Ordinary Differential Equations and the Approximation module (MA41003)

This module explores calculus, specifically to find derivatives of functions.

Derivatives give the rate of change of a function. For example, speed is the rate of change of distance, while acceleration is the rate of change of speed. Ordinary Differential Equations (ODEs) are equations involving derivative terms.

Therefore an ODE can be used to describe a relationship between distance, speed, and acceleration. ODEs are an essential modelling tool in science and engineering.

Very few ODEs can be solved exactly, and so methods have been developed to find approximate solutions.

In this module, you will learn the basic numerical methods for calculating these approximate solutions. These are methods typically used by mathematicians working in industry using computer packages to solve real-world problems.

What you will learn

In this module, you will:

• be introduced to ODEs
• learn about key types of numerical methods, including the Taylor series method, Linear Multistep Methods (LMMs), and Runge-Kutta methods.
• learn about Initial Value Problems (IVPs) and Boundary Value Problems (BVPs)
• discover the comparison principles for BVPs
• explore approximation of BVPs

By the end of this module, you will be able to:

• solve IVPs for ODEs by series methods
• derive and use Runge-Kutta methods for IVPs for ODEs
• analyse properties of Runge-Kutta methods which give a measure of the strength of the method, such as order and stability
• derive and apply linear multi-step methods for the approximate solution of ODEs
• analyse convergence and stability properties of linear multi-step methods
• solve a related theoretical problem known as eigenvalues problems for ODEs, and use them to solve BVPs
• apply basic ideas of Maximum Principles to BVPs of ODEs
• derive finite difference methods for BVPs and study their convergence properties

Assignments / assessment

• coursework (20%)
• final exam (80%)

Teaching methods / timetable

• one-hour lectures weekly
  • key points of the week's content will be discussed
  • lecture notes covering the full module content will be given before classes
  • in-class time is prioritised for interactive discussions
• two hours of tutorials weekly
  • solve problems individually and in groups
  • support with difficulties will be provided by your lecturers and peers