Relationships between 1D and space fractals and fractional integrals and their applications in physics

Abstract

In this paper, the exact relationships between the averaging procedure of a smooth function over 1D-fractal sets and the fractional integral of the RL-type are found. The numerical verifications are realized for confirmation of the analytical results and the physical meaning of these obtained formulas is discussed. Besides, the generalizations of the results for a combination of fractal circuits having a discrete set of fractal dimensions were obtained. We suppose that these new results help to deeper understand the intimate links between fractals and fractional integrals of different types, especially in applications of the fractional operators in complex systems. These results can be used in different branches of the interdisciplinary physics, where the different equations describing the complex physical phenomena, and the fractional derivatives and integrals with complex-conjugated power-law exponents are used. We consider also possibilities of applications of these results in classical mechanics. Besides these exact results, in Section 3, we consider the difficulties that can arise in attempting to generalize them for 2D and 3D fractals. We suggest one approximate approach (tested numerically) that can solve these arising difficulties.