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

9610076 Chen主要研究包含某些非线性类型的椭圆型、抛物型和双曲型二阶偏微分方程。目标是：(a)探索具有非线性边界条件的一维波动方程的时空混沌振动现象；(b)利用边界元和拟单调迭代技术分析和计算具有点观测和不等式控制约束的受控耦合半线性椭圆型和抛物型系统；(c)通过改变所涉及的几何形状来可视化环形和哑铃形状域的状态和控制，以便了解控制效果。两位研究人员将开发有效和创新的理论以及数值工具来研究所涉及的非线性问题。这项研究有望通过可视化来增强对机械振动、声学、数学生物学、化学反应器和生态学等许多领域复杂非线性现象和模式形成的理解。这反过来又将进一步有助于非线性科学的新数学技术的发展。

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.