Fuzzy Fractional Derivative: A New Definition

Abstract

In this paper, a new definition of fuzzy fractional derivative is presented. The definition does not have the drawbacks of the previous definitions of fuzzy fractional derivatives. This definition does not necessitate that the diameter of the fuzzy function be monotonic, and it does not refer to derivative of order greater than the one that is considered. Moreover, the fractional derivative of a fuzzy constant is zero based on the definition. Restrictions associated to Caputo-type fuzzy fractional derivatives are expressed. Additionally, generalized Hukuhara difference and generalized difference of perfect type-2 fuzzy numbers are defined. Furthermore, using two examples the advantages of the new definition compared with the others are borne out.

Appendix

Definition 1.

The generalized difference of two perfect type-2 fuzzy numbers $\tilde{u}$ and $\tilde{v}$ ($g_{2} - difference$) is $\tilde{w} \in E_{2}$ which is given by its $\alpha - level$ and $\tilde{\alpha } - plane$ [9] sets as follows

$$ \left[ {\tilde{w}} \right]_{{\tilde{\alpha }}}^{\alpha } = \left[ {\tilde{u}\frac{{g_{2} }}{{}}\tilde{v}} \right]_{{\tilde{\alpha }}}^{\alpha } = cl\left( {\bigcup\limits_{\gamma \ge \alpha } {\left( {\left[ {\tilde{u}} \right]_{{\tilde{\alpha }}}^{\gamma } \frac{{gH_{2} }}{{}}\left[ {\tilde{v}} \right]_{{\tilde{\alpha }}}^{\gamma } } \right)} } \right),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\alpha ,\tilde{\alpha } \in \left[ {0,1} \right] $$

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Definition 2.

Let $\tilde{f}:\left[ {a,b} \right] \subseteq {\mathbb{R}} \to E_{2}$ and its derivative be continuous and Lebesque integrable fuzzy-valued function, $\tilde{G}(x) = \frac{1}{\varGamma (1 - \beta )}\int\limits_{a}^{x} {\frac{{\tilde{f}(t)\frac{{gH_{2} }}{{}}\tilde{f}\left( a \right)}}{{(x - t)^{\beta } }}} \,dt$. The function $\tilde{f}\left( x \right)$ is the Caputo-type fuzzy fractional differentiable of order $0 < \beta < 1$ at $x_{0} \in \left( {a,b} \right]$ if there exists an element ${}^{c}D^{\beta } \tilde{f}\left( {x_{0} } \right) \in E_{2}$ such that for $h$ sufficiently near zero the following limit exists

$$ \mathop {\lim }\limits_{h \to 0} \frac{{\tilde{G}\left( {x_{0} + h} \right)\frac{{gH_{2} }}{{}}\tilde{G}\left( {x_{0} } \right)}}{h} = {}^{c}D^{\beta } \tilde{f}\left( {x_{0} } \right) $$

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