Class notes
Click here for the class notes, and
here for the class notes in handwritten form. If you find some inconsistency or something that seems wrong in the class notes, it is likely a typo. In this case, double check the handwritten class notes and let the instructor know so that the notes can be corrected.
Here are some complementary notes that cover some important background for the course.
Schedule
Below is schedule for the course which will be updated as the course progresses (for the academic calendar,
click here). Sections refer to the class notes. HW will be collected on Brightspace.
Date | Material covered | HW and remarks |
---|---|---|
Jan 9 | Introduction. Sections 1-3. | HW1 is posted on Brightspace. |
Jan 11 | Separation of variables for the Schrodinger equation. Sections 4-4.4. | |
Jan 16 | Canceled due to inclemental weather. | |
Jan 18 | Spherical harmonics. The radial Schrodinger equation for the hydrogen atom. Sections 4.4-4.6. | HW2 is posted on Brightspace. |
Jan 23 | Separation of variables for the 1d wave equation. Fourier series. Sections 5-6.1. | HW3 is posted on Brightspace. |
Jan 25 | Some function spaces. Piecewise functions. Convergence of Fourier series. Sections 6.1-6.3. | |
Jan 30 | Fourier series of periodic functions, sine and cosine Fourier series. Fourier series and the wave equation. Sections 6.4-6.5. | HW4 is posted on Brightspace. |
Fab 1 | The wave equation in ℝ. D'Alembert's forumla. Existence and uniqueness. Domains of dependence and influence. Sections 7-7.1. | |
Feb 6 | Generalized solutions to the 1d wave equation. Some tools for studying PDEs. Sections 7.2-8.4. See also the complementary notes. | HW5 is posted on Brightspace. |
Feb 8 | Formal aspects of PDEs. Section 9. | |
Feb 13 | Fundamental solution to the Laplacian in ℝ^{n}. Section 10. | HW6 is posted on Brightspace. |
Feb 15 | Solutions to Poisson's equations. Harmonic functions. Sections 10-10.1. | |
Feb 20 | The maximum principle. Brief overview of further results for Laplace's and Poisson's equations: Liouville's theorem, the Dirichlet problem, Green's function. Sections 10.1-10.2. | HW7 is posted on Brightspace. |
Feb 22 | The wave equation in ℝ^{n}: finite speed of propagaion, Euler-Poisson-Darboux equation. The reflection method. Sections 11-11.1. | |
Feb 27 | Kirchhoff's and Poisson's formulas. Existence and uniqueness of solutions to the Cauchy problem for the wave equation. Sections 11.2-11.4. | |
Feb 29 | The inhomogeneous wave equation; Duhamel's principle. Vectorfields as differential operators. Sections 11.5-11.6. | HW8 is posted on Brightspace. |
Mar 5 | The Minkowski metric and the Lorentz vectorfields. Some properties of differential operators. Section 11.7. | |
Mar 7 | Decay estimates for the wave equation. Section 11.8. | |
Mar 9-17 | Spring break. | |
Mar 19 | Decay estimates for the wave equation. Classification of PDEs in elliptic, hyperbolic, parabolic. Sections 11.8-12. | |
Mar 21 | The method of characteristics. The characteristic ODE system. Section 13. | HW9 is posted on Brightspace. |
Mar 26 | The transversality condition in the method of characteristics. Section 13. | |
Mar 28 | Local existence and uniquenes for quasi-linear equations via the method of characteristics. Section 13. | |
Apr 2 | Burgers' equation. Shocks. Sections 13.2, 13.3. | |
Apr 4 | Scalar conservation laws in one-dimension. Section 14. | HW10 is posted on Brightspace. This will be the last HW of the course. |
Apr 9 | Review for the final exam. | A study guide for the exam is posted on Brightspace. |
Apr 11 | Review for the final exam. | |
Apr 16 | The Rankine-Hugoniot conditions. Section 14.1. | |
Apr 18 | Systems of conservation laws in one dimension. | Not in the final exam. |
May 1st, 3pm | Final exam. |