1 Introduction
 1.1 Ordinary differential equations
 1.2 PDE’s
2 Classification of partial differential equations.
 2.1 Examples of PDE
 2.2 Second order PDE
  2.2.1 Reason behind names
 2.3 Quiz
 2.4 More than 2D
3 Boundary and Initial Conditions
 3.1 Explicit boundary conditions
  3.1.1 Dirichlet boundary condition
  3.1.2 von Neumann boundary conditions
  3.1.3 Mixed (Robin’s) boundary conditions
 3.2 Implicit boundary conditions
 3.3 A slightly more realistic example
  3.3.1 A string with fixed endpoints
  3.3.2 A string with freely floating endpoints
  3.3.3 A string with endpoints fixed to strings
4 Fourier Series
 4.1 Taylor series
 4.2 Introduction to Fourier Series
 4.3 Periodic functions
 4.4 Orthogonality and normalisation
 4.5 When is it a Fourier series?
 4.6 Fourier series for even and odd functions
 4.7 Convergence of Fourier series
5 Separation of variables on rectangular domains
 5.1 Cookbook
 5.2 parabolic equation
 5.3 hyperbolic equation
 5.4 Laplace’s equation
 5.5 More complex initial/boundary conditions
 5.6 Inhomogeneous equations
6 D’Alembert’s solution to the wave equation
 6.1 Background
 6.2 New variables
  6.2.1 Infinite String
  6.2.2 Finite String
 6.3 Examples
7 Polar and spherical coordinate systems
 7.1 Polar coordinates
 7.2 spherical coordinates
8 Separation of variables in polar coordinates
 8.1 Example
  8.1.1 Periodic BC
 8.2 Three cases for λ
 8.3 Putting it all together
9 Series solutions of O.D.E.
 9.1 Singular points
 9.2 *Special cases
  9.2.1 Two equal roots
  9.2.2 Two roots differing by an integer
  9.2.3 Example 1
  9.2.4 Example 2
10 Bessel functions
 10.1 Temperature on a disk
 10.2 Bessel’s equation
 10.3 Gamma function
 10.4 Bessel functions of general order
 10.5 Properties of Bessel functions
 10.6 Sturm-Liouville theory
 10.7 Our initial problem and Bessel functions
 10.8 Fourier-Bessel series
 10.9 Back to our initial problem
11 Separation of variables in three dimensions
 11.1 Modelling the eye
 11.2 Properties of Legendre polynomials
  11.2.1 Generating function
  11.2.2 Rodrigues’ Formula
  11.2.3 A table of properties
 11.3 Fourier-Legendre series
 11.4 Modelling the eye–revisited