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

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 promote the teaching practice to actively guide the students to pay attention to the physical ideas behind various methods to get the approximate solution and improve their ability to grasp the physical nature of problems and solve specific problems by physical modeling with appropriate mathematical language.