Asymptotic and qualitive analysis of nonlinear partial differential equations

非线性偏微分方程的渐近和定性分析

• 批准号: 249732-2008
• 负责人: Amundsen, David
• 金额: $ 1.09万
• 依托单位: Carleton University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2012
• 资助国家: 加拿大
• 起止时间: 2012-01-01 至 2013-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=503733
• 关键词: Asymptotic; qualitive; analysis; nonlinear; partial

Partial differential equations provide a mathematical representation of the relationship between the temporal and spatial variation of an unknown function or set of functions. Such equations arise in a natural way in numerous settings including fluid motion, chemical reactions, developmental biology, and spatial ecology. In general such equations involve nonlinear effects corresponding to, for example, underlying saturation effects and interactions between model components. Such effects while making the models more realistic, also make the solution of such equations much more challenging and indeed exact closed form solutions are typically not accessible except in certain idealized circumstances. Therefore alternative methodologies based on approximation and analysis of qualitative features in certain limiting circumstances become essential in understanding the nature of the models, their solutions and in turn the underlying physical system. Within this context, this proposal considers two classes of nonlinear partial differential equations and the analytic and computational methodologies used to study them. First I intend to continue an ongoing study of resonant solutions of nonlinear wave equations which underly waves in shallow fluids such as atmospheric flows over topography. The key goals involve both the study of solutions in generalized circumstances, as well as the development of the analytic methodology which in turn may be applicable to a wider range of problems. Second is the study of a widely applicable and increasingly prevalent class of equations which arise in the context of chemical and biological applications involving the combination of diffusive and reaction processes. The focus of this aspect of the study will be to study in general the robustness of certain solution features such as pattern formation, waves and long time behaviour under the influence of perturbative effects. In addition with the growing complexity of the models, numerical approximation of solutions is becoming an increasingly crucial analytic tool. As such this study will also investigate and develop more efficient computational methodologies suited for models of this type.

偏微分方程提供了数学表示未知函数或一组函数的空间变化之间关系的数学表示。这种方程在许多环境中以自然的方式出现，包括流体运动，化学反应，发育生物学和空间生态学。通常，此类方程涉及与类似物之间的基本饱和效应和相互作用相对应的非线性效应。这种效果在使模型更加现实时，还可以使这些方程式更具挑战性，并且确实是确切的封闭式解决方案，通常在某些理想的情况下也无法访问。因此，基于在某些限制情况下定性特征的近似和分析的替代方法对于理解模型的性质，其解决方案和基础物理系统而言至关重要。在这种情况下，该提案考虑了两类非线性偏微分方程以及用于研究它们的分析和计算方法。首先，我打算继续对非线性波方程的谐振溶液进行持续研究，这些溶液在浅流体中的基础波（例如大气流过地图上的大气流中）。关键目标既涉及在普遍情况下对解决方案的研究，又涉及分析方法的发展，而分析方法又可能适用于更广泛的问题。其次是研究广泛适用且日益普遍的方程类别的研究，该方程在涉及扩散和反应过程组合的化学和生物应用中产生。该研究的这一方面的重点将是一般研究某些解决方案特征的鲁棒性，例如模式形成，波浪和长时间行为，在扰动效应的影响下。除了模型的日益增长的复杂性外，解决方案的数值近似正在成为越来越重要的分析工具。因此，本研究还将调查并开发适合此类模型的更有效的计算方法。