Office: Stark 203I

Telephone: 507.474.5793

E-Mail: janderson@winona.edu

Web Page: course1.winona.edu/janderson

Field of Study: Partial Differential Equations

Professional Goals & Interests: The field of partial differential equations has remarkable intersection with research programs concerning applied physical models, numerical methods, and mathematical analysis.  There are suprising patterns and similarities existent in what appear to be disorganized, unrelated phenomena

from our everday experiences. Many of these have been brought to light through the study of mathematical models involving partial differential equations.

For example, models of heat flow, which could be constructed to describe the temperature inside your computer, within a curing composite material, or up in the atmosphere, involve what is known as the heat equation.  For statisticians, the heat equation arises naturally in models of random walks, in which case resulting solutions become recognizable as the so-called bell curve or its variations. The same heat equation also plays a central role in models of dispersion or diffusion phenomena, such as the movement of populations or the spread of a contaminant in our water supply.  

Since groundwater is typically contained in porous underground material, the heat equation must be generalized slightly, and its appropriate generalization is aptly called the porous medium equation.  Considering such applications of the porous medium equation to studies of flow in saturated or unsaturated soils, it is interesting to note that recent models of cancer growth also involve the same porous medium equation as well as random walk equations (i.e., heat equations).

By developing mathematical methods to analyze entire collections of analogous equations, we may obtain results that apply to a vast range of different physical situations of importance to society.  My current research program involves developing the fundamental results on unique solvability and ultimate behavior predicted by models involving a large class of porous medium type equations.  Other effects incorporated into the models to account for memory, nonlocal, and reinforced random walk phenomena, as has been motivated by studies of capillary growth in solid tumors, complicate the analysis yet further broaden the spectrum of potential applications.