Modelling in Fractal analysis is a recent and active research field since 15/20 years. In this period, our group has acquainted an expertise on this topic. Depending on the problem, different tools and techniques are necessary to give a rigorous formulation of classical operators in the Euclidean settings as well as delicate functional inequalities which are instrumental to the study of some BVPs. In view of real world applications it is also crucial to yield a numerical approximation of the solution for the problem at hand. To achieve this goal two steps are necessary. The former is to yield a constructive approach which allows to define an abstract operator in terms of smoother ones, the latter is to construct an efficient numerical approximation scheme. Irregular structures appear in different situations e.g. in problems of wettability [ZLGC], or some problems of human physiology, diffusion of sprays and gases in the lungs [Ku], fractal antennas [WG], tumor growth in biological systems, non-Newtonian fluid mechanics, reaction-diffusion problems, flows through porous media ([D] and references therein), statistical mechanics and quantum fields on fractals [AK1,AK2,AK3]. From the mathematical point of view these problems can be modeled by either scalar or vector linear or nonlinear autonomous or non-autonomous problems possibly with unusual boundary conditions (e.g. Venttsel boundary conditions) involving superposition of operators of different order.
Such problems well describe the impact of a local and a nonlocal diffusion in some concrete situations (e.g. how different types of regional or global restrictions may reduce the spreading of a pandemic disease ).
Modeling in fractal analysis is a recent and active research field since 15/20 years. Our group has acquainted an expertise on this topic. Depending on the problem, different tools and techniques are necessary to give a rigorous formulation of classical operators in the Euclidean settings, as well as delicate functional inequalities which are instrumental to the study of some BVPs. In view of real world applications, it is also crucial to yield a numerical approximation of the solution for the problem at hand. To achieve this goal two steps are necessary. The former is to yield a constructive approach which allows to define an abstract operator in terms of smoother ones; the latter is to construct an efficient numerical approximation scheme. Irregular structures appear in different situations, e.g. in problems of wettability [ZLGC], or some problems of human physiology, diffusion of sprays and gases in the lungs [Ku], spreading of a pandemic disease [E], fractal antennas [WG], tumor growth in biological systems, non-Newtonian fluid mechanics, reaction-diffusion problems, flows through porous media ([D] and references therein), statistical mechanics and quantum fields on fractals [AK1,AK2,AK3]. From the mathematical point of view, these problems can be modeled by either scalar/vector linear/nonlinear autonomous/non-autonomous problems possibly with unusual boundary conditions (e.g. of Venttsel' type) involving superposition of operators of different order.
In all those phenomena that take place in small volumes with large surfaces, fractal surfaces can efficiently model the exchange surface. From the mathematical point of view, they are non-self-similar sets and they do not enjoy many of the properties satisfied by fractal curves.
Our group has obtained many results on some scalar BVPs in fractal domains (mainly local and/or with nonlocal terms supported on the boundary). More recently, we focused on vector BVPs. In [C2] we defined a fractal boundary integrodifferential operator and in [C1] we considered the numerical approximation of a 3D Stokes flow, and we built an ad-hoc mesh allowing us to obtain an optimal rate of convergence.
Our aim is now to focus on scalar (possibly nonlinear) autonomous and non-autonomous BVPs involving fractional operators also in the bulk as well as on possibly nonlinear vector BVPs (see "Descrizione obiettivi progetto"). Moreover, we will investigate suitable numerical schemes to approximate our problems. The first examples in the literature of numerical approximation of heat-type problems in 2D fractal domains are given in [CLD] and [CLH], while that of 3D fractal problems by mixed methods are in [C2] and [C1].
These numerical schemes have a high computational cost. We will consider boundary integral formulations for some of our problems and then approximate them by boundary element methods (BEM) in order to reduce the computational costs. This is a new research direction; it will require first to investigate the properties of the boundary integral operators on fractal domains and then to give a boundary integral formulation. This topic will be investigated with Dr. C. Sorgentone, winner of SAPIEXCELLENCE2020.
The expertise of the P.I. as well as of most of the participants to the group is well-documented by the papers [L1,L2,LVe1,LVe2,LVe3,LV1,LV2,LV3,LMV,MV1,MV2,MV3,CLD,CL].
The P.I. has organized a special session on Fractals Applications in Engineering: Theoretical Aspects and Numerical Approximations at the ICIAM2019. She is editor of a special volume on Fractals Applications in Engineering, published by SEMA SIMAI Springer Series, which is in press. In 2020, she has been invited to give a talk at the 7th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals, postponed to 2021.
We think that our techniques and tools introduced up to now will be a good starting point to develop new tools or adapt existing ones to attack this new direction of research.
The collaboration with physicists and engineers will be crucial in order to construct meaningful models of interest for the applications and to interpret the numerical results.
For the bibliography, see Section "Descrizione delle attività e dei compiti dei partecipanti".