Symmetries and Differential Equations

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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. **