Mathematical Sciences: Differential Equations

数学科学：微分方程

• 批准号: 8701356
• 负责人: Stuart Hastings
• 金额: $ 13万
• 依托单位: University of Pittsburgh
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-07-01 至 1989-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8701356&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Differential; Equations

Work will be done on problems involving nonlinear differential equations. The work will focus on three areas. In the first, investigations will be carried out on the relation between Painleve transcendents and certain linear integral equations arising in inverse scattering theory. Certain ordinary differential equations in the complex plane have only poles for singularities. These equations, or transcendents, are related to nonlinear partial differential equations solvable by the method of inverse scattering. At this time only selected examples illustrate the connection but it is clear that an entire class of equations can be treated. A systematic study of the entire relationship will be undertaken. A second line of research deals with the failure or blow-up of solutions of nonlinear diffusion equations. Blow-up of solutions depends heavily on the nature of initial conditions (as well as the nonlinearity term in the equation). Since one is often dealing with functions of several space variables, blow-up can occur at different points at different times. This leads to the main conjecture of the project: blow-up must be confined to a set of space variables of negligible size (measure zero). Third, work will be done on existence, multiplicity and asymptotics for combustion theory models with complex chemistry including radicals. The models represent steady planar flame fronts in premixed combustion. This analysis goes beyond traditional simple reactions. Most models have to confront or hypothesize away what is known as the cold boundary difficulty to support traveling wave solutions. If intermediate species or radicals are introduced into a system of reactions, a scheme of the principal investigator leads to new existence proofs for flame layers. Work will proceed in expanding methods developed to cover other complex models, validate their asymptotic expansions and consider problems of multiplicity or uniqueness.

工作将涉及非线性微分问题 方程 这项工作将集中在三个领域。 在第一个， 将对Painleve之间的关系进行调查 超越和某些线性积分方程的逆 散射理论 中的某些常微分方程 复平面只有奇点的极点。 这些方程，或 超越，与非线性偏微分方程有关 可以用逆散射的方法来解决。 此时才 选定的例子说明了这种联系，但很明显， 可以处理整个方程组。 系统研究 整个关系都将得到处理。 第二类研究涉及的是 非线性扩散方程的解 解的爆破 在很大程度上取决于初始条件的性质（以及 方程中的非线性项）。 因为一个人经常要处理 函数的几个空间变量，爆破可以发生在不同的 在不同的时间点。 这导致了主要的猜想， 项目：爆破必须局限于一组空间变量， 可忽略的尺寸（测量为零）。 第三，将在存在性、多样性和 复杂化学燃烧理论模型的渐近性 包括激进分子 该模型代表稳定的平面火焰前锋 在预混燃烧中。 这种分析超越了传统的简单 反应. 大多数模型必须面对或假设什么是 称为冷边界困难，以支持行波 解决方案 如果中间物质或自由基被引入到 反应系统，主要研究者的计划导致 火焰层存在的新证据。 将继续扩大 方法开发，以涵盖其他复杂的模型，验证其 渐近展开，并考虑多重性问题或 唯一性