Seminario: “Modelling with Fractional Calculus: from Mathematical Curiosity to Real Life Processes and Materials”

Giovedì 29 Giugno 2017, ore 16:00 - Aula 2BC60 - Sergei Rogosin


Giovedì 29 Giugno 2017 alle ore 16:00 in Aula 2BC60, Sergei Rogosin (Università di Minsk - Bielorussia) terrà un seminario dal titolo “Modelling with Fractional Calculus: from Mathematical Curiosity to Real Life Processes and Materials”.

The foundation of the Fractional Calculus goes back to the end of XVII century. First, the derivative of a fractional order appeared as a game of mathematical intelligence. This game was developed by several generations of scientists. Nobody could predict an essentially rapid growth of the interest to this subject from representative of different branches of Science, Engineering, Medicine, Economics etc., which one can observe in the recent couple of decades. This interest is related to several features of the subject emphasized – the technique is fairly simple, the models built on its bases is highly applicable in different areas of science and human life, the novel models are useful to describe new features of real materials and processes, they allow to take a fresh look at well-known objects.
Here we try to answer few questions related to the title of this lecture.
First of all, we deal with attractive classical examples of applications of the fractional modeling coming from rheology, thermodynamics, quantum mechanics, nano-physics, electrochemistry, economics and medicine.
Second, we briefly describe the history of the development of the Fractional Calculus paying attention to the characteristic features of fractional derivatives and integrals.
The first part of the lecture gives an insight to the specific peculiarities of the fractional technique. The most popular and powerful constructions are outlined: via non-integer differentiation of power functions (L. Euler); via Fourier-type integral (J.-B. Fourier); via an analog of the repeated integral (B. Riemann and J. Liouville); via differences of fractional order (A.K. Grünwald and A.V. Letnikov); via non-integer powers of the Laplace operator (M. Riesz); via integrals with the weak singular kernel (M. Caputo).
Next question to answer is “how people arrive to the necessity of the use of such mathematical objects in their research?” We single out several most peculiar properties (closely related to each other, but all leading to the use of fractional derivatives). Among them are the following: hereditary law (going back to V. Volterra and including in particular the study of materials having a memory); non-locality property (with the most interesting appearance in turbulence); power-type influence (as long-time relaxation in visco-elastic materials); non-Gaussian Lèvy-stable statistics and heavy tails distributions (with particular applications of the continuous-time random walk in the study of certain problems in the theoretical economics and finance); self-similar inhomogeneties (or fractals).
We conclude the lecture with some practical recommendations of possible use of the fractional models in different subjects.