Introduction to fractional calculus I
A talk in the IRTG Block Course series by
Vanja Wagner from Zagreb
| Abstract: | The first part of the lecture (2x45min) will cover a short introduction to fractional integrals and fractional derivatives (such as Riemann–Liouville, Gruenwald–Letnikov, Caputo), their basic properties and generalizations. After introducing these basic notions we will consider some classical fractional (in time) nonlocal (in space) partial differential equations, and cover some basic setting and examples. In the second part of the lecture (2x45min), we will outline the probabilistic counterpart of these PDEs, briefly introducing subordination, inverse subordinators, and the corresponding semi-Markov processes. Some References: B. Baeumer, M.M. Meerchaert, Stochastic solutions for fractional Cauchy problems, Fract. Calc. Appl. Anal. 4 (2001) 481–500. Z.-Q. Chen, Time fractional equations and probabilistic representation, Chaos Solitons Fractals 102 (2017) 168–174. Z.-Q. Chen, P. Kim, T. Kumagai, J. Wang, Time fractional Poisson equations: Representations and estimates, Journal of Functional Analysis, 278(2), 2020. K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York, 1993. |