Symmetries and Differential Equations

挪威数学家Sophus Lie开创了对称性和微分方程（DE）的主题，目的是为了解决DE，特别是常微分方程（ODE）的大杂烩。 他基本上表明，几乎所有已知的技术都可以通过对称性分析系统化。 解决常微分方程的现代软件主要基于他的思想。常微分方程是描述曲线族的紧致方程（常微分方程的解）;偏微分方程（PDE）是描述曲面族的紧致方程（其解）。 人们可以在不知道DE的解的情况下找到它的对称性。 * DE几乎出现在所有应用领域的建模中，包括工程，科学和经济学。反过来，DE和/或其属性的解决方案提供了必要的预测和解释。 软件继续更新，以解决微分方程的分析和数字。 DE的分析方法基于找到对称性和/或守恒定律，然后使用对称性/CL来找到所提出的数据的解决方案，或者如果这是不可能的，找到相关数据的解决方案来检查数值方法的准确性。 一些最近的发展，数值方法解决DE纳入对称性和/或守恒定律的保护。 通常，不可能保持所有对称性和/或CL。 在这里，一个重要的未来研究领域是确定对于给定的数据，哪些对称性和/或CL应该被保留。我在对称性和DE领域工作了大约50年，我在这一领域的工作已经产生了四本被广泛引用的研究书籍，其中包括一名前博士生和三名前PDF。 所有这些都是由Springer出版的应用数学系列。我的工作大大扩展了李的工作，能够发现，计算和使用DE的对称性，使更多的DE可以解决。 这些扩展在中国、俄罗斯、美国和加拿大被广泛引用，并开发了许多新的软件来实现这些扩展。许多国家定期邀请我谈论我在东道国提供资金的情况下开展的工作。基本上，我的NSERC赠款的所有资金都用于支持学生和PDF。在这个提议中，目的是进一步扩展对称方法，以便可以解决或简化应用中出现的更多DE。特别是，我的工作涉及开发计算对称性的新空间。 在我所有的活动中，我培训了许多研究生和PDF-我以前的格拉德生/PDF中有很多人在加拿大工业界工作，包括量子计算，医疗诊断技术的开发，航空公司调度，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. **