Course Title: Course Title: Introduction to Applied Mathematics in Engineering Science
Course Attribute: Professional foundation courses
Hours/credits: 60 hours/3 credits
Pre-courses: Calculus, Linear Algebra
Purpose and Requirements:
The course introduces methods to solve mathematical problems that arise mainly in engineering applications. The course includes linear partial differential equations (PDEs), nonlinear PDEs, calculus of variation, asymptotic methods, geometric methods, and numerical methods. Based on classic PDEs arising from fluid dynamics, electromagnetism， solid mechanics, and quantum mechanics, a variety of solution techniques will be compared and contrasted, including separation of variables, Fourier transform, Green's function, nonlinear transformations, inverse scattering method, etc. Asymptotic techniques and geometric points of view will be used in many contexts to gain insight into the nature of solutions. The calculus of variations will enable us to understand different formulations of mechanics. Numerical methods will also provide us with necessary knowledge for further training in computational mechanics. In addition, multi-scale asymptotics will be introduced in the context of dispersive waves, where multiple space and time scales reveal the roles of dispersion and nonlinearity. Discussions will take us from classical, centuries-old techniques to open questions of active research.
Course summary:
Chapter 1. Introduction (3 hours)
1-1 Scientific frontiers of applied mathematics
1-2 Scientific applications of applied mathematics
1-3 Engineering applications of applied mathematics
Chapter 2. Superposition (12 hours）
2-1 Introduction to normed linear space and functional analysis
2-2 Duhamel’s principle
2-3 Separation of variables
2-3-1 heat conduction in a cube, in a cylinder, and in a sphere
2-3-2 flow around a circular cylinder
2-3-3Faraday waves and subharmonic instability
2-4 Integral-Transform Method
2-4-1 Fourier transform: heat equation, Schrodinger equation, water waves problem 2-4-2 Laplace transform: beam vibration
2-5 Delta Function and Green’s Function Method
2-5-1Delta function: heat kernel, Poisson kernel
2-5-2Green’s functions: electrostatic potential, quantum scattering
Chapter 3. Nonlinear Differential Equations and Transformation (6 hours)
3-1Burgers equation and Hopf-Cole transformation
3-1-1Burgers equation and turbulence
3-1-2 inviscid Burgers equation and traffic flow
3-1-3 solving Burgers’ equation：Hopf-Cole transformation
3-2 Korteweg-de Vries Equation and Inverse Scattering Method
3-2-1historical notes on the Korteweg-de Vries equation
3-2-2deriving and solving the Korteweg-de Vries equation
3-2-3introduction to the inverse scattering method
3-3 Sine-Gordon Equation and Bäcklund Transform
3-3-1 historical notes on the Sine-Gordon equation
3-3-2 solving the Sine-Gordon equation: Bäcklund transform
Chapter 4. Calculus of Variation (9 hours)
4-1 Variational principle: geodesics in Euclidean space, Fermat’s principle
4-2 Euler-Lagrange equation: minimum surface, geodesics on surface
4-3Variation problems and their equivalence with differential equations
4-4Applications of variational method in analytical mechanics
4-4-1Hamilton principle
4-4-2symmetry: Noether’s theorem
4-4-3rigid body movement
Chapter 5. Asymptotic Methods (10 hours)
5-1 Asymptotic series and their applications
5-1-1one-dimensional random walk model: Langevin equation
5-1-2asymptotic series
5-1-3Laplace’s method
5-1-4Gamma function and asymptotic Stirling series
5-2 Asymptotic solutions to differential equations
5-2-1 series method and its application in the pendulum problem
5-2-2perturbation theory and its application in the projectile problem
5-2-3Poincare’s perturbation theory
5-3Singular perturbation theory and its applications
5-3-1 a simple example: solving high-order algebraic equations
5-3-2 a further example: boundary value problem of ordinary differential equation
5-3-3 singular perturbation theory and its application in biochemical kinetics
5-3-4 Introduction to multi-scale analysis
Chapter 6.Geometric Methods (10 hours)
6-1 Geometric theory of ordinary differential equation
6-1-1 phase plane method: geometric way of thinking
6-1-2 local analysis: two-dimensional linear autonomous system, simple pendulum, limit cycle
6-1-3 global analysis: consequential function method, rotating degree method, Poincare-Bendixson theorem
6-2 High and infinite dimensional dynamical systems
6-2-1 tensor
6-2-2 dynamical system on manifolds: introduction to manifold and differential topology
6-2-3 structural stability: Harteman-Grobman theorem and center manifold theorem
6-2-3 bifurcation: logistic, circle, and Henon maps
6-2-4 chaos: Lorentz equation in atmospheric science
Chapter 7. Numerical Methods (10 hours)
7-1 Temporal discretization
7-2 Spatial discretization
7-2-1 finite difference method
7-2-2 finite volume method
7-2-3 pseudo-spectral method
7-3 Matrix calculation
7-3-1 solving system of linear equations
7-4 computing eigenvalues and eigenfunctions
7-4-1 History and prospect of computational mechanics
Assessment methods: coursework, midterm exam, finial exam
Course Text:J. Li, X. Zhou (1999) Asymptotic methods in mathematical physics. Science Press, China.
C.C. Lin, L.A. Segel (2010) Mathematics applied to deterministic problems in the natural sciences. Science Press, China.
References
R. Courant, D. Hilbert (2011) Methods of mathematical physics I. Science Press, China.
R. Courant, D. Hilbert (2011) Methods of mathematical physics II. Science Press, China.
L. Jiang, Y. Chen, X. Liu, F. Yi (1996) Lectures on equations of mathematical physics (Second Edition). Higher Education Press, China.
Discussion Session: one-hour discussion session for every chapter (except the first chapter)
Teacher's Profile:
Professor Zhan Wang received the doctor degree from University of Wisconsin - Madison in 2012, majoring in Applied Mathematics and Fluid Mechanics. After two years as a Research Associate at University College London, he became a lecturer in the Department of Mathematical Sciences at University of Bath. He is now a professor at the Institute of Mechanics of the Chinese Academy of Sciences. Dr. Wang’s research focusses on geophysical fluid mechanics, nonlinear waves, free boundary problems and fluid-structure interactions.
