Speaker : Anatoly N. Kochubei, Institute of Mathematics, National Academy of Sciences of Ukraine

Time : 15:00 - 16:00 CEST (Rome/Paris)

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

YouTube : https://youtu.be/LEtO2sxJM1c

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: General fractional calculus, Stieltjes functions, Fractional difference equations, Resolvent families, Poisson transform

Abstract: Discrete-time fractional calculus is an important branch of fractional analysis actively developed in the last decades. On the one hand, this kind of analysis generates a lot of interesting mathematical problems. On the other, there are many applications to real world problems, such as new encryption methods, fractional discrete-time neural networks and others. Up to now, all this was built on appropriate analogs of the classical Riemann--Liouville and Caputo fractional derivatives.

Meanwhile for continuous-time fractional calculus, there exists an extension called General Fractional Calculus (GFC) introduced in [1]. In GFC, the counterpart of the fractional time derivative is a differential-convolution operator whose integral kernel satisfies some additional conditions, under which the Cauchy problem for the corresponding time-fractional equation is not only well-posed, but has properties similar to those of classical evolution equations of mathematical physics.

In this work (joint with Alexandra Antoniouk), we develop the GFC approach for the discrete-time fractional calculus. We follow the idea of Lizama [2] who proposed to define discrete-time operations as the Poisson transforms of their continuous-time counterparts. In particular, we define within GFC the appropriate resolvent families and use them to solve the discrete-time Cauchy problem with an appropriate analog of the Caputo fractional derivative.

Biography: Anatoly N. Kochubei (born in 1949) is Department Head at the Institute of Mathematics of the National Academy of Sciences of Ukraine. His research interests include fractional calculus, non-Archimedean analysis, partial differential equations, operator theory, and mathematical physics.

Bibliography

[1] Anatoly N. Kochubei. “General fractional calculus, evolution equations, and renewal processes”. In: Integral Equations and Operator Theory 71 (2011), pp. 583–600.

[2] C. Lizama. “The Poisson distribution, abstract fractional difference equations, and stability”. In: Proceedings of the American Mathematical Society 145 (2017), pp. 3803–3827