Ordinary Differential Equations, Partial Differential Equations and Analysis
Faculty conduct research on theoretical and numerical issues for a variety of partial differential equations: semilinear parabolic equations including semigroup theory, elliptic equations, hyperbolic systems including systems of conservation laws, and dispersive equations. A major topic of interest is traveling waves and other aspects of wave propagation. Applications include granular flow, thin liquid films, flow in porous media, shallow water theory, shock waves, electrodynamics and nonlinear optics.
Numerical methods are important in both ordinary and partial differential equations, and many projects include novel innovations or fundamental advances in numerical analysis. For nonlinear ordinary differential equations and dynamical systems, issues of stability and bifurcation are addressed with numerical methods, and asymptotic methods are also used, especially singular perturbation techniques. Applications to geometric mechanics include nonholonomic systems.
Research topics related to control theory and applications include stabilization and controllability of partial differential equations, control of systems of ordinary differential equations, and inverse problems for partial differential equations. Applications are in nonlinear acoustics, constrained mechanical systems, control of Navier-Stokes equations, optimal control and material failure detection.
Partial Differential Equations: qualitative and quantitative properties of solutions to systems of PDEs, as well as associated stabilization and controllability issues. Applications of these problems range from nonlinear acoustics to fluid dynamics.
Implicit systems of ordinary differential equations, including numerical algorithms and control; applications to constrained mechanical systems, optimal control, and failure detection. Simulation and modeling.
Professor, Associate Director for CRSC, Department Head
Numerical methods for time-dependent partial differential equations, hyperbolic conservation laws, degenerate parabolic equations, numerical analysis, scientific computing.
Numerical ordinary differential equations, numerical linear algebra, dynamical systems, inverse problems.
Convex analysis, convex optimization, monotone operators, computational data science, applied nonlinear analysis.
Optimal control and inverse problems for partial differential equations, control of Navier-Stokes equations, numerical partial differential equations, nonlinear semigroup theory, dynamical systems in Banach spaces, stochastic differential equations and applications, applied functional analysis.
Numerical analysis and scientific computing; numerical methods for partial differential equations involving free boundary and moving interface problems, and problems on irregular domains, finite difference and finite element methods; CFD, and biological flows.
Dynamical systems generated by ordinary, delay, and partial differential equations; stability, bifurcation, singular perturbation.
Nonlinear ordinary differential equations, semi- linear parabolic and elliptic partial differential equations.
Teaching Assistant Professor
Dynamical systems, differential equations.
Tien Khai Nguyen
Nonlinear PDEs, optimal control problems and differential Games, nonsmooth analysis and geometric measure theory.
Geometric singular perturbation theory, existence and stability of traveling waves.
Numerical analysis of partial differential equations and scientific computation with applications to fluid flows and wave propagation, including acoustics, electromagnetism, optics, and plasma. Inverse problems, including active control of sound and radar imaging.