SISSA Workshops

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    Book of Abstracts: Fractional Calculus Seminar Series - 2024
    (SISSA, International School of Advanced Studies, 2024) Pranjivan Mehta, Pavan ; Fernandez, Arran; Rozza, Gianluigi
    Preface: Fractional calculus is the field of research that studies fractional derivatives and fractional integrals, which are variously defined as derivatives and integrals to noninteger orders. This is a generalised form of integer-order calculus, with more richness and variety since fractional derivatives and integrals can be defined in many different ways which are not equivalent to each other: there is no single unique answer to a question like “what is the derivative to order one-half of the identity function?” It is well known that integer-order derivatives are local operators and integerorder integrals are non-local operators. In fractional calculus, since the derivative operators are defined using the integral operators, both fractional integrals and fractional derivatives are non-local operators. Depending on the type of fractional derivative used, these operators may depend on values of the function in a finite region, or in a one-way region modelling a memory effect, or in its entire domain. The non-locality property is one of the reasons why fractional calculus has found manyapplications: real-world applications of non-local models can be found in turbulence, viscoelasticity, fracture mechanics, economic models, diffusion processes, electrical circuits, and plasma physics. The full range of applications is not yet understood, and new research is ongoing in many of these domains. The theory and applications of fractional integro-differential operators and equations has not received much attention in the wider scientific community, beyond a few specialists developing the field, so that many fundamental questions remain unanswered and the field is ripe for ongoing research in many directions. Currently, however, there is not a single united research community in fractional calculus, but rather many different groups working on it in different ways. Some research is undertaken without awareness of the established fundamentals of the field or of what other research groups are doing. Thus, the Fractional Calculus seminar series grew out of the necessity to connect different research communities and to touch on as many aspects of fractional calculus as possible. This seminar series is intended to provide deep knowledge on all aspects of fractional calculus, from analytical mathematics to numerical simulations to modelling applications. Some of the presentations are from long-standing experts who have been working in the field for decades, while some reflect new developments in particular research directions. Some of the topics are of broad interest to anyone working in fractional calculus, but there were also some focus sessions (listed below) which allowed us to drill further into particular broad topics of research in fractional calculus. Special focus sessions: • 24 May to 21 June 2024 (7 talks): fractional/nonlocal modelling of turbulence. • 05 July to 02 August 2024 (6 talks): stochastic processes and probability theory for fractional PDEs. • 09 August to 06 September 2024 (5 talks): general fractional-calculus operators. • 13 September to 11 October 2024 (5 talks): fractional inverse problems. • 18 October to 06 December 2024 (10 talks): numerical analysis, methods, and singular integral computation.