Mathematics

Partial Differential Equations

Module code: G1114
Level 6
15 credits in autumn semester
Teaching method: Lecture
Assessment modes: Unseen examination, Coursework

This module introduces you to the theory of Partial Differential Equations (PDEs) studying in detail the three fundamental linear PDEs of second-order: the Laplace equation, the heat equation, and the wave equation. These equations are, respectively, the primary examples of elliptic, parabolic, and hyperbolic PDEs and form the basis for many equations that appear in the modelling of the physical and life sciences.

The module also serves as a foundation for several subsequent modules in Analysis and PDEs.

You will learn a variety of techniques to study the above equations and learn how to construct explicit solutions. For example, topics include the separation of variables method, Greene’s identities, maximum principles, Duhamel’s principle, D’Alembert’s solution, as well as energy methods. 

Module learning outcomes

• Know the classification of second-order partial differential equations;
• Be able to solve model problems involving second-order partial differential equations;
• Be able to formulate the standard boundary-value, initial-value and boundary-initial-value problems for the Laplace, heat and wave equations;
• Be able to state and prove the uniqueness and existence theorems for the above equations using the energy method and the Maximum principle.