Partial differential equations are a central modelling tool in applied mathematics and mathematical physics. They also play an important role in pure mathematics, not least as a stimulus in the development of concepts and methods of classical and modern analysis.

This module provides an introduction to the classical analytical treatment of second-order linear partial differential equations. The essential concepts and methods are introduced and developed for prototype partial differential equations representing the three classes: parabolic; elliptic; hyperbolic.

• Use the method of characteristics to solve elementary first order partial differential equations.
• Classify second-order partial differential equations.
• Interpret the concept of well-posedness of initial/boundary value problems.
• Recognise the characteristic properties of basic partial differential equations and their solutions.
• Understand and apply classical methods of solving basic partial differential equations.

Modules will be delivered through blended learning. You will be guided through learning activities appropriate to your module, which may include:

• Weekly face to face classes (e.g. labs, lectures, exercise classes)
• Electronic resources that you work through at your own pace (e.g. videos, exercise sheets, lecture notes, e-books, quizzes)

Students are also expected to undertake self-guided study throughout the duration of the module.

Skills that will be practised and developed:

• Achieve precision and clarity of exposition.
• Recognise, formulate and solve problems in an interdisciplinary environment
• Appreciate the role of their skills in the interplay of sciences.
• Appreciate of the applicability of mathematical modelling techniques to a range of physical situations.
• Gain an awareness of important issues in the solution of differential equations.

• Methods for elementary first order equations.
• The heat equation
• physical background, fundamental solution, solution of initial-value problems by convolution and the method of separation of variables, uniqueness problems, maximum-minimum principles for initial-value problems.
• Laplace’s and Poisson’s equations
• fundamental solution, Green’s function, maximum principles, properties of harmonic functions, Newton potentials, uniqueness problems, solution of boundary-value problems by the method of separation of variables.
• The wave equation
• physical background, one-dimensional wave equation, solution and properties of the higher dimensional equation, energy integral method.