引用本文: 黄云帆, 王沫然. 力学与数理融合−浅谈相似性解方法. 力学与实践, 2024, 46(4): 1-8.
Huang Yunfan, Wang Moran. Merging of mechanics and mathematical physics——a brief discussion on similarity method. Mechanics in Engineering, 2024, 46(4): 1-8.
 Citation: Huang Yunfan, Wang Moran. Merging of mechanics and mathematical physics——a brief discussion on similarity method. Mechanics in Engineering, 2024, 46(4): 1-8.

## MERGING OF MECHANICS AND MATHEMATICAL PHYSICS——A BRIEF DISCUSSION ON SIMILARITY METHOD

• 摘要: 基于教学实践，针对半无穷大空间 Stokes 第一问题，讨论了相似性解方法与微分代数领域之间的联系。阐明了初边值条件与微分方程的李群无穷小对称性相容是相似性解存在的必要条件，并首次给出了相似性解方法能使偏微分方程化为常微分方程的严格证明。本文旨在展现热-流科学相关课程引入现代数理观点的必要性，引导学生关注各类近似方法背后的物理思想，提升学生把握核心物理图像、构建合理数学语言和建模求解具体问题的能力。

Abstract: Originated from the teaching practice with the Stokes first problem in a semi-infinite space, the applicability and principle of the similarity method are discussed. This paper points out the necessary conditions for the existence of similarity solutions, that is, the initial and boundary conditions need to be compatible with the Lie symmetry of the differential equations. Moreover, the mathematical language of Lie-group method is utilized to prove why the similarity method can transform partial differential equations to the ordinary ones for the first time, as far as we know. Through discussion of the connection between the similarity method and Lie group in differential algebra, this article aims to demonstrate the necessity of introducing the perspectives of modern mathematical physics into courses such as fluid mechanics and heat transfer, thereby promoting the teaching practice to actively guide the students to pay attention to the physical ideas behind various methods to get the approximate solution, and to improve their ability to grasp the physical nature of problems and solve specific problems by physical modeling with appropriate mathematical language.

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