Speaker: Yuri Luchko, Berlin University of Applied Sciences and Technology, Germany

Time : 15:00 - 16.00 CEST (Rome/Paris)

Hosted at: SISSA, International School of Advanced Studies, Trieste, Italy

YouTube : https://youtu.be/j6WgjZpL4Z0

Organizers : Pavan Pranjivan Mehta* (pavan.mehta@sissa.it) and Arran Fernandez** (arran.fernandez@emu.edu.tr)

* SISSA, International School of Advanced Studies, Italy

** Eastern Mediterranean University, Northern Cyprus

Keywords: anomalous diffusion, continuous time random walks, fractional master equation, regularized  general fractional derivative, Sonin kernels, fractional diffusion equation, mean squared displacement

Abstract: Nowadays, the probably most popular mathematical descriptions of anomalous diffusion processes are the Continuous Time Random Walk (CTRW) models ([1]) at the micro-level and fractional PDEs involving different types of fractional derivatives at the macro-level ([2, 3]). A strong connection between the CTRW model and fractional PDEs was first established in [4], where a fractional master equation involving the Caputo fractional derivative was embedded into the framework of the general CTRW model. In particular, the mean squared displacement (MSD) of the diffusing particles governed by this fractional master equation was shown to be proportional to a power law function of time with the exponent being equal to the order of the fractional derivative.

However, in many applications, deviations of the MSD from the power law with a fixed exponent are observed. Thus, one needs other, more general master equations that would lead to an extension of the class of functions that describe the MSD of the diffusing particles governed by these equations.

In this talk, we introduce and investigate a fractional master equation involving a regularized general fractional derivative (RGFD) with Sonin kernels [5, 6]. 

Under some conditions, this master equation is equivalent to the CTRW model with the waiting time probability density function in form of a convolution series generated by the Sonin kernel associated with the kernel of its RGFD. Then we derive a fractional diffusion equation involving the RGFDs with Sonin kernels from the CTRW model in the asymptotical sense of long times and large distances and investigate its physical characteristics and mathematical properties. In particular, a concise formula for the MSD of the particles governed by this fractional diffusion equation is deduced in terms of the Sonin kernel associated with the kernel of its RGFD. Thus, variation of the Sonin kernels in the fractional diffusion equation leads to a great diversity of possible forms of the MSD that can be fitted to the measurements data collected for a concrete anomalous diffusion process.

Finally, we discuss some important mathematical aspects of the fractional diffusion equation involving the RGFDs with Sonin kernels, including non-negativity of its fundamental solution and validity of an appropriately formulated maximum principle for its solutions on the bounded domains.

The talk is mainly based on the recent paper [7] that was published in the framework of the project PN23-16SM-1809 funded by the Kuwait Foundation for the Advancement of Sciences (KFAS).

Biography: Dr. Yuri Luchko is a Full Professor at the Faculty of Mathematics - Physics - Chemistry of the Berlin University of Applied Sciences and Technology in Germany. He studied Mathematics at the Belarussian State University in Minsk and received his PhD degree from the same University in 1993. In 1994, Yuri Luchko got a postdoc position at the Free University of Berlin, Germany, under supervision of Prof. Rudolf Gorenflo and stayed there for six years. From 2000 to 2006, he was a scientific researcher at the University in Frankfurt (Oder), Germany. In 2006, Dr. Yuri Luchko got a professorship at the Technical University of Applied Sciences Berlin, Germany. The main field of his research is Applied Mathematics with a special focus on Fractional Calculus and its applications. Yuri Luchko published about two hundred papers in international peer-reviewed scientific journals and about twenty books and books chapters as author or editor. He is an associate editor of the international journal “Fractional Calculus and Applied Analysis” and editor of several other reputable mathematical journals including ZAA (Zeitschrift fur Analysis und ihre Anwendungen).

Bibliography

[1] E. Montroll, G. Weiss. Random walks on lattices II. J. Math. Phys. 1965, 6, 167–181.

[2] Yu. Luchko. Anomalous Diffusion: Models, Their Analysis, and Interpretation. Chapter in: S.V. Rogosin, A.A. Koroleva (Eds), Advances in Applied Analysis, Series: Trends in Mathematics, Birkh¨auser, Basel, 2012, pp. 115–146.

[3] R. Metzler, J. Klafter. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Physics Reports 2000, 339, 1–77.

[4] R. Hilfer, L. Anton. Fractional Master Equations and Fractal Time Random Walk. Physical Review E 1995, 51, R848.

[5] A.N. Kochubei. General fractional calculus, evolution equations, and renewal processes. Integral Equations and Operator Theory 2011, 71, 583–600

[6] Yu. Luchko. General fractional integrals and derivatives with the Sonine kernels. Mathematics 2021, 9(6), 594.

[7] M. Alkandari, D. Loutchko, Yu. Luchko. Anomalous Diffusion Models Involving Regularized General Fractional Derivatives with Sonin Kernels. Fractal and Fractional 2025, 9(6), 363