Numerical Spectral Analysis and Approximation of Functional Traveling Waves

微分方程在科学和工程的许多领域中被用作物理和生物现象的模型。 研究者和他的同事们在这个建议中的重点是微分方程解的近似。研究人员感兴趣的是分析和计算的稳定性谱（点谱的微分和差分算子，Sacker-Sell谱，和李雅普诺夫指数）和技术的分析和近似离散模型类似于时间依赖的偏微分方程，但与差分算子，而不是空间微分算子。 离散模型在物理和生物系统的建模中起着重要的作用。 特别感兴趣的是行波解的格子微分方程。所采取的方法是将联合收割机动力系统和数值分析思想与普通、偏微分方程和格点微分方程的建模和分析相结合。 该项目关注的是开发和分析高效，准确的数值技术，这些技术对动力系统的计算和分析非常有用。 Sacker-Sell和Lyapunov谱区间是特征值的真实的部分的自然类似物，为时变微分方程提供稳定性信息。 研究人员开发，分析，并证明使用数值技术的近似这些频谱间隔。 一套计算模块的稳定性信息的计算和功能行波正在开发中。它是由分析的数值技术的形式，应证明有用的工作科学家和工程师。调查员和他的同事考虑的问题，在近似和计算的解决方案的微分方程。微分方程通常用于在科学和工程的许多领域中模拟物理和生物现象。 微分方程是一种规则，是解与解的变化率之间的关系，它决定了初始配置如何演变为未来配置。 这个项目的重点是近似的李雅普诺夫指数和相关的数量，提供信息的稳定性，附近的配置发展和保持附近的趋势，和不稳定性，附近的配置移动的趋势分开。这种类型的分析有助于理解环境中发生的复杂生物现象，并识别天气预测模型等的不稳定性。格点微分方程的分析与计算，即，在空间上离散而在时间上连续的微分方程在其中空间分量自然离散的物理和生物系统的建模中是重要的，特别是对于材料和生理学中的微观模型。

Differential equations are used as models of physical and biological phenomena in many areas of science and engineering. The focus of the investigator and his colleagues in this proposal is on the approximation of solutions of differential equations. The investigator is interested in the analysis and computation of stability spectra (point spectrum of differential and difference operators, Sacker-Sell spectrum, and Lyapunov exponents) and techniques for analysis and approximation of discrete models similar to time dependent partial differential equations but with a difference operator instead of a spatial differential operator. Discrete models play a prominent role in the modeling of physical and biological systems. Of particular interest are traveling wave solutions of lattice differential equations. The approach taken is to combine dynamical systems and numerical analysis ideas with the modeling and analysis of ordinary, partial, and lattice differential equations. This project is concerned with the development and analysis of efficient, accurate numerical techniques that are useful for the computation and analysis of dynamical systems. Sacker-Sell and Lyapunov spectral intervals are natural analogues of the real parts of the eigenvalues that provide stability information for time varying differential equations. The investigator develops, analyzes, and justifies the use of numerical techniques for the approximation of these spectral intervals. A suite of computational modules for the computation of stability information and for functional traveling waves is being developed. It is backed by analysis of the numerical techniques in a form that should prove useful to working scientists and engineers.The investigator and his colleagues consider issues in the approximation and computation of solutions of differential equations. Differential equations are commonly used to model physical and biological phenomena in many areas of science and engineering. A differential equation is a rule, a relationship between the solution and the rate of change of the solution, that determines how an initial configuration evolves into future configurations. The focus of this project is on the approximation of Lyapunov exponents and related quantities that provide information on stability, the tendency for nearby configurations to evolve and stay nearby, and instability, the tendency for nearby configurations to move apart. This type of analysis is useful in understanding complex biological phenomena that occur in the environment and in identifying instabilities in, for example, models of weather prediction. The analysis and computation of lattice differential equations, i.e., differential equations that are discrete in space and continuous in time, are important in the modeling of physical and biological systems in which the spatial component is naturally discrete, in particular for microscopic models in materials and physiology.