December 10, 2019
Fractional Order Operators and their Applications in Science and Engineering
SCEN 322, 12/10/19 at 3:30 pm to 4:50 pm
Fractional calculus has attracted considerable interest because of its ability to model complex phenomena such as continuum and statistical mechanics, viscoelastic materials, high-frequency price dynamics in financial markets, and biological systems such as population genetics. While the fractional integral has been used to describe the fractal structure of materials which leads to new thermodynamic relations, the fractional derivative could be used to describe viscoelasticity, thermal and chemical diffusion, and light-matter interactions in materials.
This area opens up an application of fractional calculus which may describe the multiscale thermomechanical material behavior of many materials. On the other hand, time-fractional generalizations of the Poisson process, which are based on the fractional Kolmogorov-Feller equation where the integer-order derivative operator is replaced by a fractional-order derivative operator, not only provides a good phenomenological model for high-frequency price dynamics financial markets, but also plays a critical role for deriving the fractional coalescent in population genetics where the order of the fractional derivative shows the environmental heterogeneity in the population.
The strong role of fractional calculus in modeling complex fractal structure-fractional property relations opens up many opportunities to advance our understanding and de- sign of novel materials, advanced structures, and intelligent systems. The mathematical tools required for these fields include a wide range of problems such as fractional differential equation, fractional partial differential equation and fractional control systems where the fractional derivative has been replaced with the integer derivative to create a new set of necessary conditions that must be satisfied.
In this talk, the fractional calculus will be introduced briefly and the application of fractional calculus in material science and population genetics will be shown.