Fractal derivative

In applied mathematics and mathematical analysis, the fractal derivative is a non-Newtonian type of derivative in which the variable such as t has been scaled according to tα. The derivative is defined in fractal geometry. In applied mathematics and mathematical analysis, the fractal derivative is a non-Newtonian type of derivative in which the variable such as t has been scaled according to tα. The derivative is defined in fractal geometry. Porous media, aquifer, turbulence and other media usually exhibit fractal properties. The classical physical laws such as Fick's laws of diffusion, Darcy's law and Fourier's law are no longer applicable for such media, because they are based on Euclidean geometry, which doesn't apply to media of non-integer fractal dimensions. The basic physical concepts such as distance and velocity in fractal media are required to be redefined; the scales for space and time should be transformed according to (xβ, tα). The elementary physical concepts such as velocity in a fractal spacetime (xβ, tα) can be redefined by: where Sα,β represents the fractal spacetime with scaling indices α and β. The traditional definition of velocity makes no sense in the non-differentiable fractal spacetime. Based on above discussion, the concept of the fractal derivative of a function u(t) with respect to a fractal measure t has been introduced as follows: