Symmetry methods and their applications to the analysis of modern mathematical models

对称方法及其在现代数学模型分析中的应用

• 批准号: RGPIN-2019-05570
• 负责人: Shevyakov, Alexey
• 金额: $ 1.24万
• 依托单位: University of Saskatchewan
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=675852
• 关键词: Symmetry; methods; their; applications; analysis

Mathematical models of physical phenomena are often stated in terms of partial differential equations (PDE), relating physical parameters to their rates of change in time and space. PDE models arise in a variety of contexts, including the description of physical continua (e.g., fluids, plasmas, mixtures, gases, and solid bodies). The majority of mathematical models of complex systems are nonlinear: a sum of two solutions of governing equations is not a solution. Examples of nonlinear phenomena include water waves, large deformations of elastic solids, and turbulent fluid flows.******The proposed research is concerned with the development of mathematical methods that can be systematically applied to analyze, simplify, and solve nonlinear PDEs. For a PDE model, one ideally would like to have a solution in terms of explicit formulas. This would allow to make computations, plot graphs, and analyze solution behaviour in a direct way. For linear models, many classical solution methods exist, but for nonlinear PDEs, usual methods are not applicable; researchers often must come up with special approaches, or use numerical simulations, which can be time consuming, and may not provide a "big picture".******The theoretical part of this proposal is devoted to the development and extension of the theory of symmetries of PDEs. Applicable to wide classes of linear and nonlinear models, symmetry methods are used to systematically extract important information about the problem, simplify it, construct exact solutions, and in some cases, completely solve a given nonlinear problem. For some PDE systems of interest, however, classical symmetry methods yield few useful results. In these cases, extensions of symmetry methods prove beneficial. The applicant will capitalize on expertise and research momentum in theory of local/nonlocal symmetries, exact solutions, and symbolic computations, as well as on Canadian and international collaborations and HQP, to develop further systematic symmetry-related methods, in particular, the theory of nonlocal and approximate symmetries, to analyze and solve previously intractable nonlinear problems.******The second part of this proposal has to do with the problem of symmetry-related computations for complex PDE systems, where computer-aided calculations are required. The applicant is the author of a Maple-based symbolic package GeM, used by researchers in many countries. In the upcoming grant cycle, it is intended to make GeM object-oriented, allowing for higher efficiency and simpler syntax, and also to implement additional routines to compute symmetry-related mathematical objects, including linearlization operators, adjoint and approximate symmetries, and Lagrangians.******The current proposal includes an applied part. With collaborators, the applicant will work on nonlinear models of wave propagation in anisotropic elastic solids, and models pertaining to turbulent flows, and study the corresponding PDEs using advanced symmetry methods. **

物理现象的数学模型通常以偏微分方程（PDE）的形式表示，将物理参数与它们在时间和空间中的变化率联系起来。PDE模型出现在各种上下文中，包括物理连续体的描述（例如，流体、等离子体、混合物、气体和固体）。大多数复杂系统的数学模型都是非线性的：控制方程的两个解之和不是一个解。非线性现象的例子包括水波、弹性固体的大变形和湍流。拟议的研究是关注的数学方法，可以系统地应用于分析，简化和解决非线性偏微分方程的发展。对于偏微分方程模型，理想情况下，人们希望有一个明确的公式的解决方案。这将允许以直接的方式进行计算，绘制图表和分析解决方案的行为。对于线性模型，存在许多经典的求解方法，但对于非线性偏微分方程，通常的方法是不适用的;研究人员往往必须想出特殊的方法，或使用数值模拟，这可能是耗时的，可能不会提供一个“大局”。这个建议的理论部分致力于发展和扩展偏微分方程对称性理论。适用于广泛的线性和非线性模型，对称性方法用于系统地提取有关问题的重要信息，简化它，构造精确解，并在某些情况下，完全解决给定的非线性问题。然而，对于一些感兴趣的偏微分方程系统，经典的对称性方法产生一些有用的结果。在这些情况下，对称方法的扩展证明是有益的。申请人将利用本地/非本地对称性，精确解和符号计算理论的专业知识和研究势头，以及加拿大和国际合作和HQP，进一步开发系统的非线性相关方法，特别是非本地和近似对称性理论，以分析和解决以前棘手的非线性问题。本建议的第二部分涉及复杂偏微分方程系统的相关计算问题，其中需要计算机辅助计算。申请人是基于Maple的符号包GeM的作者，许多国家的研究人员都在使用。在即将到来的资助周期中，它旨在使GeM面向对象，允许更高的效率和更简单的语法，并实现额外的例程来计算与几何相关的数学对象，包括线性化运算符，伴随和近似对称，以及拉格朗日。目前的提案包括应用部分。与合作者一起，申请人将致力于各向异性弹性固体中波传播的非线性模型，以及与湍流有关的模型，并使用先进的对称方法研究相应的偏微分方程。**