Mathematical Sciences: Analysis, Control, Dynamics and Visualization of Distributed Parameter Systems Containing Nonlinearities

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9610076 Chen The principal investigators will study second order partial differential equations of elliptic, parabolic and hyperbolic types containing certain types of nonlinearities. The objectives are: (a) The exploration of spatio-temporal chaotic vibration phenomena for the one dimensional wave equation with nonlinear boundary conditions; (b) The use of boundary elements and quasimonotone iteration techniques to analyze and compute controlled coupled semilinear elliptic and parabolic systems with point observations and inequality control constraints; (c) Visualization of state and control on annular and dumbbell-shaped domains by varying the geometries involved in order to understand the control effects. The two investigators will develop effective and innovative theoretical as well as numerical tools to study the nonlinear problems involved. This research promises to enhance, through visualization, the understanding of complex nonlinear phenomena and pattern formation in many fields such as mechanical vibration, acoustics, mathematical biology, chemostat and chemical reactors, and ecology. This in turn will further aid in the development of new mathematical techniques for nonlinear sciences.