×

zbMATH — the first resource for mathematics

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 \(f^{\prime}\), instead of the evaluations of \(f\) only.

MSC:
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
34A07 Fuzzy ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abbasbandy, S.; Allahviranloo, T., Numerical solution of fuzzy differential equation by runge – kutta method, Nonlinear studies, 11, 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
[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
[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)
[6] Dubois, D.; Prade, H., Toward fuzzy differential calculus: part 3, differentiation, Fuzzy sets syst., 8, 225-233, (1982) · Zbl 0499.28009
[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
[8] Gear, C.W., Numerical initial value problems in ordinary differential equations, (1971), Prentice-Hall Englewood Cliffs, NJ · Zbl 0217.21701
[9] Goeken, D.; Johnson, O., Runge – kutta with higher derivative approximations, Appl. numer. math., 39, 249-257, (2000) · Zbl 0951.65068
[10] Goetschel, R.; Voxman, W., Elementary calculus, Fuzzy sets syst., 18, 31-43, (1986) · Zbl 0626.26014
[11] Hairer, E.; Wanner, G., Solving ordinary differential equations II, (1991), Springer Berlin · Zbl 0729.65051
[12] Kaleva, O., Fuzzy differential equations, Fuzzy sets syst., 24, 301-317, (1987) · Zbl 0646.34019
[13] Kaleva, O., The Cauchy problem for fuzzy differential equations, Fuzzy sets syst., 35, 389-396, (1990) · Zbl 0696.34005
[14] Ma, M.; Friedman, M.; Kandel, A., Numerical solution of fuzzy differential equations, Fuzzy sets syst., 105, 133-138, (1999) · Zbl 0939.65086
[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
[16] Puri, M.L.; Ralescu, D., Differential for fuzzy function, J. math. anal. appl., 91, 552-558, (1983) · Zbl 0528.54009
[17] Puri, M.L.; Ralescu, D., Fuzzy random variables, J. math. anal. appl., 114, 409-422, (1986) · Zbl 0592.60004
[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
[19] Seikkala, S., On the fuzzy initial value problem, Fuzzy sets syst., 24, 319-330, (1987) · Zbl 0643.34005
[20] Wu, X.; Xia, J., Extended runge – kutta-like formulae, Appl. numer. math., 56, 1584-1605, (2006) · Zbl 1104.65075
[21] Pederson, S.; Sambandham, M., Numerical solution to hybrid fuzzy systems, Math. comput. model., 45, 1133-1144, (2007) · Zbl 1123.65069
[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
[23] Effati, S.; Pakdaman, M., Artificial neural network approach for solving fuzzy differential equations, Inf. sci., 180, 1434-1457, (2010) · Zbl 1185.65114
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.