Fractional Calculus - Fractional-order Differential Equations
Tomás Skovránek, EPFL STI LIB

Seminar • 12 May 2014 • BM 4.233

Abstract
Fractional-order differential equations (FDEs) and their numerical solutions are currently a rapidly developing field of research. They open new horizons in description of dynamical systems. Compared to classical integer-order models, FDEs provide a powerful instrument for description of memory and hereditary properties of real systems. Ordinary and partial differential equations of fractional order can be applied in modeling many physical and engineering problems. Finding accurate and efficient approximate methods and numerical techniques for solving FDEs is the goal of many research works (e.g. Blank (1996); Diethelm (1997); Diethelm and Walz (1997); Diethelm and Ford (2002); Diethelm et al. (2002, 2004); Gorenflo (1997); Podlubny (1997); Kumar and Agrawal (2006)). In opposite to methods based on iterations, the Podlubny’s “matrix approach” suitable for solving both ordinary and partial differential equations of integer and fractional order (see Podlubny (2000); Podlubny et al. (2009); Podlubny et al. (2013)) is considering the whole time interval of interest at once. The system of algebraic equations is obtained by approximating the equation in all nodes simultaneously. To demonstrate the possibilities of using FDEs in modeling real systems, some of the applications (e.g. modeling the behavior of national economies) will be presented.