Fractional Calculus Seminar Series

Fractional calculus is a generalised form of the integer-order calculus. While an integer-order derivative is a local operator, a fractional derivative is a non-local operator. The notion of Brownian motion is extended to admit Levy stable processes in the case of fractional diffusion. Many different operators have been described as fractional derivatives and integrals, with different properties and behaviours, and there are also further generalisations within non-local calculus. Real-world applications of non-local models can be found in turbulence, viscoelasticity, fracture mechanics, economics, electrical circuits, and plasma physics. Until recently, the theory and applications of fractional operators and equations did not receive much attention, so that many fundamental questions remain unanswered.

This seminar series is intended to provide deep knowledge on all aspects of fractional calculus, from analytical mathematics to numerical simulations to modelling applications. Starting in May 2024, we aim to have one talk every week, from a variety of experts in different topics related to fractional calculus.

Organisers : 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

YouTube Channel: https://www.youtube.com/@FractionalCalculusSeminar

Mailing List (subscribe/unsubscribe) : https://lists.sissa.it/mailman/listinfo/fc-seminars
Official Web-page : https://mathlab.sissa.it/fractional-calculus-seminars

All times are quoted in local timezone of Rome, ITALY

The general Fractional Calculus operators with the Sonin kernels: Basic properties, applications, and history of origins

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

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 Abstract: In this talk, we start with a discussion of origins of the general fractional integrals and derivatives with the Sonin kernels in the works by Abel and Sonin. Then some recent results regarding properties of the general Fractional Calculus operators with the Sonin kernels are presented. In particular, the first and the second fundamental theorems of Fractional Calculus for the general fractional derivatives, the regularized general fractional derivatives, and the sequential general fractional derivatives are formulated. As an application, we discuss the convolution series that are a far-reaching generalization of the power law series as well as a representation of functions in form of the convolution Taylor series. Another application is the generalized convolution Taylor formula that contains a convolution polynomial and a remainder in terms of the general fractional integrals and the general fractional sequential derivatives. Finally, we provide a survey of some other recent results devoted to the general Fractional Calculus operators with the Sonin kernels and their applications. 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 für Analysis und ihre Anwendungen). Bibliography [1] Y. Luchko. General Fractional Integrals and Derivatives with the Sonine Kernels Mathematics 2021, vol. 9(6), 594. [2] Y. Luchko. General Fractional Integrals and Derivatives of Arbitrary Order Symmetry 2021, vol. 13(5), 755. [3] Y. Luchko. Convolution series and the generalized convolution Taylor formula Fract. Calc. Appl. Anal. 2022, vol. 25, 207-228. [4] Y. Luchko. The 1st level general fractional derivatives and some of their properties J Math Sci 2022, vol. 266, 709-722.

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The general Fractional Calculus operators with the Sonin kernels: Basic properties, applications, and history of origins

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

Abstract: In this talk, we start with a discussion of origins of the general fractional integrals and derivatives with the Sonin kernels in the works by Abel and Sonin. Then some recent results regarding properties of the general Fractional Calculus operators with the Sonin kernels are presented. In particular, the first and the second fundamental theorems of Fractional Calculus for the general fractional derivatives, the regularized general fractional derivatives, and the sequential general fractional derivatives are formulated. As an application, we discuss the convolution series that are a far-reaching generalization of the power law series as well as a representation of functions in form of the convolution Taylor series. Another application is the generalized convolution Taylor formula that contains a convolution polynomial and a remainder in terms of the general fractional integrals and the general fractional sequential derivatives. Finally, we provide a survey of some other recent results devoted to the general Fractional Calculus operators with the Sonin kernels and their applications.