Representations of acting processes and memory effects: General fractional derivative and its application to theory of heat conduction with finite wave speeds

Dynamical analysis of a fractional-order eco-epidemiological model with disease in prey population

“…The fractional-order differential equations exhibit richer dynamical behavior and this is because it incorporates the memory effect in the model [28]. Zhao and Luo [29] proposed a definition of general fractional derivatives to describe the dynamics with memory effects. Bolton et al [30] conclude that the fractional-order Gompertz growth model is more realistic as regards a experimental dataset than the integer-order Gompertz model.…”

Response: Commentary: A Remark on the Fractional Integral Operators and the Image Formulas of Generalized Lommel-Wright Function

“…We recall that the classification of fractional calculus operators has been a very hot topic during the last few years. In our opinion, the author of the commentary is considering the fractional calculus in the classical way, which has comparatively lesser number of applications as compared to the other new fractional calculus operators (see for example some of new related works on this topic, e.g., [10][11][12] and the references therein).…”

On Fractional Operators and Their Classifications

“…Secondly, there exists a desire to impose criteria and strict definitions for what we call a "fractional derivative" or "fractional integral": which operators between functions should be named as such and which should not. The proposals range from strict requirements to mere suggestions, and multiple different criteria have been proposed [13,[28][29][30][31]. The motivation here is to create a mathematical framework for fractional calculus, to know the boundaries of the field.…”

“…Our aim here is related but different: instead of embedding the whole of fractional calculus into other fields of analysis, we seek to create classifications within fractional calculus itself. Some recent studies [30,[33][34][35] have proposed general classes of operators that are broad enough to cover many existing models of fractional calculus but still narrow enough to be rigorously analysed themselves. This approach is optimal for several reasons:…”