Mathematical Methods 4 (MATH0056)

This chapter focuses on ODEs, which are essential when solving PDEs. In particular it show how to solve ordinary differential equation using the Frobenius method, i.e. finding a series solution to the ODE.

Introduction to Fourier and Laplace transforms and their reverse transform and their use to solve partial differential equations

This document introduces to Sturm-Liouville eigenvalue problems, regular, singular and periodic, to find eigenvalues and eigenfunctions of partial differential equations. In introduces the concept of orthogonality of eigenfunctions. Then it focuses on how to generate the generalised Fourier series expansion of a function, and in particular the Fourier-Bessel and Fourier-Legendre series. Finally, it introduces inhomogeneous problems and how to solve them.

Providing a general approach to solve problems involving partial differential equations using the separation of variables method

This chapter introduces important PDEs such as the heat equation, wave equation, Laplace equation and Helmholtz equation and gives examples of how to solve them using separation of variables. Then it introduces the 3 coordinate systems of polars, cylindrical polars and spherical, showing how to go from one system to another. Finally it shows how separation of variables can be used to to solve some important types of equation such as the Euler-type equation, the Bessel's equation and the Legendr...