新工科背景下数学专业偏微分方程课程内容设计研究

Curriculum Content Design of Partial Differential Equations for Mathematics Majors in the Context of Emerging Engineering Education

摘要: 新工科建设对数学专业学生的数学建模、跨学科应用、科学计算和复杂问题分析能力提出了新要求。文章以本校数学专业最新版培养方案、课程大纲、现用教材及相关教改研究为依据，参照逆向设计与建构性对齐思想，对课程目标、教学内容、学习任务和评价证据进行一体化设计。课程以Hadamard适定性为主线，将模型与定解问题、经典方程的解析与定性理论、弱解与变分方法基础、数值方法与跨学科应用相互联系，并建立“能力要求–课程目标–教学任务–评价证据”的对应关系。文章给出基础、提高和拓展三个教学层次以及48学时、32学时两种配置，以一维热方程说明建模、存在性、唯一性、连续依赖和数值稳定性的组织方式。在此基础上，本文将图像处理中的Perona-Malik非线性扩散和物理信息神经网络设置为拓展项目，并提出32学时版本的课堂讲授与课外阅读的分层安排、泛函分析在线预备学习单元及单班试点教学评价资料收集方案。本文属于课程设计研究，其实施效果尚需通过后续教学实践加以检验。

Abstract: Emerging Engineering Education places new demands on mathematics majors’ abilities in mathematical modeling, interdisciplinary application, scientific computing, and the analysis of complex problems. Based on the current mathematics program, the syllabus and textbook of the Partial Differential Equations course, and relevant studies on teaching reform, this paper adopts the principles of backward design and constructive alignment to integrate course objectives, content, learning tasks, and assessment evidence. With Hadamard well-posedness as the organizing thread, the proposed curriculum connects four components: models and initial-boundary value problems, analytical and qualitative theories of classical equations, fundamentals of weak solutions and variational methods, and numerical methods and interdisciplinary applications. It also establishes an alignment among competency requirements, course objectives, learning tasks, and assessment evidence. Three instructional levels—fundamental, advanced, and extended—are specified, together with two course configurations of 48 and 32 teaching hours. A one-dimensional heat equation is used to illustrate how mathematical modeling, existence, uniqueness, continuous dependence, and numerical stability can be organized within a coherent instructional unit. The design is further supplemented with Perona-Malik nonlinear diffusion and physics-informed neural networks as extension projects, differentiated arrangements for classroom instruction and supplementary reading in the 32-hour version, an online preparatory unit in functional analysis, and a plan for collecting assessment data from a pilot class. As a curriculum design study, the effectiveness of the proposed framework remains to be evaluated through subsequent teaching practice.