ODEs and PDEs
January 13, 2021
I will serve as a TF for Harvard AM105, ODEs and PDEs, lectured by Zhigang Suo and Ethan Levian this spring term. My previous learning experience of these topics is formalized by a physics-centered course Mathematical Physics Method (数学物理方法) given in Chinese, which covered topics from complex analysis to solutions of special ODE/PDEs. AM105 is a more mathematics-centered course focusing on ODEs and PDEs. To get familiar with related English terminologies and more mathematical details, I will spend some time reviewing the textbooks Boyce, William E., Richard C. DiPrima, and Hugo Villagómez Velázquez. Elementary differential equations and boundary value problems 2017 Haberman, Richard. Applied partial differential equations 2003 myself and take notes here. Hope I could be a good TF.
The syllabus of the AM105 is, according to Zhigang’s notes:
- Intro, Boyce book 1.3
- Initial value problem (linear), Boyce book 2.1-2.3
- Existence and uniqueness, aprroximation, linearization stability. Boyce book 2.4-2.5
- Numerical methods, Boyce book 2.7, 8.1
- Numerical methods, Boyce book 8.2, 8.3
- Repeated and complex roots, Boyce book 3.4, 3.5
- Undetermined coefficients, Forcing. Boyce book 3.6, 3.8, 3.9
- Systems of ODEs, Boyce book 7.5, 7.6
- Repeated and complex eigenvalues, Boyce book 7.6, 7.8
- Nonlinear systems, Boyce book 9.1-9.3
- Series expansions and BVP, Boyce book 5.2, 5.3
- Regular Singular points and Euler equations, Boyce book 5.4, 5.5
- Solutions near a singular point. Boyce book 5.6
- Inner products and orthogonality, Fourier series, even and odd functions. Boyce book 10.1, 11.1, 10.3, 10.4
- Fourier series, SL-problem, boyce book 11.1, Haberman book 5.1, 5.3
- Intro to PDE
- Diffusion as a random walk
- Diffusion and separation of variables, Helmholtz eigenvalue problem
- Laplace equation
- Advection diffusion
- Wave Equation
- Combined effects: advection, diffusion, growth and decay