Mathematical Sciences: Theory and Applications of Functionaland Partial Differential Equations

该项目继续研究涉及泛函和偏微分方程的理论和应用问题。研究一类具有状态相关有界时滞的泛函微分方程的稳定性和Hopf分岔问题。另一个研究方向涉及奇摄动非线性微分差分方程，其目标是证明一个脉冲周期解的存在性。涉及一类偏微分方程分布解存在性的几个问题。在接近直接应用的情况下，将对人口模型的动态进行研究。它包括对Volterra人口方程的推广，以包括人口的空间分布和出生过程的延迟。还将分析性传播疾病的疾病传播模型。这里，模型中的附加元素包含了风险-行为变化。微分方程是生物科学中数学建模的基础。涉及连续变化的现象，例如在运动、材料和能量中看到的现象，都遵循某些一般规律，这些规律可以用偏导数之间的相互作用和关系来表示。数学的关键作用不仅在于说明这些关系，而且还在于从中提取定性和定量的意义。

This project continues research on theoretical and applied problems involving functional and partial differential equations. Work will be done investigating the stability and Hopf bifurcation problems for a general class of functional differential equations with state-dependent bounded delay. Another line of study concerns singularly perturbed nonlinear differential difference equations where the goal is to show the existence of a pulse-like periodic solutions. Several problems involving the existence of distributional solutions of classes of partial differential equations. Closer to immediate applications, work will be done on the dynamics of population models. It includes a generalization of Volterra's population equation to include spatially distributed population and delay in the birth process. Models for disease transmission of sexually transmitted diseases will also be analyzed. Here the additional element in the model incorporates risk-behavior change. Differential equations form the backbone of mathematical modeling in the biological sciences. Phenomena which involve continuous change such as that seen in motion, materials and energy are known to obey certain general laws which are expressible in terms of the interactions and relationships between partial derivatives. The key role of mathematics is not only to state the relationships, but also to extract qualitative and quantitative meaning from them.