International Geometry Summit-Solid and Physical Modeling
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.