On Fractional and Fractal Derivatives in Relation to the Physics of Fractals

Fractional and fractal derivatives are both generalizations of the usual derivatives that consider derivatives of non-integer orders. Interest in these generalizations has been triggered by a resurgence of clamor to develop a mathematical tool to describe “roughness” in the spirit of Mandelbrot’s (1967) fractal geometry. Fractional derivatives take the analytic approach towards developing a rational order derivative while fractal derivatives follow a more concrete, albeit geometric approach to the same end. Since both approaches alleged to extend whole derivatives to rational derivatives, it is not surprising that confusion will arise over which generalization to use in practice. This paper attempts to highlight the connection between the various generalizations to fractional and fractal derivatives with the end-in-view of making these concepts useful in various physics applications and to resolve some of the confusion that arise out of the fundamental philosophical differences in the derivation of fractional derivatives (non-local concept) and fractal derivatives (local concept).