Applied Mathematics and Differential Equations

The Applied Mathematics and Differential Equations group within the Department of Mathematics have a great diversity of research interests, but a tying theme in each respective research program is its connection and relevance to problems or phenomena which occur in the engineering and physical sciences.

Faculty

George Avalos is interested in the mathematical control and numerical analysis of those coupled partial differential equations (PDE's) which model the control design of interactive structures by means of smart material technology. By coupled PDE's, we mean those which comprise a coupling of two or more disparate dynamics; e.g., a heat equation coupled to a wave equation. Physical systems which can be modeled by such classes of PDE's include structural, structural acoustic, thermal/structure and fluid/structure interaction systems. The consideration of control theory for these coupled PDE's leads to many interesting problems, owning to (i) the pointwise-distributed (and unbounded) nature of the actuators and sensors which are embedded within the structure; (ii) The intrinsic nature of the coupling between the PDE dynamics. The analysis and resolution of control problems involving these coupled models often require PDE/microlocal analytical techniques.

Steve Cohn is generally interested in applied math and applicable analysis; in particular, his research involves work in nonlinear partial differential equations. In his many research projects, he typically collaborates with scientists and engineers in other disciplines. In fact, at the moment he is collaborating with a colleague from the Department of Chemical Engineering on work involving a model of reaction propagation in solids. His tools of research include mathematical modeling, numerical experimentation, inverse scattering theory and stochastic processes. ( In this context, "inverse scattering theory" refers to a method of solving certain nonlinear partial differential equations.)

Bo Deng is interested in dynamical systems and their applications. One of his current projects involves the analysis of various chaos-generating mechanisms within a particular food-chain model. This research will hopefully culminate in a better understanding of issues surrounding biocomplexity. Another project deals with the mathematical modeling of certain types of neuron cells, with a view towards understanding the mathematical theory of neuron-to-neuron communication. The main mathematical tools which Professor Deng uses in his research activities are the qualitative theory of differential equations and techniques of nonlinear analysis.

Mikil Foss has research interests in the multi-dimensional calculus of variations and partial differential equations. More specifically, his work focuses on the regularity (smoothness) that can be expected for solutions to variational problems, elliptic partial differential equations and parabolic partial differential equations. These studies stem from a broad range of applications including continuum mechanics, for instance, nonlinear elasticity, and modeling of diffusion processes.

Adam Larios does research in the fields of partial differential equations, fluid dynamics, numerical analysis, and computational science.  He is especially interested in problems related to turbulence modeling, geophysics (ocean/atmospheric dynamics), and magnetohydrodynamics (MHD).  In particular, his work focuses on studying the mathematical well-posedness and large-time asymptotic behavior of models based on the Navier-Stokes equations, on developing and analyzing new turbulence models, on performing large-scale massively-parallel numerical simulations on supercomputers to validate these models, and on testing mathematical and scientific hypotheses computationally.  He is also interested in phase-field models, such as the Allen-Cahn and Cahn-Hilliard equations, and also problems related to fluid-structure interaction (FSI).

Glenn Ledder works in mathematical modeling for life sciences and physical sciences. His current interests include population dynamics and dynamic energy budget models. He is also active in developing an undergraduate mathematics curriculum for biology students and in mentoring REU student groups.

Allan Peterson is mainly interested in the general area of boundary value problems for ordinary differential equations, discrete dynamical systems (difference equations), and dynamic equations on time scales. In fact, he has recently written a book concerning dynamic equations on time scales. This theory combines difference equations and differential equations, and moreover generalizes to many other interesting problems. This work has applications to problems in biology and many other fields. Research into the general theory of dynamic equations on time scales originated in 1988; consequently there are many open questions still to be investigated.

Petronela Radu works in partial and integro-differential equations. She studies qualitative aspects of solutions (existence, uniqueness, regularity) as well as longtime behavior properties for systems that arise in areas of continuum mechanics such as elasticity, dynamic fracture, diffusion. Recently, with the emergence of nonlocal theories she has focused also on convergence of nonlocal problems to their classical counterparts.

Mohammad Rammaha has research interests in applied mathematics and analysis. In particular, his research deals with issues concerning nonlinear hyperbolic partial differential equations, including global existence, blow up and long time behaviour of solutions

Richard Rebarber does research in Distributed Parameter Control Theory and in Mathematical Ecology. His Control Theory research includes control design and analysis for abstract infinite dimensional systems, and for systems of partial differential equations, such as coupled systems of partial differential equations. He studies issues such as sampled-data control, tracking and disturbance rejection, zero dynamics, and robustness. These issues are generally well understood for finite-dimensional systems, but there are many interesting and difficult issues which arise for infinite-dimensional systems.

Brigitte Tenhumberg uses stochastic, discrete time models tailored to specific biological systems to advance the understanding of ecological processes. The models she uses include stochastic dynamic programming, matrix models, and agent based simulation models. One area of research emphasis is optimal decision making of animals (foraging or life history decisions) or humans (management of wildlife populations). Recent work addresses topics in invasion ecology, in particular understanding ecological mechanisms promoting ecosystem resistance to invasions.

Professor Emeritus Steve Dunbar has research interests in nonlinear differential equations, and applied dynamical systems, particularly those which arise in mathematical biology. In conjunction with his work with differential equation models and systems of mathematical biology, he is also interested in stochastic processes, the numerical and computer-aided solution of differential equations, and mathematical modeling. He also is interested in issues of mathematical education at the high school and collegiate level. He is the Director of the American Mathematics Competitions program of the Mathematical Association of America which sponsors middle school and high school mathematical competitions leading to the selection and training of the USA delegation to the annual International Mathematical Olympiad. In addition, he has interests in documenting trends in collegiate mathematics course enrollments and using mathematical software to teach and learn mathematics.

Professor Emeritus Lynn Erbe has been interested mainly in the general area of boundary value problems and oscillation theory for ordinary differential, functional, and dynamic equations on time scales. In particular, he has long been interested in the generalized Emden-Fowler, or Thomas-Fermi equation. Such equations arise in applications in astrophysics, engineering, and other areas of applied mathematics and physics. He has also long been interested in linear systems theory, oscillations, eigenvalue problems, and asymptotic behaviour of solutions.

Professor Emeritus Wendy Hines does research in dynamical systems. She is interested in the general theory and also applications to delay equations and partial differential equations. Currently she is working on a a reaction-diffusion equation with nonlocal diffusion which models gene propogation through a population. This is a very interesting problem as very little has be one on it and it defies the application of standard reaction-diffusion methods.

Professor Emeritus David Logan works in the area of applied mathematics and ecological modeling. His interests include ordinary and nonlinear partial differential equations and their application to mathematical ecology, including nutrient cycling, physiologically-structured population dynamics, and insect ecophysiology.

Current Graduate Students

Andrew Becklin
Advised by: Mohammad Rammaha and Daniel Toundykov

Scott Gensler
Advised by: Allan Peterson

Wei Hu
Advised by: Allan Peterson

Areeba Ikram
Advised by: Allan Peterson

Jessie Jamieson
Advised by: Daniel Toundykov

William Jamieson
Advised by: Richard Rebarber

Nicholas Kass
Advised by: Mohammad Rammaha

Nathan Poppelreiter
Advised by: Richard Rebarber and Brigitte Tenhumberg

Ariel Setniker
Advised by:Allan Peterson

Kelsey Wells
Advised by: Petronela Radu

Laura White
Advised by: Petronela Radu

Cory Wright
Advised by: Petronela Radu and Mikil Foss

Recent Graduates

Kevin Ahrendt (PhD 2017)
Advised by: Allan Peterson

Christina Edholm (PhD 2016)
Advised by: Richard Rebarber and Brigitte Tenhumberg

Julia St. Goar (PhD 2016)
Advised by: Allan Peterson

Sara Reynolds (PhD 2015)
Advised by: Glenn Ledder and Chad Brassil

Jeremy Trageser (PhD 2015)
Advised by: Petronela Radu and Daniel Toundykov

Abigail Brackins (PhD 2014)
Advised by: Lynn Erbe and Allan Peterson

Tom Clark (PhD 2014)
Advised by: George Avalos

Pei Pei (PhD 2014)
Advised by: Mohammad Rammaha and Daniel Toundykov

Tanner Auch (PhD 2013)
Advised by: Lynn Erbe and Allan Peterson

Pushp Awasthi (PhD 2013)
Advised by: Lynn Erbe and Allan Peterson

Joe Geisbauer (PhD 2013)
Advised by: Mikil Foss

Ben Nolting (PhD 2013)
Advised by: David Logan

Khulud Alyousef (PhD 2012)
Advised by: Lynn Erbe and Allan Peterson

Eric Eager (PhD 2012)
Advised by: Richard Rebarber

Christopher Goodrich (PhD 2012)
Advised by: Lynn Erbe and Allan Peterson

Yangiu Guo (PhD 2012)
Advised by: Mohammad Rammaha