课程大纲

COURSE SYLLABUS

课程代码/名称

Course Code/Title

MAT8029 应用数学方法

MAT8029 Methods of Applied Math

课程性质

Compulsory/Elective

必修 Compulsory

课程学分/学时

Course Credit/Hours

3 学分/48 学时

授课语言

Teaching Language

英文

English

授课教师

Instructor(s)

张振

是否面向本科生开放

Open to undergraduates

or not

是

先修要求

Pre-requisites

MA201a 常微分方程 a，MA303 偏微分方程

Ordinary and Partial Differential Equations

教学目标

Course Objectives

掌握计算和应用数学模型建立和分析的基本方法

Master the basic methods of modelling and analysis in computational and

applied mathematics

教学方法

Teaching Methods

专题性质授课，并辅以前沿课题应用

Teaching in topics, and application to cutting edge problems

10.

教学内容

Course Contents

（如面向本科生开放，请注明区分内容。 If the course is open to undergraduates, please indicate the

difference.）

Section 1

Perturbation methods for algebraic equations:

i) Regular perturbation

ii) Singular perturbation

iii) Non-integer powers

iv) Logarithms

v) Eigenvalue problems

Section 2

Local analysis for the solutions to ODEs

i) Series solutions to ODEs

ii) Types of points of homogeneous linear ODEs

iii) Frobenius methods

iv) Method of dominant balance

Section 3

Asymptotic expansions

i) Asymptotic sequences

ii) Asymptotic power series

iii) Properties of asymptotic series

iv) Asymptotic series vs. convergent series

v) Other asymptotic expansions

Section 4

Asymptotic expansion of integrals

i) Direct expansion of integrands

ii) Integration by parts

iii) Laplace’s method

iv) Method of stationary phase

v) Method of steepest descent

Section 5

Introduction to global analysis and perturbation methods

i) Example of perturbation method

ii) Asymptotic expansion in \epsilon

iii) An example with direct expansion

iv) Regular vs. singular perturbation problems

Section 6

Boundary layer theory

i) Boundary layer problems

v) Boundary layer theory

vi) Location of boundary layer

vii) Higher order boundary layer theory

viii) Boundary layer thickness and distinguished limit

ix) Boundary layer problem including logarithm term

x) Multiple boundary layers

xi) Internal boundary layer

xii) Boundary layer in PDE problem

Section 7

WKB theory

i) Introduction

ii) WKB theory

iii) More remarks on the asymptotic expansions

iv) Problems with turning points

v) Eigenvalue problems (Storm-Liville problem)

vi) Application to wave equations

vii) Inhomogeneous linear equations

Section 8

Multiple scale analysis

i) Secular terms

ii) Method of strained coordinates

iii) Multiple scale analysis

iv) Slowly varying coefficients

v) Method of averaging

Section 9

Homogenization method

i) Background

ii) 1D problem

iii) Multi-dimensional problems

iv) Porous medium flow – Darcy’s law

Section 10

Bifurcation and stability

i) Linearized stability of steady states

ii) Limit cycle and Hopf bifurcation

iii) System of ODEs

Section 11

Basic calculus of variation

i) Introductory example – geodesic on a sphere

ii) First variation: Euler-Lagrange equation

iii) Isoperimetric problems – Catenary problem

iv) Holonomic constraints: Lagrange multipliers

v) Free boundary problems – natural boundary condition

vi) Hamilton-Jacobi system

vii) Noether’s theorem*

viii) Second variation*

Section 12

From calculus of variation to optimal control theory: an

introduction*

i) Basic formulation

ii) Pontryagin’s maximum principle

iii)

Dynamical programming and Hamilton-Jacobi-Bellman

equation

iv) Linear quadratic regulator

11.

课程考核

Course Assessment

平时作业（35%）+出勤（5%）+闭卷期中考试（20%）+闭卷期末考试（40%）

assignments (35%), attendance (5%), closed-book midterm exam (20%) and final

exam (40%)

12.

教材及其它参考资料

Textbook and Supplementary Readings

1*. (Textbook) C. M. Bender and S. A. Orszag, Advanced mathematical methods for

scientists and engineers, Springer, 1999.

2*. M. H. Holmes, Introduction to perturbation methods, Springer-Verlag, 1995.

3*. A. W. Bush, Perturbation methods for engineers and scientists, Boca Raton, 1992.

4*. E. J. Hinch, Perburbation methods, Cambridge University Press, 1991.

5^. A. Bensoussan, J.-L. Lions, G. Papanicolaou, Asymptotic analysis for periodic

structures, North-Holland, Oxford, 1978.

6^. U. Hornung, Homogenization and porous media, Springer, 1997.

7^. G. A. Pavliotis, A. M. Stuart, Multiscale methods: averaging and homogenization,

Springer, 2008.

8#. Bruce van Brunt, The Calculus of Variations, Springer-Verlag, 2004.

9#. 张恭庆，变分学讲义，高等教育出版社，2011.

10#. M. Giaquinta and S. Hildebrandt, Calculus of Variations, Vol. I and II, Springer,

1996.

11#. Daniel Liberzon, Calculus of Variations and Optimal Control Theory, Princeton

University Press, 2012.

The books with * are standard textbook about perturbation methods; books with ^

concerns mathematical homogenization methods; books with # discuss calculus of

variations.