Ma, Ming; Friedman, Menahem; Kandel, Abraham Numerical solutions of fuzzy differential equations. (English) Zbl 0939.65086 Fuzzy Sets Syst. 105, No. 1, 133-138 (1999). The authors study the explicit Euler approximation of a Cauchy problem for fuzzy (ordinary) differential equations. By the embedding approach, the original problem in a Banach space with supremum norm is replaced by two parametric, first-order ordinary differential equations. These differential equations can be solved by classical numerical algorithms. The authors present a pointwise convergence theorem for the standard Euler method in the given Banach space under uniform boundedness of second derivatives, Lipschitz-continuous right hand sides and for equidistant stepsize selection. 5 examples (both linear and nonlinear) are exhibited and discussed in view of the implementation of the explicit Euler approximation. A standard, two-dimensional error estimation is given in terms of the global Lipschitz constant, the length of integration interval \(T-t_0\), and the equidistant stepsize \(h\). Reviewer: Henri Schurz (Kaiserslautern) Cited in 1 ReviewCited in 83 Documents MSC: 65J15 Numerical solutions to equations with nonlinear operators 65L05 Numerical methods for initial value problems involving ordinary differential equations 65L20 Stability and convergence of numerical methods for ordinary differential equations 65L70 Error bounds for numerical methods for ordinary differential equations 34G20 Nonlinear differential equations in abstract spaces 65J10 Numerical solutions to equations with linear operators 34G10 Linear differential equations in abstract spaces Keywords:fuzzy differential equations; fuzzy numbers; Euler method; fuzzy Cauchy problem; fuzzy dynamical systems; Banach space; convergence; stepsize selection; linear; nonlinear; error estimation PDFBibTeX XMLCite \textit{M. Ma} et al., Fuzzy Sets Syst. 105, No. 1, 133--138 (1999; Zbl 0939.65086) Full Text: DOI References: [1] Chang, S. L.; Zadeh, L. A., On fuzzy mapping and control, IEEE Trans. Systems Man Cybernet, 2, 30-34 (1972) · Zbl 0305.94001 [2] Diamond, P.; Kloeden, P., Metric spaces of fuzzy sets, Fuzzy Sets and Systems, 35, 241-249 (1990) · Zbl 0704.54006 [3] Dubois, D.; Prade, H., Towards fuzzy differential calculus: Part 3, differentiation, Fuzzy Sets and Systems, 8, 225-233 (1982) · Zbl 0499.28009 [4] Friedman, M.; Kandel, A., Fundamentals of Computer Numerical Analysis, ((1994), CRC Press: CRC Press Boca Raton, FL), 441-444 [5] Goetschel, R.; Voxman, W., Elementary fuzzy calculus, Fuzzy Sets and Systems, 18, 31-43 (1986) · Zbl 0626.26014 [6] Kaleva, O., Fuzzy differential equations, Fuzzy Sets and Systems, 24, 301-317 (1987) · Zbl 0646.34019 [7] Kaleva, O., The Cauchy problem for fuzzy differential equations, Fuzzy Sets and Systems, 35, 389-396 (1990) · Zbl 0696.34005 [8] Kandel, A., Fuzzy dynamical systems and the nature of their solutions, (Wang, P. P.; Chang, S. K., Fuzzy Sets: Theory and Application to Policy Analysis and Information Systems (1980), Plenum Press: Plenum Press New York), 93-122 [9] Kandel, A.; Byatt, W. J., Fuzzy differential equations, (Proc. Internat. Conf. Cybernetics and Society. Proc. Internat. Conf. Cybernetics and Society, Tokyo (November 1978)), 1213-1216 [10] Kandel, A.; Byatt, W. J., Fuzzy processes, Fuzzy Sets and Systems, 4, 117-152 (1980) · Zbl 0437.60002 [11] Klement, E. P.; Puri, M. L.; Ralescu, D. A., Limit theorems for fuzzy random variables, (Proc. Roy. Soc. London Ser. A, 407 (1986)), 171-182 · Zbl 0605.60038 [12] Kloeden, P., Remarks on Peano-like theorems for fuzzy differential equations, Fuzzy Sets and Systems, 44, 161-164 (1991) · Zbl 0742.34058 [13] He, Ouyang; Yi, Wu, On fuzzy differential equations, Fuzzy Sets and Systems, 32, 321-325 (1989) · Zbl 0719.54007 [14] Puri, M. L.; Ralescu, D. A., Differentials of fuzzy functions, J. Math. Anal. Appl, 91, 552-558 (1983) · Zbl 0528.54009 [15] Seikkala, S., On the fuzzy initial value problem, Fuzzy Sets and Systems, 24, 319-330 (1987) · Zbl 0643.34005 [16] Congxin, Wu; Ming, Ma, On embedding problem of fuzzy number spaces, Fuzzy Sets and Systems, 44, 33-38 (1991), Part 1 · Zbl 0757.46066 [17] Menda, Wu, Linear fuzzy differential equation systems on \(R^1\), J. Fuzzy Systems Math, 2, 1, 51-56 (1988), (in Chinese) 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.