Abstract
We firstly present a generalized concept of higher-order differentiability for fuzzy functions. Then we interpret th-order fuzzy differential equations using this concept. We introduce new definitions of solution to fuzzy differential equations. Some examples are provided for which both the new solutions and the former ones to the fuzzy initial value problems are presented and compared. We present an example of a linear second-order fuzzy differential equation with initial conditions having four different solutions.
Highlights
The term “fuzzy differential equation” was coined in 1987 by Kandel and Byatt 1 and an extended version of this short note was published two years later 2
It soon appeared that the solution of fuzzy differential equation interpreted by Hukuhara derivative has a drawback: it became fuzzier as time goes by 6
The main shortcoming of using differential inclusions is that we do not have a derivative of a fuzzynumber-valued function
Summary
The term “fuzzy differential equation” was coined in 1987 by Kandel and Byatt 1 and an extended version of this short note was published two years later 2. The strongly generalized differentiability was introduced in 8 and studied in 9– 11 This concept allows us to solve the above-mentioned shortcoming. Boundary Value Problems generalized derivative is defined for a larger class of fuzzy-number-valued functions than the Hukuhara derivative. We use this differentiability concept in the present paper. We propose a new method to solve higher-order fuzzy differential equations based on the selection of derivative type covering all former solutions. With these ideas, the selection of derivative type in each step of derivation plays a crucial role
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