Speaker: Elina Shishkina, Voronezh State University

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

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

Zoom link : A link will appear here, one hour before the talk

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:  Bessel operator, fractional powers of Bessel operator, fractional singular differential equations

Abstract: We present the definitions and key properties of the fractional Bessel integral and derivative. We also explore fractional ordinary differential equations and the fractional Euler–Poisson–Darboux equation with fractional powers of the Bessel differential operator. By utilizing the Meijer integral transform and its modifications, we derive fundamental solutions to these equations in terms of the Fox–Wright function, the Fox H-function, and their specific cases. Additionally, we provide explicit formulas for the solutions to the corresponding initial value problems, framed in terms of the generalized convolutions introduced in this talk.

Biography: Elina Shishkina is a Full Professor at the Faculty of Applied Mathematics, Informatics, and Mechanics at the Voronezh State University in Voronezh, Russia. She graduated from the Department of Mathematics at Voronezh State University in 2004. Elina Shishkina completed her Ph.D. thesis in 2006 and her doctoral thesis in 2019. Her primary research interests lie in Mathematical Analysis and Differential Equations, with a particular emphasis on Fractional Calculus and its diverse applications.

Bibliography

[1] A. Dzarakhohov, Y. Luchko, and E.L. Shishkina. “Special Functions as Solutions to the Euler–Poisson–Darboux Equation with a Fractional Power of the Bessel Operator”. In: Mathematics 9.13 (2021), pp. 1–18.
[2] V. Kiryakova. Generalized Fractional Calculus and Applications. New York: Pit- man Res. Notes Math. 301, Longman Scientific & Technical, Harlow, Co-publ. John Wiley, 1994, p. 274.
[3] V. Kiryakova. “Obrechkoff Integral Transform and Hyper-Bessel Operators via G-function and Fractional Calculus Approach”. In: Global Journal of Pure and Applied Mathematics Proceedings of the 13th Symposium of the Tunisian Mathematical Society 1.3 (2005), pp. 321-341.
[4] I. G. Sprinkhuizen-Kuyper. “A fractional integral operator corresponding to negative powers of a certain second-order differential operator”. In: J. Math. Analysis and Applications 72 (1979), pp. 674–702.