Abstract
In this paper, we study -calculus on generalized Cantor sets, which have self-similar properties and fractional dimensions that exceed their topological dimensions. Functions with fractal support are not differentiable or integrable in terms of standard calculus, so we must involve local fractional derivatives. We have generalized the -calculus on the generalized Cantor sets known as middle- Cantor sets. We have suggested a calculus on the middle- Cantor sets for different values of with . Differential equations on the middle- Cantor sets have been solved, and we have presented the results using illustrative examples. The conditions for super-, normal, and sub-diffusion on fractal sets are given.
Highlights
It is well known that many phenomena in nature can be modeled by fractals; these shapes can be observed almost anywhere in the natural world [1]
Non-Markovian random walks and fractal dimensions which are connected to physical properties of fractal sets were studied in [11,12,13]
Using conjugacy of C ζ -C between standard calculus [33,34], we have the solution for Equation (36) as follows: ζ t−1/2
Summary
It is well known that many phenomena in nature can be modeled by fractals; these shapes can be observed almost anywhere in the natural world [1]. In the neural and vascular networks of the human body, viruses and certain tumors grow and ramify in a fractal shape [5,6,7] In these studies, researchers tried to predict and recognize osteoporosis from test results and from the fractal structure of bone texture [8]. Non-Markovian random walks and fractal dimensions which are connected to physical properties of fractal sets were studied in [11,12,13]. The scale-dependent fractal dimension for a random walk trajectory was used to derive the diffusion coefficient [22,23]. Fractional calculus has been applied to define derivatives on fractal curves [25,26,27,28,29,30,31,32].
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
