Fractional space approach to studies of physical phenomena on fractals and in confined low-dimensional systems

This paper is devoted to the development of the fractional space approach to physical problems on fractals and in confined quasi-low-dimensional systems. The aim of this development is to expand this approach by accounting for essential fractal features of the system under the study. For this purpose, the fractal properties of scale invariant and confined systems are scrutinized. This allows us to establish a set of requirements imposed by the mapping of physical problems on fractals onto boundary valued problems in the model fractional space. Accordingly, the Stillinger's definition of space with a non-integer dimension is endowed with suitable fractal attributes defined by two additional axioms. We also point out that the model fractional spaces admit different definitions of vector differential calculus. Two suitable sets of the vector differential operators in the fractional space are suggested. Furthermore, we construct several models of the fractional space enabled for studies of transport phenomena in the confined quasi-low-dimensional systems. The fractal architectures of these model spaces are highlighted.