Applied Hyperbolic Partial Differential Equations

应用双曲偏微分方程

• 批准号: 0405899
• 负责人: Jeffrey Rauch
• 金额: $ 12.52万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0405899&HistoricalAwards=false
• 关键词: Applied; Hyperbolic; Partial; Differential; Equations

Abstract: 0405899, J. Rauch, University of MichiganApplied Hyperbolic Partial Differential Equations The principal investigator will investigate mathematical problemsinvolving hyperbolic partial differential equations. These equationsappear in the description of waves which propagate with finite speed,for example acoustics, electromagnetism, fluid and aero dynamics,plasmas, lasers, telegraphy, radio, television, cell phones, shockwaves, music, power lines, .... etc. Typical mathematical problemsinvolve showing that specific problems, motivated by applications,have solutions and studying their qualitative properties, for examplestability and methods for effective computation. A first set ofproblems in the proposal concerns the damping of acoustic waves byenclosing the acoustic cavity in a diffusive medium. Two situationseasy to understand are the protection of pilots and passengers inairplanes from the noise of the engines and the absorption of sound ina submarine so that it will not escape to the sea. There are alsoapplications to the automatic control of mechanical systems immersedin fluids. A second problem concerns the scattering of solutions of astrongly nonlinearly damped wave equation. A third problem is to showthat the phenomenon of continuum generation in focused laser pulses isa consequence of nonlinear electromagnetic models without ionization.Self focused laser beams exhibit continuum generation. The band offrequencies present in the focused beam is much broader and spans acontinuum of frequencies (in contrast to a discrete set) than beforethe focusing. Current explanations are frankly not very convincing.This is a fundamental problem long recognized in the laser physicscommunity. It is a place where mathematics may add importantinsights. Focused beams are used in many situations including asurprising application to reduce resistance in hypersonic aircraft.The proposer will study the behavior of solutions of the differentialequations that govern the propagation of waves. One probleminvestigated is how well sound can be excluded from a region byenclosing the region is an absorbing layer. Here the absorption inthe layer is supposed to be of a diffusive nature, like one would havefor a gel. Applications include sonic insulation of airplanefuselages and submarines and the control of immersed mechanicalsystems. A second problem to be studied is that of white lightgeneration in focused lasers. When high power lasers focus, theircolor changes from the color of the carrier frequency to nearly whitelight. There are a variety of explanations of this, none veryconvincing. The PI will try to find a better one. There is convincingexperimental evidence that the phenomenon does not require ionizationof the propagation medium and this will help to keep the equationsunder study manageable and well founded.

摘要：0405899，J. Rauch，密歇根大学应用双曲偏微分方程 首席研究员将研究涉及双曲偏微分方程的数学问题。 这些方程出现在以有限速度传播的波的描述中，例如声学、电磁学、流体和空气动力学、等离子体、激光、电报、无线电、电视、手机、冲击波、音乐、电力线等。典型的数学问题涉及显示由应用驱动的特定问题有解决方案并研究它们的定性性质，例如稳定性和方法 以进行有效的计算。 该提案中的第一组问题涉及通过将声腔封闭在扩散介质中来阻尼声波。 有两种情况很容易理解，一是保护飞机上的飞行员和乘客免​​受发动机噪音的影响，二是吸收潜艇中的声音，使其不会逃入大海。 还应用于浸没在流体中的机械系统的自动控制。 第二个问题涉及强非线性阻尼波动方程解的散射。 第三个问题是证明聚焦激光脉冲中的连续谱生成现象是没有电离的非线性电磁模型的结果。自聚焦激光束表现出连续谱生成。 聚焦光束中存在的频带频率比聚焦之前更宽，并且跨越连续频率（与离散组相反）。 坦率地说，目前的解释不太令人信服。这是激光物理界长期以来认识到的一个基本问题。 这是数学可以添加重要见解的地方。 聚焦光束用于许多情况，包括减少高超音速飞机阻力的令人惊讶的应用。提议者将研究控制波传播的微分方程的解的行为。 研究的一个问题是，通过用吸音层包围该区域，可以将声音从该区域中排除出去。 这里，层中的吸收应该具有扩散性质，就像凝胶一样。 应用包括飞机机身和潜艇的隔音以及浸入式机械系统的控制。 要研究的第二个问题是聚焦激光器中白光的产生。 当高功率激光聚焦时，它们的颜色从载波频率的颜色变成接近白光的颜色。 对此有多种解释，但没有一个很有说服力。 PI 将尝试寻找更好的。 有令人信服的实验证据表明，该现象不需要传播介质电离，这将有助于保持所研究的方程易于管理且有充分依据。