MS004317-泛函分析与变分法

发布者:王源发布时间:2020-11-10浏览次数:458

研究生课程开设申请表

 课院(系、所): 十大澳门网站赌博


课程申请开设类型: 新开□     重开□     更名□请在内打勾,下同

课程

名称

中文

泛函分析与变分法

英文

Functional AnalysisandVariational method

待分配课程编号

  MS004317

课程适用学位级别

博士


硕士

总学时

48


课内学时

48

学分

3

实践环节


用机小时


课程类别

公共基础     专业基础     专业必修     专业选修

开课院()

十大澳门网站赌博


开课学期

秋季

考核方式

A.笔试(开卷   闭卷)      B. 口试    

C.笔试与口试结合                 D. □其他

课程负责人

教师

姓名

孙连友

职称

副教授

e-mail

lianyousun@seu.edu.cn

网页地址

/2018/0423/c19950a213721/page.htm

授课语言

汉语

课件地址

爱课程网站

适用学科范围

公共

所属一级学科名称


实验(案例)个数


先修课程

高等数学、线性代数、数学物理方程

教学用书

教材名称

教材编者

出版社

出版年月

版次

主要教材

自编讲义

孙连友




主要参考书

实变函数论与泛函分析(上、下)

夏道行等

高等教育出版社

1985

2st

索伯列夫空间引论

李立康等

上海科学技术出版社

1981

1st

泛函分析与变分法

苏家铎

中国科学技术大学出版社

1993

1st



一、课程介绍(含教学目标、教学要求等)300字以内)

本课程主要介绍工科类专业所涉及到的泛函分析概念和方法。让研究生能用泛函概念和观点理解现代分析的各种方法和应用,并进一步应用泛函方法解决各类工程问题。

教学要求:要求学生通过课程学习,理解并掌握泛函的基本概念,理解Lebesgue测度和Lebesgue积分,理解度量空间、赋范空间、Lp空间、希尔伯特空间、算子及算子空间、共轭空间、广义函数和广义函数空间以及索伯列夫空间等各类空间和其中的相关概念,以及它们的特性。了解泛函方法的一些基本应用。理解并掌握变分法,通过变分法将各种泛函方法和原理应用于解决工程问题。

二、教学大纲(含章节目录):(可附页)

绪论——数学物理方程(静电场方程)的经典解和广义解

 §1 经典解

 §2 广义解

 §3 数值解

  1. Lebesgue测度和Lebesgue积分

  §1 Lebesgue测度

一、 Lebesgue 测度

二、Lebesgue可测集

 §2 Lebesgue积分

  1. 可测函数

二、Lebesgue积分

  1. 度量空间和赋范线性空间

 §1 度量空间

 §2 赋范线性空间

  1. 线性空间

    1. 线性空间上的范数

三、 Lp空间

 §3 度量空间的完备性及Banach空间

  1. 完备空间及性质

  2. 度量空间的完备化

  3. 稠密性(稠密性概念、可析点集)

四、Banach空间

 §4 不动点原理

  1. 有界算子与泛函

 §1 有界线性算子

    1. 线性算子与线性泛函概念

    2. 线性算子的有界性与连续性

三、线性算子空间

 §2有界线性泛函与共轭空间

  1. 有界线性泛函

二、*共轭空间与连续线性泛函延拓

三、二次共轭空间与共轭算子

第四章 Hilbert 空间

 §1 内积

  1. 内积与内积空间

二、内积与范数

 §2 Hilbert空间

一、Hilbert空间

二、内积与范数

三、正交与投影

四、最小二乘法

 §3标准正交系与Fourier展开

  1. 标准正交系

  2. 正交化方法

三、Fourier级数

四、完备的标准正交基及其性质

 §4有界线性泛函的内积表示与自共轭空间

一、有界线性泛函及其表示(Riesz泛函表示)

二、自共轭空间

三、共轭算子的内积定义

 §5 双线性Hermite泛函与自共轭算子

第五章 广义函数与索伯列夫(Sobolev)空间

 §1  基本函数与广义函数

 §2  广义函数的导数

 §3  广义函数的Fourier变换

 §4  Sobolev空间

 §5  实指数Sobolev空间

   §6* 嵌入定理

   §7  迹定理

第六章 变分方法

§1  变分法原理及基本概念

一、函数的变分

二、泛函的变分

三、泛函极值

 §2  Euler方程与奥氏方程

 §3  微分方程的变分形式

 §4  变分形式与双线性泛函(内积)

 §5  有限维空间上的近似解与精确解的关系

第七章 求解电磁场问题的泛函方法

 §1  电磁场问题的泛函模型

 §2  矩量法原理

 §3  加权余量法原理

 §4  Ritz方法

 §5  Galerkin方法

 §6* 有限元方法

 §7* 矢量有限元方法(Edge Element)

注:星号部分为选讲内容。

三、教学周历

周次

教学内容

教学方式

 1

数学物理方程(静电场方程)的经典解和广义解,经典解,广义解,数值解, Lebesgue测度, Lebesgue可测集

讲课

 2

Lebesgue测度, Lebesgue可测集,可测函数, Lebesgue积分

讲课

 3

度量空间,线性空间,线性空间上的范数,

讲课

 4

赋范线性空间, Lp空间

讲课

 5

度量空间的完备性及Banach空间,完备空间及性质,度量空间的完备化,

讲课

 6

稠密性(稠密性概念、可析点集),Banach空间, 不动点原理

讲课

 7

有界线性算子,线性算子与线性泛函概念,线性算子的有界性与连续性,线性算子空间

讲课

 8

有界线性泛函与共轭空间,有界线性泛函,共轭空间与连续线性泛函延拓,二次共轭空间与共轭算子

讲课

 9

内积与Hilbert空间,内积与范数, Hilbert空间,正交与投影,最小二乘法

讲课

 10

标准正交系与Fourier展开,标准正交系,正交化方法,Fourier级数,完备的标准正交基及其性质,有界线性泛函与共轭空间。

讲课

 11

有界线性泛函及其表示(Riesz泛函表示),共轭空间与自共轭空间,共轭算子,双线性Hermite泛函与自共轭算子

讲课

 12

基本函数与广义函数,广义函数的导数,广义函数的Fourier变换,  Sobolev空间,实指数Sobolev空间,嵌入定理, 迹定理

讲课

 13

变分法原理及基本概念, 函数的变分,泛函的变分,泛函极值,  Euler方程与奥氏方程

讲课

 14

微分方程的变分形式,变分形式与双线性泛函(内积),有限维空间上的近似解与精确解的关系

讲课

 15

电磁场问题的泛函模型,矩量法原理,加权余量法原理,

讲课

 16

Ritz方法,Galerkin方法,有限元方法*,矢量有限元方法*Edge Element)

讲课

 17



 18



注:1.以上一、二、三项内容将作为中文教学大纲,在研究生院中文网页上公布,四、五内容将保存在研究生院。2.开课学期为:春季、秋季或春秋季。3.授课语言为:汉语、英语或双语教学。4.适用学科范围为:公共,一级,二级,三级。5.实践环节为:实验、调研、研究报告等。6.教学方式为:讲课、讨论、实验等。7.学位课程考试必须是笔试。8.课件地址指在网络上已经有的课程课件地址。9.主讲教师简介主要为基本信息(出生年月、性别、学历学位、专业职称等)、研究方向、教学与科研成果,以100500字为宜。


四、主讲教师简介:

孙连友:19907月毕业于复旦大学数学研究所,获硕士学位。20037月于东南大学无线电工程系获博士学位。1990年到19981月期间在东南大学数学系工作,从事教学和应用数学方面的研究。20058月至20068月由国家留学基金委公派到加拿大McGill大学从事博士后研究。19981月至今在十大澳门网站赌博工作。目前,主要从事计算电磁学数值理论和计算方法方面的教学和科研工作。在IEEE和国内核心期刊及国际会议上发表论文二十余篇,其中有十多篇篇被EISCI收录。合著有《电磁场边值问题的区域分解算法》,由科学出版社出版。



五、任课教师信息(包括主讲教师):

任课

教师

学科

(专业)

办公

电话

住宅

电话

手机

电子邮件

通讯地址

邮政

编码

孙连友

电磁场与微波技术




 lianyousun@seu.edu.cn

东南大学 十大澳门网站赌博

 210096





















六、课程开设审批意见

所在院(系)



负责人:

期:

所在学位评定分

委员会审批意见



分委员会主席:

期:

研究生院审批意见




负责人:

期:


说明:1.研究生课程重开、更名申请也采用此表。表格下载:http:/seugs.seu.edu.cn/down/1.asp

2.此表一式三份,交研究生院、院(系)和自留各一份,同时提交电子文档交研究生院。


























Application Form For Opening Graduate Courses

S

 chool (Department/Institute)


Course Type: New Open □   Reopen □   Rename □Please tick in □, the same below

Course Name

Chinese

泛函分析与变分法

English

Functional AnalysisandVariational method

Course Number


Type of Degree  

Ph. D


Master

Total Credit Hours

 48


In Class Credit Hours

48

Credit

3

Practice


Computer-using Hours


Course Type

□Public Fundamental    □Major Fundamental    □Major Compulsory     □Major Elective

School (Department)


College of Information Science & Engineering


Term

Autumn

Examination

A. □PaperOpen-book   □ Closed-bookB. □Oral    

C. □Paper-oral Combination                       D. □ Others

Chief

Lecturer

Name

Sun Lian-you

Professional Title

Associate Professor

E-mail

lianyousun@seu.edu.cn

Website

/2018/0423/c19950a213721/page.htm

Teaching Language used in Course

Chinese  

Teaching Material Website

www.icourses.cn

 Applicable Range of Discipline

General Course

Name of First-Class Discipline


Number of Experiment


Preliminary Courses

Advanced Mathematics, Linear Algebra, Mathematical Physics Equations

Teaching Books

Textbook Title

Author

Publisher

Year of Publication

Edition Number

Main Textbook

Teaching materials written   by myself  

Sun Lian-you




Main Reference Books

 Theory of Functions of Real Variable and Functional Analysis

Xia Dao-xing etc.

Higher Education Press

1985

2st

Introduction of Sobolev space

Li Li-kang etc.

Shanghai Science and Technology Press

1981

1st

Functional Analysis and  Variations Method  

Su Jia-yi

Press of University of Science and technology of China  

1993

1st





  1. Course Introduction (including teaching goals and requirements) within 300 words:

 The course mainly expatiates on the functional concepts and numerical methods applied to electromagnetical field problems. Students need understand and use variational principle, moment methods and weighted residual methods etc. to model the electromagnetical field problems. Furthermore, students need compare them to each other, and understand them relationships.


  1. Teaching Syllabus (including the content of chapters and sections. A sheet can be attached):



Introduction---Classical and Generalized Solutions of Mathematical Physics Equations

 §1 Classical Solutions

 §2Distributions

 §3Numerical Solutions

Chapter 1. Lebesgue Measure & Lebesgue Integral

  §1 Lebesgue Messure

 1. Lebesgue Measure

 2.Lebesgue Measurable Set

 §2 Lebesgue Integral

 1.Measurable function

 2. Lebesgue Integral

Chapter2.  Metric Space & Normed Linear Space

 §1 Metric Space

 §2 Normed Linear Space

 1. Linear Space

 2. Norm on Linear Space

 3. Lp Space

 §3 Completeness of Metric Space and Banach Space

 1. Completed Space and its Properties

 2. Completion of Metric Space

 3. Density(Concept of Density, Separable Metric Space)

 4. Banach Space

 §4 Fixed Point Theory

Chaper 3 Bounded Operator and Functional

 §2 bounded Linear Operators

 1. Concepts of Linear Operator and Linear Functional

 2. Continuation and bound of Linear Operators

 3. Space of Linear Operators

 §3 Bounded Linear Functional and Conjugate Space

 1. Bounded Linear Functional

 2. Conjugate Spaceand Prolongation of Linear Continuous Functional

 3. Quadratic Conjugate Space and Conjugate Operator

Chaper 4Hilbert Space

 §1 Inner Product

 1. Inner Product and Its Space

 2. Inner Product and Norm

 3. Space of Linear Operators

 §2 Hilbert Space

 1. Hilbert Space

 2. Inner Product and Norm

 3. Orthogonal Vector and Its Projection

 4. Method of Least Squares

 §3 Normal Orthogonal System and Fourier Expansion

 1. Normal Orthogonal System

 2. Methods of Orthogonalization

 3. Fourier Series

 4. Completed Normal Orthogonal System and Its Properties

 §4 Bounded Linear Functional and Its Conjugate Space

 1. Bounded Linear Functional and Its Representation

 2. Conjugate Space and Self- Conjugate Space

 3. Conjugate Operator

 §5 Bilinear Hermite Functional and Self-Conjugate Operator

Chapter 5 Distribution and Sobolev Space

 §1 Basic Function and Distribution

 §2 Derivatives of Distribution

 §3 Fourier Transformation of Distribution

 §4 Sobolev Space

 §5Sobolev Space With Real Exponent  

 §6 Embedding Theorem

 §7 Trace Theorem

Chapter 6 Variational Method for Electromagnetical Field

 §1 Variational Principle and Its Basic Concepts

     1. Variational of Function

     2. Variational of Functional

     3. Extremum of Functional

 §2 Euler Equations and OCTPOГPaДCКИЙEquations

 §3 Variational Formula of Differential Equations

 §4 Variational Formula and Bilinear Functional

 §5Relationship of Solutions Between Finite Dimensional Space and Infinite Dimensional Space

Chapter 7 Numerical Methods of Electromagnetical Field Problems

 §1. Moment Methods

 §2. Weighted Residual Method

 §3. Ritz Method

 §4.Galerkin Method

 §5*. Edge Element Method

 §6*. Finite Element Method






  1. Teaching Schedule:


 Week

 Course Content

 Teaching Method

 1

Classical Solutions of Mathematical Physics Equations(Static field), Distribution, Numerical Solution, Lebesgue MeasureLebesgue Measurable Set

 Lecture  

 2

Measurable Function, Lebesgue's IntegralMetric SpaceLinear Space, Normed linear Space

 Lecture

 3

Norm on Linear Space, Lp Space

 Lecture

 4

Completeness of Metric Space, Banach Space, Complete Space and its Property

 Lecture

 5

Completion of Metric Space,Density(Concept of Density, Separable Metric Space), Fixed Point Theory

 Lecture

 6

Bounded Linear Operator, Concepts of Linear Operator and Linear Functional

 Lecture

 7

Continuation and bound of Linear Operators, Space of Linear Operators

 Lecture

 8

Bounded Linear Functional and Conjugate Space, Bounded Linear Functional,Conjugate Spaceand Prolongation of Linear Continuous Functional,  Quadratic Conjugate Space and Conjugate Operator

 Lecture

 9

Inner Product and Its Space, Inner Product and Norm, Space of Linear Operators, Hilbert Space, Inner Product and Norm, Orthogonal Vector and Its Projection, Method of Least Squares

 Lecture

 10

Normal Orthogonal System and Fourier Expansion, Normal Orthogonal System, Methods of Orthogonalization, Fourier Series, Completed Normal Orthogonal System and Its Properties, Bounded Linear Functional and Its Conjugate Space

 Lecture

 11

Bounded Linear Functional and Its Representation, Conjugate Space and Self- Conjugate Space, Conjugate Operator, Bilinear Hermite Functional and Self-Conjugate Operator

 Lecture

 12

Basic Function and Distribution, Derivatives of Distribution,  Fourier Transformation of Distribution, Sobolev Space, Sobolev Space With Real Exponent , Embedding Theorem, Trace Theorem

 Lecture

 13

Variational Principle and Its Basic Concepts, Variational of Function, Variational of Functional, Extremum of Functional,Euler Equations and OCTPOГPaДCКИЙEquations

 Lecture

 14

Variational Formula of Differential Equations, Variational Formula and Bilinear Functional, Relationship of Solutions Between Finite Dimensional Space and Infinite Dimensional Space

 Lecture

 15

Moment Methods, Weighted Residual Method,

 Lecture

 16

Ritz Method, Galerkin Method, Edge Element Method*, Finite Element Method*

 Lecture

 17



 18



Note: 1.Above one, two, and three items are used as teaching Syllabus in Chinese and announced on the Chinese website of Graduate School. The four and five items are preserved in Graduate School.


 2. Course terms: Spring, Autumn , and Spring-Autumn term.   

 3. The teaching languages for courses: Chinese, English or Chinese-English.  

 4. Applicable range of discipline: public, first-class discipline, second-class discipline, and third-class discipline.  

 5. Practice includes: experiment, investigation, research report, etc.  

 6. Teaching methods: lecture, seminar, practice, etc.  

 7. Examination for degree courses must be in paper.  

 8. Teaching material websites are those which have already been announced.  

 9. Brief introduction of chief lecturer should include: personal information (date of birth, gender, degree achieved, professional title), research direction, teaching and research achievements. (within 100-500 words)  


  1. Brief Introduction of Chief lecturer:

Lianyou Sun received M.Sc. degree in applied mathematics from Fudan University, Shanghai, China, in 1990. PhD degree in electromagnetic field and microwave technology from Southeast University, Nanjing, China, in 2003.

He worked at Southeast University, Nanjing, China, firstly in Department of Applied Mathematics as a lecturer from 1990 to 1997, and then in Department of Radio Engineering as a associate professor from 1998 to 2005. With Financial support of China Scholarship Council, he pursued his postdoctoral studies in computational electromagnetics at McGill University, Montreal, Canada from Aug. 2005 to Aug. 2006. His current interests are in numerical methods for electromagnetic problems, especially for large scale problems. He also is a co-authorof a book:Domain Decomposition Methods for Solving the Electromagnetic field Boundary Value Problems,Beijing, China: Science Press, 2005. He has published more than ten papers on computational electromagnetics.

  1. Lecturer Information (include chief lecturer)


Lecturer

 Discipline

 (major)

 Office

Phone Number

Home Phone Number

Mobile Phone Number

 Email

Address

Postcode

 Sun Lianyou

 Electromagnetical field & Microwave Technology




 lianyousun@seu.edu.cn

 2 Si Pai Lou, School of Information Science and Engineering

 210096


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