常微分方程理论在“数学物理方程”课程中的应用

Application of Ordinary Differential Equation Theory in“Mathematical Physics Equations”Course

偏微分方程求解既是“数学物理方程”课程教学的主体内容,又是课堂教学的重难点。求解偏微分方程的方法有很多,如特征线法、波的反射原理、分离变量法、格林函数法、傅里叶变换、拉普拉斯变换等。学会求解一些简单的偏微分方程是数学专业学生学好“数学物理方程”课程乃至为以后继续深造打下基础的关键。因此,揭示偏微分方程求解方法中所蕴含的数学思想,帮助学生系统而深入地掌握求解的方法显得尤为重要。以方程特殊形式解的求解、分离变量法、傅里叶变换法为例,结合具体的定解问题求解来阐述将偏微分方程问题化归为常微分方程问题这一思想在偏微分方程求解中的应用。

Solving partial differential equations is the main content of the“Mathematical Physics Equations”course,and it is also the most challenging point in classroom teaching.There are many methods for solving partial differential equations,such as characteristic line method,wave reflection principle,variable separation method,Green function method,Fourier transformation,Laplace transformation,etc.Learning to solve some simple partial differential equations is the key for students in mathematics to learn the“Mathematical Physics Equations”course and to lay the foundation for further studies.Therefore,it is crucial to reveal the mathematical ideas in solving partial differential equations and help students master the technique of solving systematically and deeply.Taking the solution of equation in special form,separating variables and Fourier transformation as examples,this paper combines with the solution of specific definite solution problem and expounds the application of the idea of transforming partial differential equation problem into ordinary differential equation problem in the solution of partial differential equation.