Fractional integral and its physical interpretation

A relationship is established between Cantor's fractal set (Cantor's bars) and a fractional integral. The fractal dimension of the Cantor set is equal to the fractional exponent of the integral. It follows from analysis of the results that equations in fractional derivatives describe the evolution of physical systems with loss, the fractional exponent of the derivative being a measure of the fraction of the states of the system that are preserved during evolution time $t$. Such systems can be classified as systems with “residual” memory, and they occupy an intermediate position between systems with complete memory, on the one hand, and Markov systems, on the other. The use of such equations to describe transport and relaxation processes is discussed. Some generalizations that extend the domain of applicability of the fractional derivative concept are obtained.