### 课程介绍

This course discusses Green function methods for solving partial differential equations, focusing on the Laplace equation but also extending to other PDEs. Starting from the Dirac delta function as a formal symbol to denote a point source, we begin a discussion of generalized functions (distributions), including weal derivatives and the distributional Fourier transform, leading to weak (distributional) solutions of ordinary and partial differential equations.

Pre-requisites: Applied Calculus IV, Honors Mathematics IV, or consent of instructor

### 课程大纲

----------------------------
Week 1: Introduction
----------------------------
The Classical Heat Equation
The Equilibrium Heat Equation and Classical Solutions
Point Sources and Green Functions
A Solution Formula using the Green Function
Proof of the Solution Formula
Further Approaches to the Green Function

----------------------------
Week 2: Distributions
----------------------------
Smooth, Compactly Supported Functions
Null Sequences
Test Functions and Distributions
Elementary Operations on Distributions
The Weak Derivative
Two Applications of the Weak Derivative

----------------------------
Week 3: Families of Distributions and the Fourier Transform
----------------------------
Families of Distributions
Delta Families
The Fourier Transform and Functions of Rapid Decrease
Continuity of the Fourier Transform
The Fourier Inversion Formula and Properties of the Fourier Transform

----------------------------
Week 4: Differential Eqautions
----------------------------
Tempered Distributions
Application of the Fourier Transform to Partial Differential Equations
Linear Differential Operators and Green’s Formula
Classical and Weak Solutions
Fundamental Solutions
Initial Value Problems, Independence and the Wronksian
The Homogeneous Equation with Non-Vanishing Initial Conditions
The Inhomogeneous Equation

----------------------------
Week 5: Boundary Value Problems for ODEs
----------------------------
Second-Order Boundary Value Problems
Second-Order Boundary Value Problems with Separated Boundary Conditions
Green’s Function and a Solution Formula for Second-Order Boundary Value Problems
The Adjoint Second-Order Boundary Value Problem
The Adjoint Green Function for a Second-Order Problem

----------------------------
Week 6: More on Boundary value Problems for ODEs
----------------------------
Boundary Value Problems of General Order
Solvability Conditions
Modified Green Functions
Solution Formula via Modified Green Functions

----------------------------
Week 7: Partial Differential Equations
----------------------------
Second-Order Equations and Boundary Value Problems
The Elliptic Boundary Value Problem
The Parabolic Boundary Value Problem
Solution Formula for the Parabolic BVP
Causal Fundamental Solution for the Parabolic BVP

----------------------------
Week 8: Eigenfunction Expansions
----------------------------

The Eigenvalue Problem for the Elliptic Operator
Full Eigenfunction Expansions
Partial Eigenfunction Expansions

----------------------------
Week 9: The Method of Images
----------------------------
The Method of Images
Exploiting Symmetries for Image Charges
Series of Image Charges
Green functions for Dirichlet, Neumann and Robin problems on the Upper Half Space
The Boundary Integral Solution
The Boundary Element Method

### 学习要求

Students are expected to have taken courses in single- and multi-variable calculus and in ordinary differential equations. Some knowledge of classical solution methods of PDEs (e.g., separation of variables) will be helpful.