Justin Crum Abstracts

Justin Crum

Ph.D. Candidate

Applied Mathematics

International Geometry Summit-Solid and Physical Modeling

Vancouver, Canada

June 17-21, 2019

Fractional partial differential equations (FDEs) are used to describe phenomena that involve a “non-local” or “long-range” interaction of some kind. Accurate and practical numerical approximation of their solutions is challenging due to the dense matrices arising from standard discretization procedures. In this paper, we begin to extend the well-established computational toolkit of Discrete Exterior Calculus (DEC) to the fractional setting, focusing on proper discretization of the fractional derivative. We define a Caputo-like fractional discrete derivative, in terms of the standard discrete exterior derivative operator from DEC, weighted by a measure of distance between p-simplices in a simplicial complex. We discuss key theoretical properties of the fractional discrete derivative and compare it to the continuous fractional derivative via a series of numerical experiments.

Abstract for Lay Audience

Differential equations are a common way to mathematically model the real world. These are used in applications such as biological modeling, computer graphics and fluid flow, amongst many others. When studying these applications using mathematics, one main aspect of focus is how well the equations capture the true behavior. This involves comparing experimental data to the results of computer simulations, where the simulations are based off the underlying differential equations. In some applications, this works. Mechanical engineers design vehicles based off these principles. Astronomers predict the trajectories of comets. Biologists model population growth. However, some differential equations fail to accurately capture the behavior exhibited in the real world of some physical phenomenon. For example, how fluid flows through porous media (such as rocks or cement), does not quite match the typical behavior that a differential equation would model in this scenario. Thus we need something more. Hence researchers have worked on extending differential equations to fractional differential equations. Fractional differential equations allow us to account for nonlocal behaviors, something a normal differential equation fails to be able to do. For example, a fractional differential equation can be used to model the probability of a drug addict relapsing today given that they have been sober for some amount of time previous to today. This sort of behavior fails to be properly exhibited in the normal differential equation case. Unfortunately, the mathematics behind solving fractional differential equations gets complicated because of the nonlocal nature. In a computer, the computational costs and storage costs are much higher for a fractional differential equation than they are for a normal differential equation. Studying how to efficiently solve fractional differential equations is an important step in being able to model and understand these sort of applications. This current work is about extending a method (discrete exterior calculus) for solving differential equations on computers to solving fractional differential equations. Discrete exterior calculus is a method for solving differential equations on a mesh that approximates a real surface. While there are many other methods that have already been extended to the fractional differential equation setting, these methods struggle to work efficiently. We believe that extending discrete exterior calculus to solve fractional differential equations may give a way to efficiently solve these equations on the computer.