Differential Equations and their Approximation module (MA42009)

Many differential equations cannot be solved exactly. This module develops the analytical and numerical ideas needed to understand what can still be learned from them, and how reliable approximations can be constructed.

Building on MA31002 Differential Equations and MA32005 Scientific Computing and Numerical Methods, you will study methods for initial value and boundary value problems involving ordinary differential equations. The emphasis is not just on using a method, but on understanding why it works, how accurate it is, and when it can be trusted. You will learn how concepts such as local truncation error, stability, zero-stability, A-stability, and convergence explain the behaviour of numerical approximations.

The module also develops analytical ideas for boundary value problems. You will study eigenvalues and eigenfunctions, orthogonality, Green’s functions, maximum principles, and finite difference methods. These ideas help connect the qualitative behaviour of differential equations with practical methods for approximating their solutions.

You will then move on to partial differential equations, including the wave, heat, and Poisson equations. You will study classification, the method of characteristics, boundary conditions, finite difference methods, and stability ideas such as von Neumann analysis.

By combining differential equations with numerical analysis, this module prepares you to judge approximation methods critically and to understand how mathematical models can be solved when exact formulae are not available.

What you will learn

In this module, you will:

• analyse how error, stability, and convergence affect the reliability of numerical solutions to ODEs and PDEs
• solve boundary value problems using eigenvalue methods, Green’s functions, and finite differences
• study qualitative properties of differential equations, including maximum principles
• analyse model partial differential equations such as the heat, wave, and Poisson equations
• investigate finite difference methods, boundary conditions, and von Neumann stability analysis.

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

• compare analytical and numerical approaches to solving differential equations
• apply numerical methods to initial value problems for ODEs
• assess whether an approximation method is accurate, stable, and convergent
• formulate and solve boundary value problems for ODEs
• construct and analyse numerical schemes for model PDEs
• explain how error, stability, and convergence affect the reliability of approximations

Assignments / assessments

• Coursework (20%)
• Written exam (80%)

Teaching methods / timetable

• lectures, introducing analytical and numerical methods for ordinary and partial differential equations
• worked examples, showing how approximation methods are constructed, analysed, and applied to model problems
• tutorials, where you will practise solving problems involving stability, convergence, error analysis, and finite difference methods.