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Fractional Differential Equations

Participation in INI programmes is by invitation only. Anyone wishing to apply to participate in the associated workshop(s) should use the relevant workshop application form.

Programme
4th January 2021 to 30th April 2021
Organisers: 
Vassili Kolokoltsov University of Warwick
Jozsef Lörinczi Loughborough University
Eulalia Nualart Universitat Pompeu Fabra
Michael Roeckner Universität Bielefeld
Laura Sacerdote

Programme Theme

Modelling based on differential equations of fractional order and related heavy-tailed probability distributions has recently witnessed a surge of interest due to their ability to model complex phenomena. Applications include modelling contamination of groundwater flow, the electrical dynamics of the heart and the design of new materials.  Fractional differential equations capture effects going well beyond the range tractable by conventional concepts and tools, and it is increasingly recognised that this framework is on the way of becoming a new paradigm in scientific modelling.  This programme sets out to investigate the multiple facets of space-time fractional dynamics and make significant progress in the description and deep understanding of this new phenomenology.

A common feature of the problems the programme aims to address is space-time non-locality. In a mathematical formulation, spatial non-locality means that the evaluation of the action of an operator on a function at a particular point in space generally depends also on its effect in far-away positions. A typical example is the fractional Laplacian, however, we will consider a whole range of possible non-local operators, which generally may have different qualitative behaviours. Temporal non-locality means that the future evolution depends on its past, which is described in terms of fractional (such as Caputo) time-derivatives.

A particular aspect in the attempt to mathematically capture these problems is that several complementary approaches are at hand. A natural framework is offered by integro-differential equations featuring integral kernels in space and/or time variables, and one path to follow is then an analysis in terms of singular integrals and related objects. Another possibility is a more operator-based analysis emphasising spectral properties of pseudo-differential operators.  A third way is a probabilistic formulation in terms of Feynman-Kac-type representations, involving random processes with jump discontinuities. One of the main aims of the programme is to explore the interaction of these multiple approaches, and develop a common framework and new methodology.

The programme will focus on the following themes within three workshops: (1) Deterministic and Stochastic Fractional Differential Equations and Jump Processes, (2) Fractional Kinetics, Hydrodynamic Limits and Fractals, (3) Optimal Control in Fractional Dynamics. Within the final two workshops there will also be sessions addressing applications and computational aspects of fractional differential equations. These sessions will cover themes such as physics (transport phenomena, quantum optics), chemistry and biology (chemotaxis, population dynamics/foraging), environmental sciences (groundwater contamination, fractal solute transport), and technology (image reconstruction and signal analysis, financial engineering).

The programme intends to offer to scientists working broadly in the interdisciplinary area of fractional calculus, non-local equations, random processes with jumps, and their applications, a platform to interact and communicate their research results in a vibrant research environment. To strengthen intra- and inter-disciplinary cross-fertilisation, these topics will be organised in a way to combine key subjects of research with a double focus, including both theoretical aspects with more applied/computational interests. Participants to the programme and the workshops are expected from six continents, including a significant proportion of early career scientists. We also encourage PhD students/post-doctoral researchers to attend some of the proceeds or outreach activities.

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons