People began to study partial differential equations shortly after differential calculus came into being. For example,/kloc-the research on the transverse vibration of strings at the beginning of the 0/8th century, and the research on heat conduction theory, fluid mechanics and counterpoint function later, all obtained corresponding mathematical and physical equations, which are believed to be effective solutions. By the middle of19th century, the general theory of partial differential gradually formed from the in-depth study of individual equations, such as the classification and characteristic theory of equations, which is the category of classical partial differential equation theory.
However, in the 20th century, with the continuous development of science and technology, new mathematical and physical equations were put forward in scientific practice, and the appearance of electronic computers provided a powerful means to realize the research results of mathematical and physical equations. Because other branches of mathematics (such as functional analysis, topology, group theory, differential geometry, etc. ) has also developed rapidly, which provides a powerful tool for further study of partial differential equations. Therefore, in the 20th century, the research on mathematical and physical equations made unprecedented progress, and these developments showed the following characteristics and trends:
1. Many mathematical descriptions of problems raised in natural science and engineering technology are mostly nonlinear partial differential equations. Even if some linear partial differential equations are treated approximately, with the deepening of research, nonlinear effects must be reconsidered. It is much more difficult to study nonlinear partial differential equations, but it will provide many useful inspirations to the existing results of linear partial differential equations.
Second, in practice, many factors combine and influence each other. Therefore, its mathematical models are mostly nonlinear partial differential equations. Such as reaction-diffusion equation, fluid dynamics equation, electromagnetic fluid dynamics equation, radiation fluid equation, etc. Mathematically, it is called hyperbolic-parabolic equation.
Third, mathematical physical equations are no longer just mathematical forms that describe engineering processes such as physics and mechanics. At present, some very important partial differential equations have been put forward in the fields of chemistry, biology, medicine, agriculture, environmental protection and even economy.
Four, the mathematical description of an actual model, in addition to the equation (or equation) describing the process, should also have certain solving conditions (such as initial conditions and boundary conditions). Traditionally, these conditions are linear and expressed point by point. However, many conditions for definite solutions are nonlinear, especially nonlocal. The study of nonlocal boundary value problems is a new and very meaningful field.
5. Relationship with other branches of mathematics. For example, many important nonlinear partial differential equations are put forward in geometry, such as minimal surface equation, harmonic mapping equation, equation and so on. Modern tools such as functional analysis, topology and group theory are widely used in the theoretical study of partial differential equations. For example, space provides a powerful framework and tool for studying linear nonlinear partial differential equations. The application of generalized function makes the classical linear differential equation theory more systematic and perfect. Then, with the wide application of computers and the rapid development of calculation methods, especially the wide application of finite element, the study of partial differential equations has been realized and tested in practice.
When dealing with application problems by mathematical methods, we must first establish a reasonable mathematical model, which is a partial differential equation in many cases. The establishment of the model is a rather complicated process.
Basic requirements of syllabus and chapters
Chapter 1 Wave Equation
Teaching points:
Through the teaching of this chapter, students can initially understand the methods and characteristics of mathematical equations, master the solution of equations, and express the physical meaning.
1. Make students understand the deduction method of wave equation.
2. Understand the conditions and significance of the definite solution.
3. Master the method of separating variables to solve the initial-boundary value problem.
4. Can solve Cauchy problem of high-dimensional wave equation.
5. Make clear the meaning of wave propagation and attenuation.
6. Using energy inequality to determine the uniqueness and stability of the solution of the equation.
Teaching hours: 20 hours.
Teaching content:
The derivation and definite solution conditions of the first section equation
Section 2 D'Alembert Formula and Wave Propagation
In the third quarter, the method of separating variables for initial-boundary value problems
Section 4 Cauchy problem of high-dimensional wave equation
Propagation and attenuation of The 5th Wave.
Uniqueness and stability of the sixth energy saving inequality and wave equation solution
Evaluation requirements:
The derivation and definite solution conditions of the first section equation (understanding and application)
Section 2 D'Alembert Formula and Wave Propagation (Understanding)
Section III Separation of Variables for Initial-boundary Value Problems (Understanding and Application)
The fourth quarter Cauchy problem of high-dimensional wave equation (understanding and application)
Section 5 Wave Propagation and Attenuation (Understanding)
Uniqueness and stability of the sixth energy-saving inequality and wave equation solution (understanding and application)
Chapter II Heat Conduction Equation
Teaching points:
Through the teaching of this chapter, students can understand that the heat conduction equation is established by physical principles, the initial-boundary value problem can be solved by separating variables, the Cauchy problem can be solved by Fourier transform, and the uniqueness and stability of the solution of the definite solution problem can be determined by extreme value principle.
Teaching hours: 15 hours.
Teaching content:
Section 1 heat conduction equation and derivation of its definite solution
Separation of variables method for initial-boundary value problems in the second quarter
Section 3 Cauchy problem
Section 4 Extreme Value Principle, Uniqueness and Stability of Definite Solution
Evaluation requirements:
The first section heat conduction equation and the derivation of its definite solution (understanding)
Section 2 Separation of Variables for Initial-boundary Value Problems (Understanding and Application)
Section 3 Cauchy Problem (Understanding and Application)
Section IV Extreme Value Principle, Uniqueness and Stability of Definite Solution (Understanding and Application)
Chapter III Harmonic Equation
Teaching points:
Through the teaching of this chapter, students can establish harmonic equations, clearly determine the conditions of solutions, master Green's formula and its application, understand Green's function, and judge the uniqueness of the solution of the second boundary value problem by using the strong extremum principle.
Teaching hours: 15 hours.
Teaching content:
The first section establishes the equation and the conditions of definite solution
Section 2 Green's Formula and Its Application
Section 3 Green's Function
Section four: the strong extremum principle, the uniqueness of the solution of the second boundary value problem
Evaluation requirements:
The establishment and definite solution conditions of the first section equation (application)
Section 2 Green's Formula and Its Application (Understanding and Application)
Section 3 Green's Function (Understanding)
The fourth section, the strong extremum principle, the uniqueness of the solution of the second boundary value problem (understanding and application)
The fourth chapter is the classification and summary of second-order linear partial differential equations.
Teaching points:
Through the teaching of this chapter, students can master the classification method of second-order linear equations, the characteristic theory of second-order linear equations and the characteristics of three kinds of equations.
Teaching hours: 12 hours.
Teaching content:
Section 1 Classification of Second Order Linear Equations
The characteristic theory of second-order linear equations in the second quarter
Comparison of three kinds of equations in the third quarter
Evaluation requirements:
Section 1 Classification of Second Order Linear Equations (Memory and Understanding)
Characteristic theory of second-order linear equations (memory and understanding)
Section 3 Comparison of Three Equations (Memory and Understanding)
Chapter V Integral Theory
Teaching points:
Through the teaching of this chapter, students can understand the concept and characteristic theory of first-order partial differential equation, make clear the Cauchy problem and definite solution of binary linear hyperbolic equation, and master the solution of two series.
Teaching hours: 10 hour.
Teaching content:
Section 1 Introduction 1. Example 2 of first order partial differential equation. The relationship between first-order equation and higher-order equation,
The second section is the characteristic theory of first-order linear partial differential equations with two independent variables.
Section 3 Cauchy problem of linear hyperbolic equation with two independent variables
Other definite solutions of linear hyperbolic equations with two independent variables in the fourth quarter
Section 5 Two Series Solutions (Application)
Evaluation requirements:
The first section introduces 1. Example 2 of first order partial differential equation. The relationship between first-order equation and higher-order equation, (understanding)
The second section is the characteristic theory of first-order linear partial differential equations with two independent variables. (memory and understanding)
Section 3 Cauchy problem of hyperbolic linear equations (memory and understanding)
Section 4 Other definite solutions of binary linear hyperbolic equations (memory and understanding)
Three. Recommended teaching materials and reference numbers
1. Equations of Mathematical Physics, edited by Gu Chaohao, 2nd edition, Higher Education Press, 2002.
2. Equations of Mathematical Physics, edited by tikhonov, translated by Huang Kegu, 2nd edition, Higher Education Press, 196 1.
3. Methods of Mathematical Physics, edited by Mathematics Teaching and Research Group of Nanjing Institute of Technology, Higher Education Press, 1982.
4. Advanced Mathematics, edited by Mathematics Department of Sichuan University, 4th edition, People's Education Press, 1979.