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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.