Math 199C – PDE Reading Course

In this reading course, we’ll study the three most famous PDEs: Laplace’s equation, the heat equation, and the wave equation, and study them from a more theoretical perspective: How do you derive their ‘fundamental’ solutions? What properties do they satisfy? What more can you say about them?


Syllabus

Here is the Syllabus for the course: Syllabus


Course Schedule 

Note: The readings are based on the book “Partial Differential Equations” (Second Edition) by Lawrence C. Evans, GSM Volume 19, American Mathematical Society, ISBN 978-0-8218-4974-3

Lecture Notes Section Title YouTube Videos
Week 1 Chapter 1 Introduction What is a PDE?
Pages 20-21 Derivation of Laplace Derivation of Laplace
See Notes Applications of Laplace Applications of Laplace
Section 2.2.1a Derivation of Fundamental Solution Fundamental Solution of Laplace
Week 2 See Notes Dominated Convergence Theorem
See Notes Polar Coordinates Formula
See Notes Integration by Parts
Section 2.2.1b Poisson’s Equation Poisson’s Equation
Week 3 See Notes Change of Variables The Jacobian
Section 2.2.2 Mean Value Formula Mean Value Formula
Section 2.2.3a Strong Maximum Principle and Uniqueness
Section 2.2.3f Harnack’s Inequality
Week 4 Section 2.2.3c (Optional) Local Estimates for Harmonic Functions (only the case k = 1)
Section 2.2.3d Liouville’s Theorem
Appendix C.5 Convolution and Smoothing (Skip the proofs)
Section 2.2.3b Regularity
Section 2.2.5 Energy Methods Calculus of Variations
See Notes Calculus of Variations
Week 5 Page 44 Derivation of the Heat Equation
See Notes Applications of the Heat Equation
Section 2.3.1a Fundamental Solution Fundamental Solution of Heat
See Notes Gaussian Integral Gaussian Integral
Gauss Playlist
Gauss Cubed
Section 2.3.1b Initial-Value Problem Initial Value Problem
Week 6 See Notes Infinite Speed of Propagation
Section 2.3.1c Non-homogeneous Problem
Section 2.3.2 Mean-Value Formula
Section 2.3.3a Strong Maximum Principle (up to and including page 56)
Week 7 Section 2.3.4 Energy Methods
See Notes Convexity
Section 2.3.a Strong Maximum Principle (pages 57-59)
Lecture 8 Section 2.1 Transport Equation Transport Equation
Section 2.4 The Wave Equation (pages 65-66)
Section 2.4.1a D’Alembert’s Formula D’Alembert’s Formula
See Notes Some Consequences Reflections of Waves
Section 2.4.1b Spherical Means
Week 9 Section 2.4.3 Energy Methods
Section 2.4.1c Solution for n = 3 (pages 71-72)
Section 2.4.1c (Optional) Solution for n = 2 (pages 73-74)
Section 2.4.2 (Optional) Non-homogeneous Problem
Week 10 See Notes Chemical Reactions and Diffusions The PDE that gave me the PhD

Suggested Homework

Note: The homework and old exams are for extra practice only, you do not have to do them or hand them in. Don’t do the whole assignment, just the problems outlined below. Also, there will not always be suggested HW every week; in that case, you’ll have to find a different way of getting credit for the week, such as sending me a question or posting/answering on campuswire.

 

Suggested HW Problems Solutions
Homework 1 Chapter 1: 5 Solutions
Chapter 2: 2
Homework 2 See Assignment Solutions
Homework 3 Chapter 2: 3 Solutions
See Assignment
Homework 4 Chapter 2: 4, 5, 6 Solutions
Homework 5 Chapter 2: 12, 13 Solutions
See Assignment
Homework 6 Chapter 2: 14 (in HW 5) 16, 17 Solutions
See Assignment
Homework 7 Mock Midterm: 3, 4 Solutions
Homework 8 Chapter 2: 1, 18 Solutions
Homework 9 Chapter 2: 19(a)(b)(c), 21 Solutions
Homework 10 Chapter 2: 22 (in HW 9), 24, AP1 Solutions
Mock Final Exam Optional Mock Final (1, 2, 4, 8, 9, 10) Solutions