysis and applications 91, 552-558 (1983) · Zbl 0528.54009 · doi:10.1016/0022-247X(83)90169-5 [12] Seikkala, S.: On the fuzzy initial value problem, Fuzzy sets and systems 24, 319-330 (1987) · Zbl 0643.34005 · doi:10.1016/0165-0114(87)90030-3