Program

Recent developments in modeling of different physical and biological systems undoubtedly confirmed that fractional calculus is not just an exotic mathematical theory, as it might appeared at the dawn of its emergence. The present work aims to demonstrate this through physical examples of diffusion and quantum motion in two-dimensional comb-like structures, leading to fractional calculus description of the corresponding processes. Comb geometry is one of the most simple paradigm of a two-dimensional structure, where anomalous diffusion can be realized. The comb model, for the first time, has been applied to investigate anomalous diffusion in low-dimensional percolation clusters, and nowadays it has many applications in description of transport properties in porous dielectrics and low dimensional composites, turbulent diffusion on a comb, tracer dynamics in subsurface aquifers, and anomalous diffusion in spiny dendrites, to name but a few. In this presentation we will show how the time fractional diffusion and Schrödinger equations can be derived by projection of the two dimensional comb dynamics in the one-dimensional configuration space. The anomalous character of the corresponding process can be described by the fact that the particle which moves along the main backbone channel can get trapped in the fingers of the comb. We will also show how the anomalous diffusion along the main backbone depends on the fractal structure of the comb, resulting in natural appearance of fractional derivatives in the corresponding equations for the probability density function. The connection between the anomalous diffusion exponent and the fractal dimension of the comb/mesh structures has been recently experimentally observed in the anomalous transport through porous structurally inhomogeneous media.

We consider a one-dimensional nonlinear diffusion equation with nonlocal time, i.e. the temporal derivative is calculated via the Riemann-Liouville operator. This problem set on a half-line models moisture percolation in certain porous media such as construction materials.

The efficient and convergent numerical method can be constructed as follows. We cast our equation into the self-similar form with the use of the Erdelyi-Kober operator. Then, by a suitable transformation the free-boundary problem is transformed into the initial-value one. Lastly, the resulting integro-differential equation can be written as a nonlinear Volterra equation. In this way, a nonlinear PDE with free boundary can be solved by finding a solution of an ordinary Volterra equation. This quickens the calculations tremendously.

We also prove that a certain family of numerical schemes is convergent to the solution of the aforementioned Volterra equation. The main difficulty lies in the non-Lipschitzian character of the governing nonlinearity. In that case we cannot use the classical theory and have to proceed in other ways. In the talk we will describe the details of the proof.

Using two simple examples, the continuous-time random walk as well as a two state Markov chain, the relation between generalized anomalous relaxation equations and semi-Markov processes is illustrated. This relation is then used to discuss continuous-time random statistics in a general setting, for statistics of convolution-type. Two examples are presented in some detail: the sum statistic and the maximum statistic.

We prove Hardy-type inequalities for fractional powers of the sublaplacian in the Heisenberg group. In order to get these inequalities, we study the extension problem associated to the sublaplacian. Solutions of the extension problem are written down explicitly and used to establish a trace Hardy inequality that will lead to a Hardy inequality with sharp constants.

Several new results concerning the extension problem in the Heisenberg group are also attained, including characterisations of all solutions of the extension problem satisfiying \(L^p\) integrability, and the study of the higher order extension problem.

This is a joint work with S. Thangavelu (Indian Institute of Science of Bangalore, India).

We consider a nonlocal equation driven by a perturbed periodic potential. We construct multibump solutions that connect one integer point to another one in a prescribed way.

In particular, heteroclinic, homoclinic and chaotic trajectories are constructed.

This result regarding symbolic dynamics in a fractional framework is part of a study of the Peierls-Nabarro model for crystal dislocations. The associated evolution equation can be studied in the mesoscopic and macroscopic limit. Namely, the dislocation function has the tendency to concentrate at single points of the crystal, where the size of the slip coincides with the natural periodicity of the medium.

These dislocation points evolve according to the external stress and an interior potential, which can be either repulsive or attractive, depending on the relative orientations of the dislocations. For opposite orientations, collisions occur, after which the system relaxes exponentially fast.