Mathematical Sciences: Some Nonlinear Problems In Differential and Integral Equations

在这个项目中，主要研究人员将分析涉及非线性微分方程组和积分方程组的四个问题。第一个问题是一个积分-微分方程式，它描述了楔子进入水浴的过程。第二个问题涉及使用同伦方法来研究应用中出现的积分和微分方程行波解的存在性。第三个问题涉及使用打靶法来研究混沌轨道的行为。最后，第四个问题涉及寻找一些二维形式的Navier-Stokes方程的相似解。许多描述日常现象行为的数学模型都涉及积分方程式或微分方程式或这两类方程式的组合。通常，定性地研究这类方程的解是最容易的，也就是说，人们寻找或多或少包含确切解的本质的解的近似表示，而不是寻找可能非常复杂的精确的解析或数值解。在这个项目中，主要研究人员将使用定性的方法来研究流体动力学、生物学和力学中出现的积分方程解和微分方程式的解。

In this project the principal investigators will analyze four problems involving nonlinear differential and integral equations. The first problem is an integro-differential equation that describes the entry of a wedge into a bath of water. The second problem involves the use of homotopy methods to study the existence of travelling wave solutions of integral and differential equations that arise in applications. The third problem involves the use of shooting methods to study the behavior of chaotic orbits. And finally the fourth problem concerns finding similarity solutions of some two-dimensional versions of the Navier-Stokes equations. Many of the mathematical models that describe the behavior of everyday phenomena involve integral equations or differential equations or combinations of these two types of equations. Very often it is easiest to study the solutions of such equations qualitatively, that is, instead of looking for exact analytical or numerical solutions that can be quite complicated, one looks for approximate representations of the solutions that contain, more or less, the essence of the exact solution. In this project the principal investigators will use qualitative methods to study the solutions of integral and differential equations that arise in fluid dynamics, biology and mechanics.