Lecture Recordings

Revision 3

Summary.

  • Revision

Revision 2

Summary.

  • Revision

Revision 1

Summary.

  • Revision

Lecture 19

Summary.

  • Solving the heat equation on an infinitely long rod using Fourier transforms

Lecture 18

Summary.

  • Fourier transform

  • Consideration of functional spaces

  • The Fourier transform of \(f \in L^1(\mathbb{R})\) is not necessarily in \(L^1(\mathbb{R})\) itself

  • Schwartz space

  • Different conventions/definitions for the Fourier transform pair

Lecture 17

Summary.

  • Fourier transform

  • Convolution theorem

  • Fourier inversion theorem

  • Example of using Fourier transforms to solve Laplace’s equation in a strip

Lecture 16

Summary.

  • Fourier transform

  • Existence, linearity, examples of computing the Fourier transform

  • Shifting theorem, scaling theorem

  • Derivative theorem

Lecture 15

Summary.

  • Proving the maximum principle

  • Uniqueness of heat equation solutions (via maximum principle)

  • Introducing Fourier transforms

Lecture 14

Summary.

  • Uniqueness of wave equation solutions

  • Using energy integrals

  • Introducing the heat equation

  • The parabolic boundary and maximum principle

Lecture 13

Summary.

  • Wave equation on a finite length string

  • Sturm-Liouville problem; seeking a separable solution

  • Arriving at the solution

  • Finding the Fourier coefficients

Lecture 12

Summary.

  • Inhomogeneous wave equation with inhomogeneous initial conditions

  • Wave equation on a finite length string

  • Sturm-Liouville problem; seeking a separable solution

Lecture 11

Summary.

  • The inhomogeneous wave equation

  • Derivation of Duhamel’s formula

  • Example of application

Lecture 10

Summary.

  • The wave equation on a semi-infinite string (continued)

  • The inhomogeneous wave equation

Lecture 09

Summary.

  • Causality: domains of dependence and influence

  • The wave equation on a semi-infinite string

Lecture 08

Summary.

  • Derivation of d’Alembert’s formula

  • Example

  • Discussion of causality

Lecture 07

Summary.

  • Well-posed problems

  • Hadamard’s example of a well-posed problem

  • General solution of the wave equation: \(u(x,t)=F(x+ct)+G(x-ct)\)

Lecture 06

Summary.

  • Second order linear PDEs

  • Classification: hyperbolic, elliptic, parabolic

  • Reduction to canonical form

Lecture 05

Summary.

  • Examples of solving the general case: \(a(x,y)u_x + b(x,y)u_y + c(x,y)u = f(x,y)\)

Lecture 04

Summary.

  • Method of Characteristics

  • Example of \(a u_x + b u_y = 0\), for \(a, b \in \mathbb{R}\)

  • Variable coefficients: example of the form \(a(x,y) u_x + b(x,y) u_y = 0\)

  • Parameterising curves

  • The general case: \(a(x,y)u_x + b(x,y)u_y + c(x,y)u = f(x,y)\)

Lecture 03

Summary.

  • Integrating factor example

  • Vector calculus refresher

  • Gradient

  • Directional derivative

  • First order linear PDEs of the form \(a u_x + b u_y = 0\), for \(a, b \in \mathbb{R}\)

Lecture 02

Summary.

  • Linear operators

  • Linear PDEs

  • Order

  • Linear homogeneous vs linear inhomogeneous

  • The superposition principle

  • Simple PDEs… PDEs have arbitrary functions in their general solutions.

Lecture 01

Summary.

  • How the module works

  • Why study PDEs?

  • Notation

  • Introducing classification