Darlithoedd

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Lecture 13

Summary.

  • Uniquess of wave equation solutions - the proof

  • The heat equation

  • The maximum principle

Lecture 12

Summary.

  • The wave equation on a finite-length string continued

  • The problem for \(T(t)\)

  • Finding the Fourier coefficients

  • Uniquess of wave equation solutions

Lecture 11

Summary.

  • The wave equation on a finite-length string

  • Seeking a separable solution

  • The Sturm-Liouville problem for \(X(x)\)

  • Seeking non-trivial solutions

  • Eigenvalues and eigenfunctions

Lecture 10

Summary.

  • Derivation of Duhamel's formula (continued)

  • Example of its application

  • Solving the inhomogeneous wave equation with inhomogeneous initial conditions

Lecture 09

Summary.

  • Finalising the solution to the wave equation Cauchy problem on a semi-infinite string

  • The inhomogeneous wave equation

  • Starting to derive Duhamel's formula

Lecture 08

Summary.

  • Causality (domains of dependence and influence)

  • The wave equation on a semi-infinite string

  • Solving this using odd reflection

Lecture 07

Summary.

  • The wave equation

  • Derivation of the one-dimensional wave equation's general solution

  • The Cauchy problem for the wave equation

  • Derivation of d'Alembert's formula (solution to aforementioned Cauchy problem)

Lecture 06

Summary.

  • Proving the canonical forms theorem

  • Well-posed problems

  • Hadamard's example of an ill-posed problem

  • Dirichlet and Neumann boundary conditions

  • Examples of well-posed problems for elliptic, hyperbolic, and parabolic PDEs

Lecture 05

Summary.

  • Case 3: \(a(x,y) u_x + b(x,y) u_y + c(x,y)u = f(x,y)\) (solution reduces to ODE along characteristic curves). Another example.

  • Classification of second order linear PDEs: elliptic, hyperbolic, parabolic.

  • A theorem on canonical forms (proof next time).

Lecture 04

Summary.

  • Parameterising curves

  • First order linear PDEs - the method of characteristics

  • Case 3: \(a(x,y) u_x + b(x,y) u_y + c(x,y)u = f(x,y)\) (solution reduces to ODE along characteristic curves)

Lecture 03

Summary.

  • Vector calculus refresher

  • First order linear PDEs - the method of characteristics

  • Case 1: \(a u_x + b u_y = 0\) (solution is constant along characteristic lines)

  • Case 2: \(a(x,y) u_x + b(x,y) u_y = 0\) (solution is constant along characteristic curves)

Lecture 02

Summary.

  • Linearity

  • Order

  • Homogeneous and inhomogeneous

  • The Superposition Principle

  • Some simple PDEs: ODEs dressed up as PDEs

  • Directional derivative

Lecture 01

Summary.

  • How the module works

  • Why study PDEs?

  • Notation

  • Introducing classification: linearity