Schedule
Below is an ongoing schedule for the course (for the academic calendar,
click here).
Click here for the class notes, and
here for the class notes in handwritten form.
The class notes will be updated and typeset as the course progress. Since typesetting
them might take some time, typically the most up-to-date version of the notes will be
in the handwritten form. Non-math majors taking the course can consult
these notes
on preliminary notions for some background on mathematical concepts, conventions,
and notation that will be used in the course.
Date | Material covered | Remarks |
---|---|---|
Jan 18 | Introduction. Examples of PDEs. | HW1 is posted on Brightspace. Due: 1/25, 12pm. |
Jan 21 | More examples. Notations and conventions. | |
Jan 25 | Separation of variables for the Schrödinger equation with a radially symmetric potential. Solutions to the Θ equation. | HW2 is posted on Brightspace. Due: 2/1, 12pm. Solutions to HW1 are posted on Brightspace. |
Jan 27 | Schrödinger equation: solutions to the Φ equation. Legendre polynomials and spherical harmonics. | |
Feb 1 | Schrödinger equation: solutions to the radial equation. Final solution to the Schrödinger equation for the hydrogen atom. | HW3 is posted on Brightspace. Due: 2/8, 12pm. Solutions to HW2 are posted on Brightspace. |
Feb 3 | Separation of variables for the 1d wave equation. Fourier series. | |
Feb 8 | Properties and convergence of Fourier series. | HW4 is posted on Brightspace. Due: 2/15, 12pm. Solutions to HW3 are posted on Brightspace. |
Feb 10 | Sine and cosine Fourier series. Fourier series of periodic functions. Back to the wave equation: existence of solutions. | |
Feb 15 | The 1d wave equation on the real line. D'Alembert's formula. | HW5 is posted on Brightspace. Due: 2/22, 12pm. Solutions to HW4 are posted on Brightspace. |
Feb 17 | The characteristics of the 1d wave equation. Domains of dependence and influence. Singularities and generalized solutions. | |
Feb 22 | Some formal aspects of PDEs. | HW6 is posted on Brightspace. Due: 3/1, 12pm. Solutions to HW5 are posted on Brightspace. |
Feb 24 | Laplace's and Poisson's equation in ℝ^{n}: fundamental solution. | |
Mar 1 | Existence of solutions to Poisson's equation in ℝ^{n}. | HW7 is posted on Brightspace. Due: 3/15, 12pm. Solutions to HW6 are posted on Brightspace. |
Mar 3 | Harmonic functions, mean value, and maximum principle. | |
Mar 8 | Spring break, no class. | |
Mar 10 | Spring break, no class. | |
Mar 15 | The wave equation in ℝ^{n}. Domains of dependence and influence. Euler-Poisson-Darboux equation. | HW8 is posted on Brightspace. Due: 3/22, 12pm. Solutions to HW7 are posted on Brightspace. |
Mar 17 | Existence and uniqueness of solutions to the Cauchy problem for the wave equation in ℝ^{n}. Kirchhoff's and Poisson's formulas. | |
Mar 22 | The inhomogeneous wave equation; Duhamel's principle. Lorentz field and decay estimates for the wave equation. | HW9 is posted on Brightspace. Due: 3/29, 12pm. Solutions to HW8 are posted on Brightspace. |
Mar 24 | The Cauchy problem for the heat equation in ℝ^{n}. | |
Mar 29 | The method of characteristics: derivation of the characteristic system of ODEs. | HW10 is posted on Brightspace. Due: 4/5, 12pm. Solutions to HW9 are posted on Brightspace. |
Mar 31 | The method of characteristics: examples and the transversality condition. | |
Apr 5 | Existence and uniqueness of solutions to first-order quasilinear PDEs in one dimension satisfying the transversality condition. | HW11 is posted on Brightspace. Due: 4/12, 12pm. Solutions to HW10 are posted on Brightspace. |
Apr 7 | The Burgers equation. | |
Apr 12 | Blow-up of solutions to Burgers' equation; shock waves. Scalar conservations laws in one dimension. | HW12 is posted on Brightspace. Due: 4/19, 12pm. Solutions to HW11 are posted on Brightspace. |
Apr 14 | Weak solutions to scalar conservation laws in one dimension. | |
Apr 19 | The Rankine-Hugoniot conditions. Strictly hyperbolic systems of conservation laws in one dimension. | HW13 is posted on Brightspace. Due: 4/26, 12pm. Solutions to HW12 are posted on Brightspace. |
Apr 21 | Simple and rarefaction waves. The Riemann problem. | |
Apr 26 | Breakdown of solutions to 2x2 systems of conservation laws. Entropy solutoins. | Solutions to HW13 are posted on Brightspace. |
Apr 28 | Guest lecturer, Xinyue Zhao: applications of PDEs in biology. |