ncts

Abstract: Fractional-time derivatives have been widely used in mathematical modeling of physical phenomena such as anomalous diffusions and memory effects of materials. However, an efficient time stepping scheme for solving fractional-time differential equations (FTDE) remains a challenge because the time-fractional derivative of a function

u

is a time-convolution between

u^{'}

and the kernel function. Traditionally, FTD is handled by approximating the time-domain kernel function with a sum of exponential functions. The limitation of this approach is that many terms are needed due to the weak singularity of the kernel function at

t = 0

. In this talk, we will take a different approach and will focus on the Caputo FTDE

D^{α}

with

0 < α < 1

because the Caputo time derivative of a constant function is 0. We will show how a simple time-stepping algorithm for FTDE can be developed by using the theory of Heglotz-Nevanlinna functions and their subclasses, a well-studied field in operator theory and complex analysis. The key is in identifying the Stieltjes-Markov function structure hidden in the Laplace transform of the kernel function, and the rational interpolation scheme for these functions. Compared to most of the current methods in literature, our result contains a Dirac measure with carefully calculated strength in the approximation of the time-convolution term. As a result, the number of exponential terms needed for approximating the time-domain kernel function drastically decreases for the same level of accuracy. I will explain the theory behind the time-stepping algorithm and end the talk by showing some numerical examples. This is joint work with Prof. Gamze Tangnolu and Prof. Jiangming Xie. This research is partially funded by Project EXPOWER via EU Research and Innovation Staff Exchange (RISE) funding in mathematics under H2020 Marie Skłodowska-Curie Actions.