A NEW GENERAL CONFORMABLE FRACTIONAL DERIVATIVE AND SOME APPLICATIONS

Document Type : Regular research papers

Authors

1 Department of Mathematics university of Blida 1

2 Department of Mathematics, Ecole Normale Sup´erieure de Bou Sa^ada

Abstract

This paper contributes to the ongoing development of fractional calculus by introducing a novel class of local fractional derivatives, referred to as the \emph{$\mathcal{M}$-conformable derivative}. This new formulation offers a general framework that encompasses and extends several existing conformable fractional derivatives. The derivative is defined by the following limit:
\[
\mathfrak{D}_{q\left( \cdot \right) ,\mathcal{M}}^{\rho }\vartheta
(t)=\lim_{\epsilon \rightarrow 0}\frac{\vartheta \left( t+\epsilon q\left(
t\right) ^{1-\rho }\mathcal{M}\left( \vartheta \left( t\right) \right)
\right) -\vartheta (t)}{\epsilon },
\]
where $\mathcal{M}(\cdot)$ and $q(\cdot)$ are suitably chosen functions satisfying regularity conditions, and the parameter $\rho$ lies in the interval $(0,1)$.

We begin by establishing the fundamental properties of the proposed derivative, laying the groundwork for further analysis. Classical results such as Rolle’s theorem, the mean value theorem, and L’Hôpital’s rule are then extended within this new fractional context. Furthermore, we introduce the associated \emph{$\tilde{\mathcal{M}}$-conformable integral}, derived from a modified version of the fundamental theorem of calculus, and we develop an appropriate integration by parts formula.

To highlight the practical relevance and robustness of this operator, we conclude the paper by solving a selection of $\mathcal{M}$-conformable fractional differential equations. These examples demonstrate that the proposed derivative is a powerful tool for modeling and solving a broad class of dynamic systems.

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