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: 2009
• 资助国家: 加拿大
• 起止时间: 2009-01-01 至 2010-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=426049
• 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.

偏微分方程提供了未知函数或函数集的时间和空间变化之间关系的数学表示。这些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。一般而言，此类方程涉及对应于例如潜在饱和效应和模型组件之间的相互作用的非线性效应。这种效应虽然使模型更加真实，但也使此类方程的求解更具挑战性，并且实际上，除非在某些理想化情况下，否则通常无法获得精确的封闭形式解。因此，在某些限制情况下基于定性特征的近似和分析的替代方法对于理解模型的性质、它们的解决方案以及底层物理系统变得至关重要。在此背景下，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，我打算继续研究非线性波动方程的共振解，该方程是浅层流体（例如地形上的大气流动）中的波浪的基础。主要目标包括研究一般情况下的解决方案，以及开发分析方法，从而适用于更广泛的问题。其次是对广泛适用且日益流行的一类方程的研究，这些方程出现在涉及扩散和反应过程组合的化学和生物应用中。这方面研究的重点是总体上研究某些解决方案特征的鲁棒性，例如在扰动效应影响下的模式形成、波浪和长时间行为。此外，随着模型日益复杂，解的数值近似正成为越来越重要的分析工具。因此，本研究还将研究和开发适合此类模型的更有效的计算方法。