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常微分方程教学大纲

课程名称常微分方程(基地) (Ordinary Differential Equations)

一、课程目的常微分方程是数学专业的一门基础课,是整个课程体系中不可缺少的重要组成部分。本课程是大学本科基地班的一门必修课,课程目的是围绕基本概念、基本理论、具体求解和实际应用三条主线,培养学生正确掌握常微分方程的各种基本概念,理论和方法,以及处理微分方程问题的思维方式。培养学生具有初步分析和解决实际问题的能力,为培养高素质的教学和科研人才打好基础。

二、课程任务:学生通过学习本课程不仅可加深对数学分析,高等代数中已学过的基础知识的理解,还可以提高应用能力,为后继的数学和应用数学各课程提供解决问题的方法和工具,使这门课成为通向各个应用学科和工程技术的重要桥梁。

三、教学方式:  以3+1的模式教学,以课堂教学、辅导为主,结合自学展开。3课时的课堂教学主要讲解基本概念,基本理论和基本方法,并将应用这些理论解决的实际问题的范例融入基本原理的讲解,使同学们更好地理解这个学科、提高对常微分方程这个学科的兴趣、初步了解它的理论体系、思维方式和研究方法。 在方法上,由浅入深,循序渐进,理论联系实际,把问题背景交代清楚。1课时的习题课用来处理学生作业中和平时学习中的问题。

四、教材及参考书目
教材:Martin Braun: Differential Equations and Their Applications, 4th edition. Springer, 1994..

参考书目:
1、 C. H. Edwards and D. E. Penny: Differential Equations and Linear Algebra, 3rd edition, Prentice Hall – Pearson, 2010.
2、林武忠、汪志鸣、张九超编著,常微分方程(第一版), 科学出版社, 2003年9月。
3、王树禾编著《微分方程模型和混沌》,中国科学技术大学出版社,1999。
4、丁同仁,李承治主编《常微分方程教程》,高等教育出版社,1991。
5、王高雄等主编:《常微分方程》,第二版,高等教育出版社,1983。
6、叶彦谦主编:《常微分方程讲义》,第二版,人民教育出版社,1984。

五、考核方式与评价结构比例:
平时成绩占10%,期中闭卷考试,考试成绩占30%;期末闭卷考试,考试成绩占60%。

六、教学内容:
Chapter 1 First order differential equations
1.2 First order linear differential equations
1.3 The Van Meegeren art forgeries(自学)
1.4 Separable equations
1.5 Population models
1.6 The spread of technological innovations(自学)
1.7 An atomic waste disposal problem(自学)
1.8 The dynamics of tumor growth mixing problems and orthogonal trajectories
1.9 Exact equations and why we cannot solve very many differential equations
1.10 The existence/uniqueness theorem (Picard iteration)
?Supplementary materials: Decay, Half Life, Air Resistance Problem, Newton’s Cooling Law, Pursuit problem
Chapter 2 Second order linear differential equations
2.1 Algebraic properties of solutions
2.2 Linear equations with constant coefficients
2.3 The nonhomogeneous equation
2.4 The method of variation of parameters
2.5 The method of judicious guessing
2.6 Mechanical vibrations: The Tacoma Bridge disaster
2.7 A model for the detection of diabetes (自学)
2.8 Series solutions: 2.81 Singular points Euler equations;2.82 Regular singular points the method of Frobenius;2.83 Equal roots and roots differing by an integer
2.9 The method of Laplace transforms
2.10 Some useful properties of Laplace transforms
2.11 Differential equations with discontinuous right hand sides
2.12 The Dirac delta function
2.13 The convolution integral Consider the initial value problem
2.15 Higher order equations
Supplementary materials Difference equations
Chapter 3 Systems of differential equations
3.1 Algebraic properties of solutions of linear system
3.2 Vector spaces (已学过内容,略)
3.3 Dimension of a vector space(已学过内容,略)
3.4 Applications of linear algebra to differential equations(已学过内容,略)
3.5 The theory of determinants(已学过内容,略)
3.6 Solutions of simultaneous linear equations(已学过内容,略)
3.7 Linear transformations(已学过内容,略)
3.8 The eigenvalue/eigenvector methods
3.9 Complex roots
3.10 Equal roots
3.11 Fundamental matrix solutions
3.12 The nonhomogeneous equation variation of parameters

Chapter 4 Qualitative theory of differential equations
4.2 Stability of linear systems
4.3 Stability of equilibrium solutions
4.4 The phase plane
4.6 Qualitative properties of orbits
4.7 Phase portraits of linear systems
4.8 Long time behavior of solutions the Poincare Bendixson Theorem
4.10 Predator-prey problems
4.11 The principle of competitive exclusion in population biology
4.12 The Threshold Theorem of epidemiology
4.13 A model for the spread of gonorrhea

七、教学时数分配:

章次

复习、练习

 

 

 

学时

16

16

12

20

8

 

 

 

 

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