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# Partial Differential Equations I - MAT00053M

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• Department: Mathematics
• Module co-ordinator: Dr. Konstantin Ilin
• Credit value: 10 credits
• Credit level: M
• Academic year of delivery: 2019-20

## Module summary

Pre-requisite Module(s)

For Natural Sciences Students: Mathematics for the Sciences III MAT00019I

• None

• None

## Module will run

Occurrence Teaching cycle
A Autumn Term 2019-20

## Module aims

A partial differential equation (PDE) is a differential equation that contains an unknown function and its partial derivatives. PDEs are used to describe a wide range of natural processes. Examples include fluid mechanics, elasticity theory, electrodynamics, quantum mechanics, etc. PDEs also play an important role in other areas of mathematics such as analysis and differential geometry.

The aim of this course is to give an introduction to the basic properties of PDEs and to the basic analytical techniques to solve them.

## Module learning outcomes

At the end of the module students should:

• Be able to determine the type of a second order PDE
• Be able to solve simplest first order PDEs
• Understand what are well-posed initial (and/or boundary) value problems for classical PDEs such as the wave equation, the Laplace equation and the heat (diffusion) equation
• Know basic analytical techniques for solving the above classical equations

## Module content

[Pre-requisite modules:

• Natural Sciences students: Maths for the Sciences III MAT00019I
• MSc Mathematical Sciences students: knowledge of Vector calculus and elementary complex function theory.]

Syllabus

• Introduction: what is a PDE, first-order linear PDEs, initial and boundary conditions, well-posed problems, types of second-order PDEs.
• Wave equation: d'Alembert’s formula, causality and energy, reflection of waves.
• Heat (diffusion) equation: maximum principle, heat equation on the whole line and on the half-line
• Laplace equation: maximum principle, Poisson’s formula, rectangular domain.
• Periodic solutions for wave and heat equations. (This topic is not taught in the H-level variant of this module.)

• Academic skills: the techniques taught are used in many areas of pure and applied mathematics.

• Graduate skills: through lectures, examples, classes, students will develop their ability to assimilate, process and engage with new material quickly and efficiently. They develop problem solving-skills and learn how to apply techniques to unseen problems. Students on this module will learn to work more independently and assimilate advanced material at a greater rate than those on the H-level variant.

## Assessment

Task Length % of module mark
Online Exam
Partial Differential Equations I
2 hours 100

None

### Reassessment

Task Length % of module mark
Online Exam
Partial Differential Equations I
2 hours 100

## Module feedback

Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy.