General Information
For a description of the course, including the grading policy, consult the course syllabus. Students are responsible for reading the syllabus and being aware of all the course and university policies. If you are taking this course for graduate credit, consult the MATH 5120 syllabus.  

This course will be taught through a hybrid online/in-person model. See the course syllabus for details.

Textbook: No textbook will be required. See the course syllabus for suggested references.

Classes meet on TR, 2:20–3:35pm at Buttrick Hall 206.

Due to the COVID-19 outbreak, we will be following all guidelines stipulated by Vanderbilt University. See the return to campus website, the Vanderbilt COVID-19 response website, and the Vanderbilt COVID-19 dashboard for more information.

Contact Information and Office Hours
Instructor's office: Stevenson Center 1515 (5th floor of the Mathematics Building).
Instructor's email: marcelo.disconzi@vanderbilt.edu.
Instructor's office hours: Tuesdays, 4-5pm, Thursdays, 1-2pm and 5-6pm, or by appointment. Office hours will be held virtually. If you want to meet in person, please schedule an appointment.
Instructor's office phone: (615) 322-7147.

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.

Date	Material covered	Remarks
Jan 26	Introduction. Examples of PDEs.	HW 1 is posted on Brightspace.  Due: Feb 2, 11:59pm.  Solutions to HW 1.
Jan 28	Separation of variables for the Schrodinger equation with a radially symmetric potential.
Feb 2	More on the Schrodinger equation. The angular equation (Legendre polynomials and spherical harmonics). The radial equation.	HW 2 is posted on Brightspace.  Due: Feb 9, 4pm.  Solutions to HW 2.
Feb 4	Final remarks on the Schrodinger equation. Separation of variables for the one-dimensional wave equation.
Feb 9	More on the wave equation. Fourier series.	HW 3 is posted on Brightspace.  Due: Feb 16, 4pm.  Solutions to HW 3.
Feb 11	Convergence of Fourier series.
Feb 16	Sine and cosine Fourier series. Back to the initial-value-boundary problem for the 1d wave equation. The 1d wave equation on the real line; D'Alembert's formula.	HW 4 is posted on Brightspace.  Due: Feb 22, 11:59pm.  Solutions to HW 4.
Feb 18	Domains of influence and dependence for the 1d wave equation. Generalized solutions to the 1d wave equation and propagation of singularities. Some general tools for the study of PDEs.	HW 5 is posted on Brightspace.  Due: Mar 2, 4:00pm.  Solutions to HW 5.  Extra credit 1 is posted on Brightspace.  Due: Mar 2, 4:00pm.
Feb 23	Formal aspects of PDEs (calculus facts, multi-index notation, some useful conventions, etc.)	In-class reading day.
Feb 25	The fundamental solution to the Laplacian in R^n. Solutions to Poisson's equation in R^n.	In-class reading day.
Mar 2	Harmonic functions. The mean value theorem and its converse.	HW 6 is posted on Brightspace.  Due: Mar 9, 4:00pm.  Solutions to HW 6.  Extra credit 2 is posted on Brightspace.  Due: Mar 11, 11:59pm.
Mar 4	The wave equation in R^n. Finite propagation speed.
Mar 9	Toward solutions to the wave equation in R^n. Spherical averages and the Euler-Poisson-Darboux equation. Reflection method.	HW 7 is posted on Brightspace.  Due: Mar 16, 4:00pm.  Solutions to HW 7.
Mar 11	Existence and uniqueness of solutions to the Cauchy problem for the wave equation in n=2,3. Kirchhoff's and Poisson's formulas.
Mar 16	Solutions to the inhomogeneous wave equation in R^n; Duhamel's principle. Vectorfields as differential operators. The Minkowski metric.	HW 8 is posted on Brightspace.  Due: Mar 23, 4:00pm.  Solutions to HW 8.  HW 9 (Heat eq. project) is posted on Brightspace.  Due: Mar 30, 4:00pm.  Solutions to HW 9.
Mar 18	Lorentz vectorfields. Differential operators and commutators. Sobolev inequality.
Mar 23	Decay estimates for solutions to the wave equation in R^n.
Mar 25	The canonical form of second order linear PDEs: elliptic, parabolic, and hyperbolic equations. General strategy for the study of PDEs. Well-posedness. The method of characteristics.	HW 10 is posted on Brightspace.  Due: Apr 6, 4:00pm.  Solutions to HW 10.
Mar 30	Local existence and uniqueness for quasilinear first-order PDEs via the method of characteristics.
Apr 1	Bugers' equation. Blow-up and shocks.	HW 11 is posted on Brightspace.  Due: Apr 13, 4:00pm.  Solutions to HW 11.
Apr 6	Scalar conservation laws in one dimension. Weak solutions. The Rankine-Hugoniot conditions.	The class projects are posted on Brightspace.  Due: May 5, 4:00pm.
Apr 8	Systems of conservation laws in one dimension. Simple waves. Rarefaction waves.	In-class reading day.
Apr 13	The Riemann problem. Riemann invariants.	HW 12 is posted on Brightspace.  Due: Apr 20, 4:00pm.  Solutions to HW 12.
Apr 15	Breakdown of solutions to 2x2 systems of conservation laws. Non-uniqueness of weak solutions. Entropy solutions.
Apr 20	Review/class project.
Apr 22	Review/class project.
Apr 27	Review/class project.
Apr 29	Review/class project.

Anonymous feedback
Students are encouraged to bring suggestions and discuss with the course instructor any concerns they may have, including something they think is not being properly handled in the course. But if you do not feel comfortable doing that, here you have the opportunity to send anonymous feedback.