Speaker : Rudolf Hilfer, ICP, Universität Stuttgart, Allmandrng 3, 70569 Stuttgart, Germany

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

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

Zoom :  Cancelled due to medical emerency

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: fractional calculus, distributions

Abstract: Fractional derivatives and integrals are generalized from functions [6, 9] to measures [10] and to distributions [5, 7, 8]. The focus is on translation invariant fractional operators, their domains of definition and their co-domains. It is found that translation-invariant fractional operators interpreted as suitably generalized distributional convolution operators have the largest domains of definition. The extension from domains of functions to distributions also leads to unifications of many previously existing definitions of fractional integrals and derivatives, including discretized fractional calculus operators and fractional operators for periodic distributions [4].

Biography: Rudolf Hilfer has worked on Fractional Calculus and its Applications in Physics since 1985 [1]. In 1990 he introduced fractional derivatives into thermodynamics and into the theory of critical phenomena by identifying the order of the derivative with the order of phase transitions [2]. In 1992 he established time fractional diffusion as the continuum limit of certain continuous time random walks (stochastic processes). This discovery was published during his stay at SISSA in 1994 together with L. Anton [3].

Bibliography

[1] R. Hilfer. Applications of Fractional Calculus in Physics. Singapore: World Scientific Publ. Co., 2000. isbn: ISBN: 978-981-02-3457-7. doi: https://doi.org/10.1142/3779. url: https://www.worldscientific.com/worldscibooks/10.1142/3779.

[2] R. Hilfer. “Multiscaling and the Classification of Continuous Phase Transitions”. In: Physical Review Letters 68 (1992), p. 190. doi: 10 . 1103 / PhysRevLett . 68 . 190. url: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.68.190.

[3] R. Hilfer and L. Anton. “Fractional Master Equations and Fractal Time Random Walks”. In: Physical Review E, Rapid Communication 51 (1995), R848. doi: https: //doi.org/10.1103/PhysRevE.51.R848. url: https://journals.aps.org/pre/abstract/10.1103/PhysRevE.51.R848.

[4] R. Hilfer and T. Kleiner. “Fractional Calculus for Distributions”. In: Fractional Calculus and Applied Analysis (2024), accepted.

[5] R. Hilfer and T. Kleiner. “Maximal Domains for Fractional Derivatives and Integrals”. In: Mathematics 8 (2020), p. 1107. doi: https://doi .org /10.3390 /math8071107. url: https://www.mdpi.com/2227-7390/8/7/1107.

[6] T. Kleiner and R. Hilfer. “Convolution Operators on Weighted Spaces of Continuous Functions and Supremal Convolution”. In: Annali di Matematica Pura ed Applicata 199 (2020), pp. 1547–1569. doi: https://doi.org/10.1007/s10231-019-00931-z. url: https://link.springer.com/article/10.1007/s10231-019-00931-z.

[7] T. Kleiner and R. Hilfer. “Fractional glassy relaxation and convolution modules of distributions”. In: Analysis and Mathematical Physics 11 (2021), p. 130. doi: https://doi.org/10.1007/s13324-021-00504-5. url: https://link.springer.com/article/10.1007%2Fs13324-021-00504-5.

[8] T. Kleiner and R. Hilfer. “On extremal domains and codomains for convolution of distributions and fractional calculus”. In: Monatshefte f ̈ur Mathematik 198 (2022), pp. 122–152. doi: https://doi.org/10.1007/s00605-021-01646-1. url: https://link.springer.com/article/10.1007/s00605-021-01646-1.

[9] T. Kleiner and R. Hilfer. “Sequential generalized Riemann–Liouville derivatives based on distributional convolution”. In: Fractional Calculus and Applied Analysis 25 (2022), pp. 267–298. doi: https://doi.org/10.1007/s13540-021-00012-0. url: https://link.springer.com/article/10.1007/s13540-021-00012-0.

[10] T. Kleiner and R. Hilfer. “Weyl Integrals on Weighted Spaces”. In: Fractional Calculus and Applied Analysis 22 (2019), pp. 1225–1248. doi: DOI:10.1515/fca-2019-0065. url: https://www.degruyter.com/view/j/fca.