Fractal and Fractional

Fractal and Fractional are two words referring to some characteristics and fundamental problems which arise in all fields of science and technology. These problems are characterized by having some non-integer order features. Fractals are geometrical objects with non-integer dimension, while fractional is the non-integer order of differential operators. Fractals and fractional calculus have been deeply studied since the beginning of calculus, however non-integer order objects were mainly considered as singular objects or outstanding topics and they were usually approached as geometrical or analytical oddities. Indeed, in the last century, it became more and more evident that when dealing with real problems we have to face the complexity with sophisticated mathematical models. As a consequence, scholars became more familiar with fractal objects and found that the real world is more like a fractal object than they initially thought. Fractals are geometric objects, usually defined by a recursive law, which are characterized by some peculiar properties such as self-similarity and scale. According to self-similarity, a small part of a fractal looks like the whole. The scale property can be explained, in short, by the fact that by increasing the unit length by an integer factor, the measure of the fractal increases by a non-integer factor. In the last Century, due to developing technologies and growing interest, many fractal-like objects were discovered nearly everywhere, at the microand nano-scale of matter as well as at the cosmological scale. The most striking result was the observation of self-similarity and scale dependence in almost all fields of investigation, from biology and medicine to geography and astronomy. Moreover, the human morphology of organs, e.g., of the brain, lungs, vessels, and the activity of heart, neurons, and cells, are fractals. Fractals and fractal-like objects were investigated in almost all fields of science and technology, e.g., in mathematics, computer science, image analysis, communications, medicine, seismology, structural mechanics, nano-composites, nano-fluidics, and fracture analysis. New findings about the fractal nature in some specific topics led to unexpected new results and progress in research. As a consequence, we became more and more aware both of the fractal nature of the world and of the importance of the fractal models to describe the evolution of the world around us. We have gone from the original idea that fractals were some picturesque singularities to the modern point of view that sees them at the core of knowledge. It has been a tortuous path, at each step of which new discoveries appear in different unexpected fields. In recent years, there has also been increasing interest in the analytical properties of geometrical fractal-like objects. However, when dealing with fractals, the fundamental characteristic of being differentiable is missing, therefore it is a challenging problem to define operators on fractal sets. Smoothness and differentiability are the two fundamental pillars of any analytical structure. Moreover, the integer order of derivatives was considered the necessary condition for building differential operators. Nevertheless, since the origin of differential calculus, Cauchy was proposing a definition of fractional order repeated integrals, thus opening new perspectives in differential calculus. Since then, there has been increasing interest in such a kind of operators and differential problems with fractional order operators. The main idea in fractional calculus is to define a suitable differential operator which depends on a fractional order parameter. The fundamental properties of these operators are the following: when the parameter is integer, then each fractional operator must reduce to the ordinary differential operators, moreover, the basic rules of calculus, e.g., linearity,