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, \(r\in [0,1]\), are given by \(x'(t) \in [f(t, x(t))]_r, x(0)\in [x_0]_r \) where the \(r\)-level set of \(u\) is \([u]_0= \text{supp}(u)\) and \( [u]_r = \{x\in\mathbb R^n; u(x)\geq r \}\) for \(r\in (0,1]\) and \( [f(\cdot, \cdot)]_r :[0,T] \times\mathbb R^n \to \kappa_c^n\) with \(\kappa_c^n\) the space of nonempty convex compact subsets of \(\mathbb R^n\).
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 \( r \in [0,1]\), comparing with those obtained with explicit Euler’s method.


65L05 Numerical methods for initial value problems involving ordinary differential equations
26E50 Fuzzy real analysis
34A60 Ordinary differential inclusions
34A07 Fuzzy ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
Full Text: DOI


[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] Dubois, D.; Prade, H., Towards fuzzy differential calculus: part 3, differentiation, Fuzzy sets and systems, 8, 225-233, (1982) · Zbl 0499.28009
[3] Puri, M.L.; Ralescu, D.A., Differentials of fuzzy functions, J. math. anal. appl., 91, 321-325, (1983)
[4] Goetschel, R.; Voxman, W., Elementary fuzzy calculus, Fuzzy sets and systems, 18, 31-43, (1986) · Zbl 0626.26014
[5] Kaleva, O., Fuzzy differential equations, Fuzzy sets and systems, 24, 301-317, (1987) · Zbl 0646.34019
[6] Kaleva, O., The Cauchy problem for fuzzy differential equations, Fuzzy sets and systems, 35, 389-396, (1990) · Zbl 0696.34005
[7] S. Seikkala, On the fuzzy initial value problem, Fuzzy Sets and Systems, {\bf24}, 319-330 (24) · Zbl 0643.34005
[8] Nieto, J.J., The Cauchy problem for continuous fuzzy equations, Fuzzy sets and systems, 102, 259-262, (1999) · Zbl 0929.34005
[9] Diamond, P., Brief note on the variation of constants formula for fuzzy differential equations, Fuzzy sets and systems, 129, 65-71, (2002) · Zbl 1021.34048
[10] Abbasbandy, S.; Viranloo, T. Allah, Numerical solution of fuzzy differential equations by Taylor method, Comput. methods appl. math., 2, 113-124, (2002) · Zbl 1019.34061
[11] O’Regan, D.; Lakshmikantham, V.; Nieto, J.J., Initial and boundary value problems for fuzzy differential equations, Nonlinear analysis, 54, 405-415, (2003) · Zbl 1048.34015
[12] Hiillermeier, E., An approach to modelling and simulation of uncertain dynamical systems, Uncertainty, fuzziness & knowledge-based systems, 5, 117-137, (1997)
[13] Diamond, P., Stability and periodicity in fuzzy differential equations, IEEE trans. fuzzy systems, 8, 583-590, (2000)
[14] Diamond, P.; Kloeden, P., Metric spaces of fuzzy sets, (1994), World Scientific Upper Saddle River, NJ · Zbl 0843.54041
[15] Fox, L.; Mayers, D.F., Numerical solution of ordinary differential equations, (1987), Chapman & Hall Singapore · Zbl 0643.34001
[16] Lambert, J.D., Numerical methods for ordinary differential systems: the initial value problem, (1991), John Wiley & Sons, Ltd. London · Zbl 0745.65049
[17] de Blasi, F.S.; Myjak, J., On the solution sets for differential inclusions, Bull. acad. polon. sci., 33, 17-23, (1985) · Zbl 0571.34008
[18] Diamond, P., Time dependent differential inclusions, cocycle attractors and fuzzy differential equations, IEEE trans. fuzzy systems, 7, 734-740, (1999)
[19] Vorobiev, D.; Seikkala, S., Towards the theory of fuzzy differential equations, Fuzzy sets and systems, 125, 231-237, (2002) · Zbl 1003.34046
[20] Ma, M.; Friedman, M.; Kandel, A., Numerical solutions of fuzzy differential equations, Fuzzy sets and systems, 105, 133-138, (1999) · Zbl 0939.65086
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.