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 that 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 (PDEs) that model the control design of interactive structures by means of smart material technology. By coupled PDEs, we mean those that comprise a coupling of two or more disparate dynamics, e.g., a heat equation coupled to a wave equation. Physical systems that can be modeled by such classes of PDEs include structural, structural-acoustic, thermal/structure, and fluid/structure interaction systems. The consideration of control theory for these coupled PDEs 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.

Animesh Biswas is interested in the area of partial differential equations (PDEs) and harmonic analysis. Specifically, he studies the regularity of solutions for nonlocal elliptic and parabolic equations. These problems occur in many different physical applications, such as semipermeable membrane problems, in the phenomenon of osmosis, and biological population models. He is also interested in the nonlinear PDEs and free boundary problems. In particular, his work focuses on studying the nonlocal form of Monge-Ampere equations. More information can be found on his website https://sites.google.com/view/animesh-biswas/home. 

Javier Cueto's research interests include Calculus of variations and the analysis of nonlocal problems, which are driven by applications, mainly those related to continuum mechanics and engineering, but also other areas, usually related to integro-partial differential equations. In particular, recent work has focused on studying nonlocal frameworks suitable for nonlinear elasticity, motivated by the theory of peridynamics, as well as the localization of these operators and models to their counterparts in the classical (local) case. This involves the study of fractional and nonlocal gradients.

Associate Professor Emeritus 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 that 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 problems related to fluid-structure interaction (FSI).

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, and diffusion. Recently, with the emergence of nonlocal theories, she has also focused on convergence of nonlocal problems to their classical counterparts.

Professor Emeritus 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-term behavior 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 that 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 that 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 an interest 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 the asymptotic behavior of solutions.

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

Professor Emeritus 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.

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.

Professor Emeritus 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.

Current Graduate Students

Molly Creager
Advised by: Richard Rebarber and Brigitte Tenhumberg

Matt Enlow
Advised by: Adam Larios

Alex Heitzman
Advised by: Mikil Foss

Scott Hootman-Ng
Advised by: Petronela Radu

Dylan McKnight
Advised by: George Avalos and Mohammad Rammaha

Sara (Myers) McKnight
Advised by: George Avalos

Isabel Safarik
Advised by: Adam Larios

Lawrence Seminario-Romero
Advised by: Richard Rebarber and Brigitte Tenhumberg

Anh Vo
Advised by: Petronela Radu