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.

In this talk, we present a complete spectral research of generalized Cesàro operators on Sobolev-Lebesgue sequence spaces. The main idea is to subordinate such operators to suitable \(C_0\) -semigroups on these sequence spaces. We introduce that family of sequence spaces using the fractional finite differences and we prove some structural properties similar to classical Lebesgue sequence spaces. In order to show the main results about fractional finite differences, we state equalities involving sums of quotients of Euler’s Gamma functions. Finally, we display some graphical representations of the spectra of generalized Cesàro operators. Main results of this talk are included in a joint paper with L. Abadias ([1]).

References
[1] L. Abadias and P.J. Miana. Generalized Cesàro operators, fractional finite differences and Gamma functions. J. Funct. Anal., 274 , (2018), 1424–1465.

Partially supported by Project MTM2016-77710-P, DGI-FEDER, of the MCYTS and Project E-64, D.G. Aragón, Universidad de Zaragoza, Spain.

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.

Anomalous diffusion is widely observed in biological systems. Lot of efforts have been dedicated
to derive models in agreement with all the statistical features emerging from data [1, 2], but an exhaustive description is still missing.
We derive a stochastic diffusion model based on a Langevin approach, characterized by additive noise and linear friction force [3].
The complexity of the medium is parametrized by a population of the parameters, the relaxation time and diffusivity.
For proper distributions of these parameters, both Gaussian anomalous diffusion,
fractional diffusion and its generalizations can be retrieved, but characterized by a superdiffusive regime by model construction.
The inclusion of a confining potential, for example by considering the harmonic Langevin oscillator, permits to switch the process to a subdiffusive regime.
This anomalous diffusive behaviour can be reflected in the motion of the center of mass of an heterogeneous ensamble of particles [4]
and the motion of an inert tracer globally connected with such heterogeneous mesoscopic surrounding.

References
[1] M. Mura, G. Pagnini. 2008 Characterizations and simulations of a
class of stochastic processes to model anomalous diffusion. J. Phys.
A: Math. Theor. \textbf{41}, 285003.

[2] D. Molina-Garciá, T. M. Pham, P. Paradisi, C. Manzo, G. Pagnini.
2016. Fractional kinetics emerging from ergodicity breaking in random
media Phys. Rev. E \textbf{94}, 052147.

[3] S. Vitali, V. Sposini, O. Sliusarenko, P. Paradisi, G. Castellani, G.
Pagnini. 2018. J. R. Soc. Interface \textbf{15}: 20180282.
http://dx.doi.org/10.1098/rsif.2018.0282

[4] M. D’Ovidio, S. Vitali, V. Sposini, O. Sliusarenko, P. Paradisi,
G. Castellani, G. Pagnini. 2018. Centre-of-mass like superposition
of Ornstein-Uhlenbeck processes: a pathway to non-autonomous
stochastic differential equations and to fractional diffusion. Submit-
ted. https://arxiv.org/abs/1806.11351.

Arbitrary order calculus is a natural generalisation of usual calculus in which the order of differentiation and integration operators in not restricted to integer numbers. In engineering fractional order derivatives are used for describing the behaviour of materials with memory (i.e. viscoelastic materials) due to the fact that these materials lay somewhere in between Hookean springs and Newtonean fluids[1,2]; there are also many dynamical systems which can be better described when arbitrary order derivatives are included [3].

Lanskin [4] formulated the first Fractional Schrödinger Equation (FSE) along with the Fractional Continuation Equation in 2002; however, we are still far from fully understanding the effect pf the FSE on physical properties such as: Tunnelling [5], Diffraction [6] and Scattering [7]. Due to the properties of fractional derivatives, many jobs have been done in which relativistic properties and effects of extrinsic magnetic fields are obtained by incorporating an arbitrary order to the kinetic energy in the Hamiltonian [8,9].

Further studies of the FSE applied on astrophysically interesting systems such as \(H_2^+\) [10] and even hydrogen atom [11] seem to be promising. We shall take the FSE for a particle in a ring (1) as a first step into this world for which the eigenvalues are (2) and the eigenfunctions (3)

\(
\left[\frac{1}{2mr^2}\right]^{\alpha-1}\left[i\hslash \partial_\theta \right]^\alpha \Psi_\alpha(\theta;r)=\lambda_\alpha\Psi_\alpha(\theta;r) \tag{1}
\)

\(
\lambda_\alpha = \left[\frac{1}{2mr^2}\right]^{1-\alpha} N^\alpha \hslash^\alpha \exp \left[ i \pi \alpha \left(n+1\right)\right]\; | \; N, \; n \in \mathbb{N}+\{0\} \tag{2}
\)

\(
\Psi_\alpha(\theta;r)=C\exp \left(-i N^\alpha\left[ \frac{\hslash}{2mr^2}\right]^{\alpha-1}\exp \left(i \pi \alpha \left(n+1\right)\right)\theta \right). \tag{3}
\)

References
[1] M. Stiassnie, 1979, Appl. Math. Modelling, 3, 300.

[2] M. Du et al., 2013, Scientific Reports, 3, 3431.

[3] V. E. Tarasov, 2013, Int. J. Mod. Phys. B, 2013, 9, 1330005.

[4]N. Laskin, Physics Review E, 2000, 66, 056108.

[5] E. Capelas et al., 2011, J. Phys. A 44, 185303.

[6] Y. Zhang et al., 2015, Scientific Reports 6, 23645.

[7] A. Liemert, 2016, Mathematics, 4, 31.

[8] J. Lorinczi and J. Malecki, 2012, J. Diff. Eq., 253, 2846.

[9] J. Blackledge and B. Babajanov, 2013, Math. Aeterna, 3, 601.

[10] A. Turbiner et al., 1999, JETP Letters, 11, 69.

[11] A. I. Arbab, 2012, J. Modern Physics, 3, 1737.

We develop a pseudo-spectral method to solve initial-value problems associated to PDEs involving the fractional Laplacian operator acting on the whole real line (see [1]). After a suitable representation of the operator, we perform the change of variable \(x = L\cot(s)\), \(L > 0\), to transform the real line \(\mathbb{R}\) into the interval \([0, \pi]\), where a Fourier expansion of the solution \(u(x(s))\) can be applied. We approximate the fractional Laplacian by means of the midpoint quadrature rule, improving the results with Richardson’s extrapolation, similarly as in [2]. This method deals accurately and efficiently with problems posed on \(\mathbb{R}\), and avoids truncating the domain (which requires introducing artificial
boundary conditions). In order to illustrate its applicability, we have simulated the evolution of the following non-local Fisher-KPP [3] and ZFK-Nagumo [4] models.

Keywords: Fractional Laplacian, Pseudo-spectral methods, Fourier transform, Chebyshev Polynomials.

References

[1] M. Kwaśnicki. Ten equivalent definitions of the fractional Laplace operator. Fractional Calculus and Applied Analysis, 20(1):7–51, 2017.

[2] F. de la Hoz and C. M. Cuesta. A pseudo-spectral method for a non-local KdV-Burgers equation posed on \(\mathbb{R}\). Journal of Computational Physics, 311:45–61, 2016.

[3] R. A. Fisher. The wave of advance of advantageous genes. Annals. of Eugenics, 7:355–369, 1937.

[4] Y. B. Zel’dovich and D. A. Frank-Kamenetsky. Towards the theory of uniformly propagating flames. Doklady AN SSSR, 19:693–697, 1938.

[5] J. P. Boyd. Chebyshev and Fourier Spectral Method. Springer–Verlag, XVI, 1989.

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).