Catherine A. Roberts

My training is as an applied mathematician. Below you can read descriptions of my research. As you will see, some of these projects provide opportunities for students to participate in research. If you are interested in any of these projects, please email me.

You can also read about several projects that were conducted by students prior to my arrival at Holy Cross in Fall 2001. Brief descriptions are included here at the bottom of this page.

**Explosion, Quenching, and Absorption**

`The objective of these studies is to investigate the phenomena of explosion,
quenching and absorption within a diffusive medium in the context of nonlinear
Volterra integral equations. These problems are characterized by spatially
localized nonlinearities, which distinguishes them from previous work. Such
nonlinearities are motivated, for example, by applications where the event
occurs within a very confined area (e.g. from an internal electrode energy
stimulation within a flame or on a heated surface). `

`When explosion, quenching or absorption occurs within a diffusive medium,
the system undergoes dramatic change. In explosion, the solution becomes unbounded
in finite time. In quenching, the solution remains bounded, whereas its derivative
in time blows-up at some finite moment. In absorption, the solution vanishes
in finite time in a region called a dead-core. These phenomena are of wide
importance in applications. `

`Studies of explosion, and to a lesser extent quenching and absorption,
have been conducted on theoretical models that are originally presented as
nonlinear partial differential equations. Unlike previous studies, which have
required a smoothness property for spatially dependent nonlinearities, future
research replaces such requirements by strongly localized behavior. In this
case, conversion of the theoretical models from partial differential equations
to integral equations represents a very effective format for the analysis.
This reformulation permits a more direct and efficient inquiry into the challenging
scenario of spatially localized nonlinear behavior. `

`Prior work (see vita) on the phenomena of explosion
solutions for Volterra equations inform the development of viable techniques
to address the newer circumstances of quenching and absorption. We have treated
systems of Volterra equations, as well as the influence of non-local nonlinearities
on explosion. In addition, we have examined several classes of quenching problems
using similar analytic techniques. To date, for certain classes of nonlinear
Volterra equations, we have demonstrated the existence of a quenching solution,
established lower and upper estimates for the critical time to quenching,
and characterized the solution growth rates for various nonlinearities. The
resulting integral equations have corresponded to one-dimensional models with
a spatially localized nonlinearity as either the source or boundary condition.
`

`Future research will derive the appropriate Volterra equation formulation
for a variety of new problems. For example, we will consider quenching problems
posed in more general domains and in two or three spatial dimensions. A wider
class of nonlinearities, such as the inclusion of a logarithmic component
and the incorporation of spatial derivative dependence, will be examined.
The influence of several source terms will also be explored. We will characterize
the classes of nonlinearities leading to various quenching or absorption solutions.
Such models are physically relevant yet have not been explored mathematically.
Moreover, the analytic techniques under development for explosion problems
are modifiable for use in these new problems. Future research will characterize
sufficient conditions to guarantee quenching and estimate absorption rates
for a variety of nonlinearities. The results will characterize the spatial
results for each of the physical applications because it is useful to determine
conditions for when quenching or absorption occurs at a single point, a finite
region or everywhere in the domain. Hyperbolic partial differential equations
with a similar strongly spatially localized nonlinearity have not been considered
in the explosion literature.
Most recently, I have been investigating questions related to anomalous diffusion, where blow-up is impacted by the nature of the diffusive material. This has led to several new publications with my collaborators, W. E. Olmstead and C. M. Kirk. `

`The methods used above are currently being modified to address problems
of this sort. Interestingly, analysis leads to a nonlinear Volterra equation
that can also be represented in terms of a delay differential equation. This
class of delay differential equations has not been previously examined for
unbounded solutions. It is anticipated that contributions along these lines
will naturally follow from examination of the hyperbolic problem. The primary
tools for this research come from functional, numerical and asymptotic analyses.
A new direction of interest for us is to examine these equations from a numerical
point of view. Contraction mappings are used to establish a lower limit on
existence of a continuous bounded solution. Techniques of comparison and contradiction
are applied to prove nonexistence of a continuous bounded solution beyond
a certain point in time. Asymptotic expansions of integrals are used to determine
the growth rate behavior of the solutions in the key limit. New tools will
be developed in this research to strengthen the bounds for the critical time
and to examine new questions regarding the solutions of these integral equations.
`

`This work is conducted collaboratively. Current collaborators include
W. Okrasinski (Univ. of Zielona Gora, Poland), W. E. Olmstead (Northwestern
University), Colleen Kirk (Montclair State University), and Kelly Fuller (Nazarene
College).`

`STUDENT PARTICIPATION: One doctoral student,
Kelly Fuller, received her Ph.D. from me in 1996. She started working on her
project when I was a faculty member at the University of Rhode Island.`

** Mathematical Modeling: Human-Environment Interactions**

`This research seeks to characterize complex human-environment interactions
on a river. Using mathematics and statistics, the intent of this research
is to develop a novel computer program that simulates the responses of humans
to an ever-changing natural environment. `

`Already, we have developed the Grand Canyon River Trip Simulator. The
program uses artificial intelligence and statistical techniques to capture
the unique nature of this interaction. The computer simulation program is
being used by managers at the Grand Canyon
National Park as part of their Colorado River Management Plan. ``This
project is the result of a cooperative effort between researchers at Northern
Arizona University and the University
of Arizona.`

`The next step in this research is to re-engineer our program for other
environmental settings, sharpen the artificial intelligence algorithms and
examine more carefully the robustness of these mathematical approaches to
modeling the real-world. ``The Grand Canyon River Trip Simulator Project
(GCRTSim) involves a team at Northern Arizona University working in cooperation
with a team from the University of Arizona to assist the Grand Canyon National
Park in the management of rafting traffic on the Colorado River. Taking over
three years to develop and test, our software was recently turned over to
the park (version 1.0 in October 1999 and version 2.0 in August 2000). This
computer-implemented model estimates the movement of and interactions among
raft-trip parties on the Colorado River within the Grand Canyon National Park.
This modeling system employs complementary statistical analyses and mathematical
models based upon exiting raft trip itinerary data, as well as data collected
during the 1998 rafting season. GCRTSim is an interactive computer simulation
system providing dynamic visual displays of individual raft-trip progress
as well as interactions among the multiple trips on the river. The goal of
this model system is to provide Park managers with an effective decision support
tool for representing and evaluating alternative trip/launch scheduling scenarios.
`

`Current research efforts include expanding the database by incorporating
additional river trip diaries collected in 1999 and 2000. In addition, the
computer program will be generalized to enable simulations to be run under
two different Glen Canyon dam flow regimes -- the "typical" flow rate represented
by the 1998 and 1999 seasons, and the recent "low" flow rate represented by
the 2000 season. Current and future work consists of a re-direction of the
Grand Canyon River Trip Simulation Project to new settings. I intend to investigate
the plethora of mathematical questions that have arisen out of this earlier
work. This research project will move away from the specific setting of the
Grand Canyon and, instead focus on the development of new integrated mathematical
and statistical techniques for modeling complex human-environment interaction
using artificial intelligence and other methods. `

`Mathematical modeling of human-environment interactions is quite new
-- from our perspective of simulation modeling, we are confident that this
work will represent a significant new contribution to science. Expanding this
project to encompass the broader question of exploring the interaction between
humans and the environment is, clearly, a new direction for my research. I
plan to capitalize on what I learned with the Colorado River project, but
I also recognize that the demands of this project require that I gain expertise
in new methodologies and new mathematical modeling techniques. My aim is to
accelerate my learning in this new area of research. This summer I spoke at
an international conference addressing new progress in Complex Modeling Systems.
I have also joined the Resource Modeling Association to help expand my network.
Social scientists and ecosystem management science has addressed some of the
issues related to this project, but the mathematical and statistical artificial-intelligence
based computer programming perspective is entirely new. `

`One aim of this research is to continue to involve undergraduates in
mathematical research. This project has already given six undergraduates (as
well as two graduate students) the valuable experience of working on a real-world
project. One student was so inspired by his work on this project that he is
now pursuing his doctoral degree in the applied math program at the University
of Colorado. Another student just had an article published by the Association
for Women in Science in a special issue devoted to young scientists. Techniques
used in developing this model include intelligent agent design, artificial
intelligence, fuzzy logic and statistical optimization algorithms. The simulation
engine uses object oriented programming (VisualBasic). The objective is to
develop a robust computational tool that not only simulates the current human-environment
scenarios but that can also be modified to consider the implications of various
policy changes in managing the resource. While we develop a simulation engine
along with an adaptive learning and statistical analysis module, we are developing
the artificial intelligence algorithms required for such complex, multi-variable
scenarios.
`

`STUDENT PARTICIPATION: The computer simulation
was designed and coded by undergraduates at Northern Arizona University
including Doug Stallman (Computer Science & Engineering, May 2000), Rian
Bogle (Mathematics, May 2001, and a master's in applied mathematics from University
of Colorado, Boulder in 2003), Rob Allred (Computer Science &
Engineering, May 2001) and Joanna Bieri (Mathematics and Physics & Astronomy,
May 2002 and doctorate in applied mathematics at Northwestern
University in 2009). A graduate student in Geology, Gary O'Brien, conducted a
study of beach site capacity in 2001. Funding was from the Grand
Canyon National Park, the NAU/NASA
Space Grant and the NAU Organized
Research and Hooper Awards programs.`

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