ACM 30020 Applied Analysis

I am teaching this module in Spring 2025.

Description: The purpose of this module is to learn a variety of mathematical methods for deriving useful approximate solutions of the differential equations and integrals found in the Mathematical Sciences. Topics to be covered in the module include existence and uniquess theory for ordinary differential equatins; integral equations, includinding the Volterra Integral Equation and the Fredholm Integral Equation; Sturm--Liouville Theory, including basic properties, unboundedness of eigenvalues, and completeness in the approriate sense of the set of eigenfunctions. Time permitting, we will also look at the theory of infinite-dimensional vector spaces.

Higher-Level Aim of Module: Often, in Applied Mathematics, there is the temptation to view a module such as this one as "just another maths methods module". The blame for this misconception lies with the Lecturer and not the students. There is the risk that such a misconception could arise in this module, because of the smorgasbord of techniques and methods covered. However, there is real substance in this module, as it is an introduction into a branch of Mathematics called Applied Analysis. Here, the aim is to take an equation inspired by applications, and to submit it to rigorous analysis. To ask questions like, "does a solution to this equation even exist?" Or, "how smooth is the solution?" Or, "if this model is a mathematical model for a population (say), is there a way a priori to say that the solution remains positive?" This module will equip students with the tools to answer such questions. Indeed, by the end of this module, students will be familiar with some of the issues involved in determining whether or not the Navier-Stokes equations in Fluid Dynamics have a unique globally defined solution in three dimensions. This is the main question tackled in the book below (see the screenshot); by the end of this module, students will have the background to read and appreciate this book. To emphasize this higher purpose of the module, the module has been given a name change in 2025.

Learning Outcomes: On completion of this module students should be able to:

  1. Understand conditions guaranteeing existence and uniqueness results for ordinary differential equations and recognize examples where those conditions do not hold.
  2. State and prove Picard's theorem.
  3. Transform between an initial value problem and the corresponding Volterra integral equation.
  4. Transform between a boundary value problem and the corresponding Fredholm integral equation.
  5. State the axiomatic properties of the Green function for a second order initial value problem and boundary value problem.
  6. Understand the concept of the adjoint differential operator.
  7. Recognise a Sturm-Liouville eigenvalue problem and prove the basic properties of eigenvalues and eigenfunctions.
  8. Understand the fundamental properties of infinite-dimensional vectors spaces (time permitting).
  9. Understand the application of these techniques to standard problems in Applied Mathematics.

Lecture Notes:

Course Documents:

Examinable results:

  • Week 1 / 2: Well-posedness of IVPs: existence, uniqueness, and continuous dependence on initial conditions
  • Week 2: Gronwall's Inequality (Lemma 2.1)

Problem Classes: