Partial Differential Equations and their Approximation module (MA42003)

One of the fundamental mathematical operations in calculus is finding derivatives of functions. Derivatives give the rate of change of a function. If functions have more than one input variable then this gives partial derivatives. Partial differential equations (PDEs) are equations involving partial derivative terms. PDEs are used to model a wide variety of applications. These include chemical reactions, population genetics, and airflow around vehicles.

Very few PDEs can be solved exactly, and so methods have been developed to find approximate solutions. In this module, you will learn numerical methods for calculating these approximate solutions.

You will be given a broad introduction to PDEs and will learn about how PDEs can be classified into different types. You will encounter classical solution methods, and techniques for constructing approximate solutions. You will develop your skills in applying PDEs to the wider world by focusing on model examples. This module will develop your analytical and numerical analysis skills. These skills will be for simulations of scientific and engineering problems. It will also prepare you to explore further areas of computational mathematics.

What you will learn

In this module, you will:

• learn about classification of second order PDEs into three main types. These are called elliptic, parabolic, and hyperbolic
• explore techniques called maximum principles, and separation of variables
• learn methods of solution called finite difference methods
• be introduced to the importance of concepts called convergence and stability for solution techniques.
• develop an understanding of a solution technique called the method of characteristics

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

• solve simple instances of first order PDEs by the method of characteristics
• solve model second order PDEs
• apply Green’s functions to solve PDEs
• be familiar with the basic qualitative properties of different types of PDE
• use finite difference techniques to approximate first and second order PDEs
• analyse the convergence properties of finite difference methods

Assignments / assessment

• coursework (20%)
• final exam (80%)

Teaching methods / timetable

• two one-hour lectures weekly
  • week's content will be discussed in detail
  • lecture notes covering the full module content will be available before classes
• one hour of tutorials weekly
  • solve problems individually and in groups
  • support with difficulties will be provided by your lecturers and peers