Symmetries and Differential Equations

挪威数学家Sophus Lie开创了对称与微分方程（DEs）这一学科，其目的是将求解微分方程（尤其是常微分方程）的特殊技术的大杂烩整理出来。他基本上表明，几乎所有已知的技术都可以通过对称分析系统化。解决ode的现代软件很大程度上是基于他的想法。ODE是描述曲线族（ODE的解）的紧化方程；偏微分方程（PDE）是描述一类曲面（其解）的紧化方程。人们可以在不知道微分方程解的情况下找到它的对称性。de出现在几乎所有应用领域的建模中，包括工程、科学和经济学。反过来，微分方程的解和/或它们的性质提供了基本的预测和解释。软件继续更新，以解决微分方程的解析和数值。微分方程的分析方法是基于寻找对称性和/或守恒定律，然后使用对称性/守恒定律找到所提出数据的解，或者，如果这是不可能的，找到相关数据的解来检查数值方法的准确性。最近在求解偏微分方程的数值方法中有了一些进展，这些进展包括对称性的保持和/或守恒定律。通常，不可能保留所有的对称性和/或CLs。在这里，一个重要的未来研究领域是确定对于给定的数据应该保留哪些对称性和/或CLs。***我在对称和DEs领域工作了大约50年，我在这个领域的工作产生了四本被广泛引用的研究书籍，其中包括一位前博士生和三份前pdf文件。所有这些都被b施普林格发表在其应用数学系列中。我的工作大大扩展了Lie的工作，能够发现，计算和使用DEs的对称性，从而可以解决更多的DEs。这些扩展已经被广泛引用，并且在中国、俄罗斯、美国和加拿大已经开发了许多新的软件来实现这些扩展。我经常受到许多国家的邀请，在东道国提供的资助下谈论我的工作。基本上我所有的NSERC基金都用于支持学生和pdf。***本建议的目的是进一步扩展对称方法，以便解决或简化应用中出现的更多微分方程。特别是，我的工作涉及到为计算对称性开发新的空间。在我所有的活动中，我培养了许多研究生和pdf -我以前的许多研究生/ pdf都在加拿大工业界工作，包括量子计算，医疗诊断技术的发展，航空调度，卫星成像和风险分析。**

The Norwegian mathematician Sophus Lie pioneered the subject of symmetries and differential equations (DEs) with the aim of putting order to the hodgepodge of ad-hoc techniques for solving DEs, especially ordinary differential equations (ODEs). He essentially showed that almost all known techniques could be systematized through symmetry analysis. Modern software for solving ODEs are based largely on his ideas.***An ODE is a compact equation describing a family of curves (the solutions of the ODE); a partial differential equation (PDE) is a compact equation for describing a family of surfaces (its solutions). One can find the symmetries of a DE without knowing its solutions. ***DEs arise in modelling in almost all applied fields, including engineering, science and economics. In turn, solutions of the DEs and/or their properties provide essential predictions and interpretations. Software continues to be updated for solving differential equations both analytically and numerically. Analytical methods for DEs are based on finding symmetries and/or conservation laws and then using symmetries/CLs to find solutions for posed data or, if this is not possible, to find solutions for related data to check accuracy of numerical methods. Some recent developments in numerical methods for solving DEs incorporate the preservation of symmetries and/or conservation laws. Normally, it is not possible to preserve all symmetries and/or CLs. Here an important future research area is to determine which symmetries and/or CLs should be preserved for given data.***I have worked in the field of symmetries and DEs for about 50 years and my work in this field has resulted in four well-cited research books with co-authors that have included a former PhD student and three former PDFs. All have been published by Springer in its Applied Math series. My work has significantly extended the work of Lie in being able to discover, calculate and use symmetries of DEs so that more DEs can be solved. These extensions have been well-cited and much new software has been developed to implement these extensions in China, Russia, USA and Canada. I am regularly invited by many countries to talk about my work with funding provided by the host country. Essentially all funding from my NSERC grant is used for supporting students and PDFs.***In this proposal, the aim is to further extend symmetry methods so that more DEs arising in applications can be solved or simplified. In particular, my work involves developing new spaces for computing symmetries. In all of my activities I train many graduate students and PDFs--a large number of my former grad students/PDFs have worked in Canadian industry, including quantum computing, development of medical diagnostic techniques, airline scheduling, satellite imaging, and risk analysis. **