MA 523 Fall 2024

Lecture Notes   Click on the three books icon to collapse the big left margin. Zoom in or out for a comfortable view.

Homework

• HW1 due 9/2 (Mon) on Gradescope
• HW2 due 9/9 (Mon) on Gradescope 
• HW3 due 9/25 (Wed) on Gradescope 
• HW4 due 10/4 (Fri) on Gradescope
• HW5: p.110, #2, #3, #4, #5 (see this scanned page), and this problem.  Due 11/3 on Gradescope.
• HW6 due 11/15 (Fri) on Gradescope
• HW7:  p.145: #2, #4, #7. Due 12/1 on Gradescope

Other recommended books: 

• Folland, Introduction to Partial Differential Equations 
• Evans, Partial Differential Equations 
• F. John, Partial Differential Equations 

Problem sets:

Weekly problem sets will be posted on this page.

Midterms

Three evening midterm exams will be given, as follows.

• Exam 1: Thu, 9/12 at 8:00pm in ARMS 1109. Will cover Chapter 1 (1st order PDEs). 
• Exam 2: Thu, 10/10 at 8:00pm in ARMS 1109. Will cover Chapter 2 and Chapter 3 up to Kirchhoff’s formula in 3D. 
• Exam 3: Thu, 11/14 at 8:00pm in ARMS 1109.  Will cover the material starting from Chapter 3 (the wave equation in dimensions 1, 2, 3) up to the Dirichlet problem for the Laplace equation in a half-plane and in a ball and properties of harmonic functions. In other words, up to what I covered last week.

Final exam

A Final Exam will be given during the Final Exam Week.

Course grade

Your course grade will be determined using the following distribution: HW: 30%; Midterms 16% each (48% total); Final: 22%.

Schedule 

1. First Order Equations
Quasi-linear equations. Method of Characteristics and the Cauchy problem for 1st order PDEs.
Transport equations, shocks formations, conservation laws.
General non-linear equations: the eikonal equation only.

2. Higher (mainly second) Order Equations
The Cauchy Problem,  The Cauchy-Kowalevski Theorem, The Lewy example, 
Characteristics. See this for characteristics in any dimension. 
Linear 2nd order PDEs, normal form in two dimensions
Weak solutions
Distributions and fundamental solutions

3. The Wave Equation
The 1D wave equation
Methods of Spherical Means, Kirchhoff’s formula in 3D
Hadamard’s Method of Descent, solution formula in 2D
Duhamel's Principle, Energy

4. The Laplace Equation
Separation of Variables, Green's identities
Fundamental Solution, Poisson's Formula,
Maximum Principle, Mean Values
Fundamental solution, Green's function and Poisson Kernel
The Dirichlet problem in a half-plane and in a ball
Existence theorems  

5. The Heat Equation
Heat Kernel, Maximum Principle and uniqueness
Separation of variables, smoothing property
Infinite speed of propagation.