Description  % of the final grade  Date 
Location and Time 
Remarks 
Midterm  30%  Thu, Feb 27  In class  Review questions. 
Final  35%  Wed, Apr 29 
3pm, location TBA  In view
of the coronavirus outbreak, students have the option to use
their grade from the midterm as their grade for the final exam,
as discussed in class. If you still want to take the final exam,
please contact the instructor to discuss details. 
HW assignments  35%  on a regular basis (see below)  posted on webpage 
Date  Material covered  Homework  Remarks 

Jan 7  Introduction. Examples of PDEs. 
HW 1. HW 1 Solutions. 
HW 1 due on 1/16 at 5pm. 
Jan 9  The Schrodinger equation and separation of variables. 
Notes on the
Schrodinger equation. 

Jan 14  More on Schrodinger's equation. Separation of variables for the
1d wave equation. 
HW 2. HW 2 Solutions. 
HW 2 is due on 1/23 at 5pm. 
Jan 16  Fourier series. 
Summary of
theorems. 

Jan 21  The 1d wave equation on the real line. D'Alembert's formula. 

Jan 23  More on 1d waves: generalized solutions and propagation of
singularities. Some general tools for the study of PDEs. 
HW 3. HW 3 Solutions. 
HW 3 is due on Jan 30 at 5pm. 
Jan 28  Formal aspects of PDEs: general definitions and notation.
Laplace's and Poisson's equation in R^n. Fundamental solution to
Laplace's equation. 

Jan 30  Existence of solutions to Poisson's equation. Harmonic functions
and their properties. 
HW 4. HW 4 Solutions. 
HW 4 is due on 2/13 at 5pm. 
Feb 4  Project: heat equation. 
Project. Project Solutions. 
Project due on 2/13 at 5pm. 
Feb 6  Project: heat equation. 

Feb 11  More on harmonic functions: maximum principle and selected
results. The wave equation in R^n: finite propagation speed. 

Feb 13  Solutions to the wave equation in R^2 and R^3. 
HW 5. HW 5 Solutions. 
HW 5 is due on 2/20 at 5pm. 
Feb 18  Duhamel's principle. The Minkowski metric, Lorentz fields, and
commutator properties. 
HW 6. HW 6 Solutions. 
HW 6 is due on 2/27 at 5pm. 
Feb 20  Decay of solutions for the wave equation in R^n. 

Feb 25  Review for the test. 

Feb 27  Midterm  
Feb 29  Mar 8  Spring break  
Mar 10  Canceled due to the coronavirus outbreak 

Mar 12  Canceled due to the coronavirus outbreak 
Date  Material covered  Homework  Remarks 

Mar 17  The canonical form of second order linear PDEs: elliptic,
parabolic, and hyperbolic equations. General strategy for the
study of PDEs. Wellposedness. The method of characteristics. 
HW 7. HW 7 Solutions. 
HW 7 is due on 3/26 at 5pm. 
Mar 19  Local existence and uniqueness for firstorder quasilinear
equations via the method of characteristics. 

Mar 24  Burgers' equation and shockwaves. 

Mar 26  Scalar conservation laws. Weak solutions. 
HW 8. HW 8 Solutions. 
HW 8 is due on 4/9 at 5pm. 
Mar 31  No class (makeup class on Apr 7). 

Apr 2  No class (makeup class on Apr 9). 

Apr 7  The RankineHugoniot conditions. 

Apr 7 (makeup class) 
Systems of conservation laws. Simple waves.  
Apr 9  Rarefaction waves. 

Apr 9 (makeup class) 
The Riemann problem. Riemann invariants. 
HW 9. HW9 Solutions. 
HW 9 is due on 4/29 at 5pm. 
Apr 14  Blowup of solutions for 2x2 systems of conservation laws. 

Apr 16  Nonuniqueness of weak solutions. Entropy solutions. 