Mathematical Sciences: Research in Differential Equations

9302737 Hastings这个项目是关于非线性常微分方程的数学问题的研究。目前有六个领域正在调查中。它们包括由逆散射获得的解的渐近性和痛苦级超越，应用于混沌研究的射击方法，特别是在洛伦兹方程的分析中，以及Barenblatt湍流爆发衰减的四阶模型的自相似解。工作还将进行扩展最近的研究稳定性在模拟固态扩散在半导体中，在五阶方程的所有阶的渐近性和在解析函数的渐近展开的斯托克斯不连续的平滑。微分方程是物理世界数学建模的基础。数学分析的作用与其说是创建方程，不如说是提供关于解的定性和定量信息。这可能包括回答关于独特性、平滑性和增长性的问题。此外，分析常常发展出解的近似方法和对这些近似精度的估计。＊＊＊

9302737 Hastings This project concerns investigations into mathematical problems of nonlinear ordinary differential equations. There are six areas under investigation. They include asymptotics and painleve transcendents involving solutions obtained by inverse scattering, shooting methods applied to the study of chaos, specially in the analysis of Lorenz equations, and self-similar solutions of Barenblatt's fourth order model for the decay of turbulent bursts. Work will also be done expanding recent studies of stability in modeling solid state diffusion in semiconductors, asymptotics beyond all orders in fifth order equations and on the smoothing of Stokes discontinuities in the asymptotic expansions of analytic functions. Differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. ***