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: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=738377
• 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式(PDE)的形式表述，将物理参数与它们在时间和空间中的变化率联系起来。PDE模型产生于各种背景下，包括对物理连续体(例如，流体、等离子体、混合物、气体和固体)的描述。复杂系统的大多数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。所提出的研究涉及可系统地应用于分析、简化和求解非线性偏微分方程组的数学方法的发展。对于PDE模型，理想情况下，人们希望有一个显式公式方面的解决方案。这将允许以直接的方式进行计算、绘制图表和分析溶液行为。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组，通常的方法并不适用；研究人员往往必须想出特殊的方法，或者使用数值模拟，这可能很耗时，而且可能无法提供全局。本提案的理论部分致力于对偏微分方程组对称性理论的发展和扩展。对称方法适用于广泛的线性和非线性模型，它被用来系统地提取问题的重要信息，简化问题，构造精确解，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组，经典的对称性方法几乎没有产生有用的结果。在这些情况下，对称方法的扩展被证明是有益的。申请者将利用局部/非局部对称性、精确解和符号计算理论方面的专业知识和研究势头，以及加拿大和国际合作和HQP，进一步发展与对称有关的系统方法，特别是非局部和近似对称理论，以分析和解决以前难以解决的非线性问题。这项建议的第二部分涉及复杂PDE系统的对称性相关计算问题，其中需要计算机辅助计算。申请者是以枫树为基础的象征性宝石包的作者，许多国家的研究人员都在使用这种宝石。在即将到来的赠款周期中，它的目的是使GEM面向对象，从而实现更高的效率和更简单的语法，并实现额外的例程来计算与对称相关的数学对象，包括线性化算子、伴随和近似对称以及拉格朗日。目前的提案包括适用部分。申请者将与合作者一起研究波在各向异性弹性固体中传播的非线性模型，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组。