APMA 0360

Welcome to APMA 0360 – partial differential equations, but a whole lot of fun! On this website, you can find the syllabus, lecture notes, homework assignments, and study materials.

Syllabus

APMA 0360 – Syllabus (Summer 2025)

Homework Extension Form

Lecture Notes

Note: The lecture notes will be frequently updated. Please refresh your webpage so that you have the most up-to-date version of the notes

#
Date
Lecture Title
 YouTube Videos
0 Jun 16 Introduction Welcome to APMA 0360
1 Jun 16 What is a PDE? Partial Derivatives
What is a PDE?
What is a solution?
2 Jun 16 Classification of PDE Simple PDE
Classification of PDE
Linear PDE
Second-Order Types
3 Jun 17 First-Order Linear PDE Directional Derivatives
Constant-Coefficient PDE
Variable-Coefficient PDE
First-Order Example
4 Jun 17 Transport Equation The Chain Rule
Coordinate Method
Transport Equation Derivation
Transport Equation Solution
Coordinate Method (practice)
Transport Equation Demo (demo)
5 Jun 18 Wave Equation Derivation Wave Equation Intro
Wave Equation Derivation
6 Jun 18 Wave Equation Solution Undetermined Coefficients Review
Wave Equation Solution
Wave Coordinate Method
Wave Equation Demo (demo)
7-8 Jun 23 D’Alembert’s Formula D’Alembert’s Formula 
D’Alembert Interpretation
Oscillating Wave
The Plucked String
Oscillating Wave (demo)
The Plucked String (demo)
9 Jun 24 Heat Equation Derivation Heat Equation Intro
Heat Equation Derivation
Behavior of Solutions
Heat Equation (demo)
Heat Equation Visual (demo)
10  Jun 24 Fourier Transform Gaussian Integral
Fourier Transform
Fourier Transform Example
Schwartz Class (optional)
exp(kt) (demo)
Fourier Transform (demo)
11 Jun 25 Fourier and Heat PDE (I) Fourier Transform Miracle
Laplace and ODE
Solving the Heat Equation
Convolution
Convolution Intuition (optional)
Convolution (demo)
12 Jun 25 Fourier and Heat PDE (II) Solving the Heat Equation
Heat Kernel 
Heat Kernel (demo)
Heat Equation Example
13 Jun 26 Inverse Fourier Transform Inverse Fourier Transform
Inverse Fourier and PDE
Shifting Property
14 Jun 26 Exam 1 – Review First-Order PDE
Coordinate Method
Fourier Transform
Transform Method
Factoring Method
Jun 27 Exam 1 opens
15 Jun 30 Heat Equation Properties  Heat Equation Properties
16 Jun 30 Energy Methods (I) Energy Method (Wave)
Application: Uniqueness
Jul 1 Exam 1 closes at 11:59 pm
17 Jul 2 Energy Methods (II) Energy Method (Heat)
Handout Summary: Heat vs Wave Heat vs Wave
18 Jul 2 Separation of Variables (I) Boundary-Value Problems
Separation of Variables
Easier Problem
Separation of Variables (demo)
Separation of Variables (demo) 
19 Jul 7 Separation of Variables (II) Separation of Variables Wave
Inhomogeneous Problem
Easier Problem Wave (practice)
20 Jul 7 Fourier Sine Series Orthogonality
Fourier Sine Series
Sine Series Example
Fourier series (demo)
Heat equation (demo)
21 Jul 8 Fourier Cosine Series Fourier Cosine Series
Cosine Series Example
Full Fourier Series
22 Jul 8 Complex Fourier Series Complex Fourier Series
Complex Fourier Example
Convergence of Fourier Series
23 Jul 9 Parseval’s Formula Parseval’s Formula
Sum of 1/m²
Sum of 1/m^4
Sum of 1/(m²+1)
24 Jul 9 Laplace Equation Laplace Equation Intro
Laplace Applications
OMG Application
25 Jul 10 Separation of Variables (III) Separation of Variables Laplace
Variation
26 Jul 10 Exam 2 – Review Separation of Variables
Parseval
Energy Method Laplace
Jul 11 Exam 2 opens
27 Jul 14 Rotation Invariance Rotation Invariance
Polar Laplace (watch until 15:00)
28 Jul 14 Fundamental Solution Fundamental Solution
Jul 15 Exam 2 closes at 11:59 pm
29 Jul 16 Laplace Properties Mean-Value Formula
Maximum Principle
Laplace Uniqueness
Positivity
Maximum Principle Heat
30 Jul 16 Method of Characteristics Method of Characteristics
Characteristics Example
31 Jul 17 Ecology Application Ecology Application
Turing Model
Exponential Cosine (demo)
32 Jul 17 Wealth and Income Distributions Economics Application
Simplifying our Model
Macro Model
33-34 Jul 21 Epidemiology: COVID Model COVID Model
Rescaling Time
Traveling Waves
Transition Fronts
Two Cases
Transition Front (demo)
Transition Front Effect (demo)
35-36 Jul 22 Fluid Flow Application Vector Calculus Review
Incompressible Fluid
Navier-Stokes Derivation
37 Jul 23 The PDE that got me the PhD The PDE that got me the PhD
38 Jul 23 Exam 3 – Review Prologue
Dirac Delta
Characteristics Practice
Laplace Practice
Epilogue
Jul 25 Exam 3 opens
Jul 29 Exam 3 closes at 11:59 pm

Optional Handout: Calculus of Variations (Video)

Old Recitation Notes: Recitation Notes (courtesy Ethan Brady)

Assignments

Homework
Due Date
Solutions
Homework 1 Jun 20 Solutions
Homework 2 Jun 27 Solutions
Homework 3 Jul 4 Solutions
Non-Homework Never Solutions
Homework 4 Jul 11 Solutions
Homework 5 Jul 18 Solutions
Non-Homework Never Solutions
Homework 6 Jul 25 Solutions
Mini-Project Jul 25

YouTube Playlists

Chapter Playlist
1 Introduction
2 First-Order PDE
3 Wave Equation
4 Heat Equation
5 Energy Methods
6 Fourier Series
7 Laplace Equation
8 Applications

Exams

Exam 1: Friday, June 27 until Tuesday, July 1 at 11:59 pm, covers Lec 1-14

Practice Exam Solutions Comments
Study Guide
Mock Midterm 1 Solutions
Mock Midterm 2 Solutions Fall 2019 Midterm 1
Mock Midterm 3 Solutions Spring 2023 Midterm 1
Mock Midterm 4 Solutions Fall 2024 Midterm 1
Mock Midterm 5 Solutions Spring 2025 Midterm 1
Exam 1 Solutions Summer 2025 Midterm 1
Exam 1 – Review

Exam 2: Friday, July 11 until Tuesday, July 15 at 11:59 pm, covers Lec 15-26

Practice Exam Solutions Comments
Study Guide
Mock Midterm 1 Solutions
Mock Midterm 2 Solutions Summer 2012 Midterm 2
Mock Midterm 3 Solutions Fall 2019 Midterm 2
Mock Midterm 4 Solutions Spring 2023 Midterm 2
Mock Midterm 5 Solutions Fall 2024 Midterm 2
Mock Midterm 6 Solutions Spring 2025 Midterm 2
Exam 2 Solutions Summer 2025 Midterm 2
Exam 2 – Review
Extra Problems Solutions 18 practice problems
PDE Handout Old PDE Handout

Exam 3: Thursday, July 25 until Tuesday, July 29 at 11:59 pm, covers the entire course

Note: Please also check out the practice midterms above for more study material

Practice Exam Solutions Comments
Study Guide
Mock Final 1 Solutions
Mock Final 2 Solutions
Mock Final 3 Solutions Fall 2019 Final
Mock Final 4 Solutions Spring 2023 Final
Mock Final 5 Solutions Fall 2024 Final
Exam 3 Solutions Summer 2025 Final
Exam 3 – Review

Calendar