Diffusion on Middle-ξ Cantor Sets

Abstract

In this paper, we study C ζ -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 C ζcalculus on the generalized Cantor sets known as middle-ξ Cantor sets. We have suggested a calculus on the middle-ξ Cantor sets for different values of ξ with 0 < ξ < 1. Differential equations on the middle-ξ 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.

On stability of a class of second alpha-order fractal differential equations

“…In Figure 1d we have ploted the characteristic function for the middle-µ choosing µ = 1/5. The C α -limit of a function h : → as z → t is defined in [25,26,28] by…”

“…On the other hand, we modify and adopt the ordinary calculus conditions in fractal calculus [38]. The main results are obtained using the generalized Lyapunov function with the fractal sets support [25,26,28,37,38]. Let us consider the following second α-order fractal differential equation…”

“…In view of this new measure, the C µ -Calculus was formulated for totally disconnected fractal sets and non-differentiable fractal curves [25,26,27,28] During the last decade, several researchers have explored in this area and applied it in different branches of science and engineering [29,30]. Fractal differential equations (FDEs) were solved and analogous existence and uniqueness theorems were suggested and proved [31,32].…”

Refinement multidimensional dynamic inequalities with general kernels and measures

“…For the general fractional derivative, we refer the reader to [9,32]. We also refer the reader to [11,12,23], which give a unification of calculus of the functions on totally disconnected and continuous real line.…”

Analogues to Lie Method and Noether’s Theorem in Fractal Calculus

“…Motivated by these ideas, mathematicians developed the theory of "analysis on fractals", with branches including the fractional calculus approach, the probabilistic approach, the measure theory approach, and F α -calculus [6,[15][16][17][18][19][20][21][22][23][24].…”