zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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 solutions of fuzzy differential equations by extended Runge-Kutta-like formulae of order 4. (English) Zbl 1238.65068
Summary: We apply a numerical algorithm for solving the fuzzy first order initial value problem, based on extended Runge – Kutta-like formulae of order 4. We use Seikkala’s derivative. The elementary properties of this new solution are given. We use the extended Runge – Kutta-like formulae in order to enhance the order of accuracy of the solutions using evaluations of both f and f ' , instead of the evaluations of f only.
65L06Multistep, Runge-Kutta, and extrapolation methods
34A07Fuzzy differential equations
65L05Initial value problems for ODE (numerical methods)
[1]Abbasbandy, S.; Allahviranloo, T.: 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 syst. 110, 43-54 (2000) · Zbl 0947.34049 · doi:10.1016/S0165-0114(98)00141-9
[3]Chakrivat, P. C.; Kamew, M. S.: Stiffly stable second multi-step methods with higher order and improved stability regions, Bit 23, 75-83 (1983) · Zbl 0507.65034 · doi:10.1007/BF01937327
[4]Chang, S. L.; Zadeh, L. A.: On fuzzy mapping and control, IEEE trans. Syst. man cybernet. 2, 30-34 (1972) · Zbl 0305.94001
[5]Cong-Xin, W.; Ming, M.: On embedding problem of fuzzy number spaces: part: I, Fuzzy sets syst. 44, 33-38 (1991) · Zbl 0757.46066 · doi:10.1016/0165-0114(91)90030-T
[6]Dubois, D.; Prade, H.: Toward fuzzy differential calculus: part 3, differentiation, Fuzzy sets syst. 8, 225-233 (1982) · Zbl 0499.28009 · doi:10.1016/S0165-0114(82)80001-8
[7]Enright, W. H.: Second derivative multi-step methods for stiff ordinary differential equations, SIAM J. Numer. anal. 11, 321-331 (1974) · Zbl 0249.65055 · doi:10.1137/0711029
[8]Gear, C. W.: Numerical initial value problems in ordinary differential equations, (1971) · Zbl 1145.65316
[9]Goeken, D.; Johnson, O.: Runge – Kutta with higher derivative approximations, Appl. numer. Math. 39, 249-257 (2000)
[10]Goetschel, R.; Voxman, W.: Elementary calculus, Fuzzy sets syst. 18, 31-43 (1986) · Zbl 0626.26014 · doi:10.1016/0165-0114(86)90026-6
[11]Hairer, E.; Wanner, G.: Solving ordinary differential equations II, (1991)
[12]Kaleva, O.: Fuzzy differential equations, Fuzzy sets syst. 24, 301-317 (1987) · Zbl 0646.34019 · doi:10.1016/0165-0114(87)90029-7
[13]Kaleva, O.: The Cauchy problem for fuzzy differential equations, Fuzzy sets syst. 35, 389-396 (1990) · Zbl 0696.34005 · doi:10.1016/0165-0114(90)90010-4
[14]Ma, M.; Friedman, M.; Kandel, A.: Numerical solution of fuzzy differential equations, Fuzzy sets syst. 105, 133-138 (1999) · Zbl 0939.65086 · doi:10.1016/S0165-0114(97)00233-9
[15]Palligkinis, S. Ch.; Papageorgiou, G.; Famelis, I. Th.: Runge – Kutta methods for fuzzy differential equations, Appl. math. Comput. 209, 97-105 (2009) · Zbl 1161.65058 · doi:10.1016/j.amc.2008.06.017
[16]Puri, M. L.; Ralescu, D.: Differential for fuzzy function, J. math. Anal. appl. 91, 552-558 (1983) · Zbl 0528.54009 · doi:10.1016/0022-247X(83)90169-5
[17]Puri, M. L.; Ralescu, D.: Fuzzy random variables, J. math. Anal. appl. 114, 409-422 (1986) · Zbl 0592.60004 · doi:10.1016/0022-247X(86)90093-4
[18]Rosenbrock, H. H.: Some general implicit processes for the numerical solution of ordinary differential equations, Comput. J. 23, 329-330 (1963) · Zbl 0112.07805 · doi:10.1093/comjnl/5.4.329
[19]Seikkala, S.: On the fuzzy initial value problem, Fuzzy sets syst. 24, 319-330 (1987) · Zbl 0643.34005 · doi:10.1016/0165-0114(87)90030-3
[20]Wu, X.; Xia, J.: Extended Runge – Kutta-like formulae, Appl. numer. Math. 56, 1584-1605 (2006)
[21]Pederson, S.; Sambandham, M.: Numerical solution to hybrid fuzzy systems, Math. comput. Model. 45, 1133-1144 (2007) · Zbl 1123.65069 · doi:10.1016/j.mcm.2006.09.014
[22]Pederson, S.; Sambandham, M.: The Runge – Kutta method for hybrid fuzzy differential equations, Nonlinear anal. Hybrid syst. 2, 626-634 (2008) · Zbl 1155.93370 · doi:10.1016/j.nahs.2006.10.013
[23]Effati, S.; Pakdaman, M.: Artificial neural network approach for solving fuzzy differential equations, Inf. sci. 180, 1434-1457 (2010) · Zbl 1185.65114 · doi:10.1016/j.ins.2009.12.016