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Keywords:
fuzzy differential equations; interval differential equations; initial value problem; boundary value problem; bunch of functions; linear differential equations
fuzzy differential equations; interval differential equations; initial value problem; boundary value problem; bunch of functions; linear differential equations
Summary:
Dynamics containing deterministic uncertainties can be modeled with fuzzy differential equations. Unlike classical differential equations, fuzzy differential equations lack a unified interpretation and theoretical foundation, as researchers adopt different approaches to fuzziness, solution concepts, and underlying mathematical structures. The main reason is whether the fuzzy function derivative is used in the equation in question and, if it is used, what meaning it carries. Researchers who do not involve a derivative of a fuzzy number-valued function either use the extension principle, an alternative concept of fuzzy function, or transform the problem into a differential inclusion. Various definitions have been used in studies involving the derivatives of fuzzy number-valued functions. The main reason is that none of the known derivatives can fully meet the requirements: either the fuzziness increases excessively, or it becomes impossible to solve higher-order equations, or unnatural assumptions must be made. In this study, we tried to classify almost all studies on fuzzy differential equations. We compared the results of studies conducted in relatively recent years, particularly in initial value and boundary value problems, using examples. We discussed the possible direction of future research on fuzzy differential equations.
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