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 |