# LINEAR PDES - 2022/3

Module code: MAT2011

## Module Overview

The Linear PDEs Module introduces students to linear partial differential equations, mainly in one and two space dimensions. The classical linear PDEs such as the heat, wave and Laplace equation will be analyzed in details. Questions of existence and uniqueness of solutions will be addressed and their physical and mathematical meaning emphasized.

### Module provider

Mathematics

NOBILI Camilla (Maths)

### Module cap (Maximum number of students): N/A

Independent Learning Hours: 106

Lecture Hours: 11

Seminar Hours: 11

Guided Learning: 11

Captured Content: 11

Semester 2

## Prerequisites / Co-requisites

Calculus MAT1030, Ordinary Differential Equations MAT2007

## Module content

The contents of the module will include:

• Linear PDEs: Examples, classification of PDEs and their physical interpretation.

• First-order and Second-order linear PDEs:  Method of characteristics.

• Introduction to Fourier series and Fourier transform. Solution of initial and  boundary-value problems. Method of separation of variables.

• The heat equation. The wave equation, d’Alembert’s solution.  Interpretation of solutions.

• Laplace’s equation: mean-value theorem, maximum principle, Poisson formula.Existence and Uniqueness of solutions for the canonical PDEs. Energy methods.

## Assessment pattern

Assessment type Unit of assessment Weighting
Online Scheduled Summative Class Test ONLINE TEST 20
Examination Online ONLINE EXAM 80

N/A

## Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate:

• Understanding of and ability to interpret and manipulate various methods for finding the solutions of linear partial differential equations.

• Subject knowledge through the recall of key definitions, theorems and their proofs.

• Analytical ability through the solution of unseen problems in the test and exam.

Thus, the summative assessment for this module consists of:

• One final examination worth 80% of the module mark.

• One in-semester test; worth 20% module mark.

Formative assessment and feedback

Students receive written feedback via a number of marked coursework assignments over an 11 week period.  In addition, verbal feedback is provided by the lecturer during tutorial lectures.

## Module aims

• The aim of this module is to study both qualitative and quantitative aspects of linear PDEs in one and two space dimensions. Students will be introduced to the very important method of characteristics and they will understand and use the method of separation of variables for solving initial-boundary value problems of linear PDEs.  The maximum principle will be proved and its power and importance in the analysis of solutions of PDEs conveyed to students.

## Learning outcomes

 Attributes Developed 1 Classify linear PDEs and choose the appropriate method to solve them. C 2 Solve linear PDEs using the method of characteristics, Fourier transform, and separation of variables. C 3 Interpret solutions and critically relate them to physical settings. C 4 Understand the use of the maximum principle and energy methods for uniqueness and well-posedness. C

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

## Methods of Teaching / Learning

The learning and teaching strategy is designed to:

• Present a detailed introduction to linear partial differential equations and the most common techniques for finding their solutions.

• Give students experience (through demonstration) of the methods used to interpret, understand and solve problems in linear partial differential equations.

The learning and teaching methods include:

• 3 x 1 hour lectures per week x 11 weeks, with written notes to supplement the module handbook and Q + A opportunities for students.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

Upon accessing the reading list, please search for the module using the module code: MAT2011

## Programmes this module appears in

Programme Semester Classification Qualifying conditions
Mathematics and Physics BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics MPhys 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics MMath 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics MMath 2 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics with Statistics MMath 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics with Statistics BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics BSc (Hons) 2 Compulsory A weighted aggregate mark of 40% is required to pass the module
Financial Mathematics BSc (Hons) 2 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics with Music BSc (Hons) 2 Compulsory A weighted aggregate mark of 40% is required to pass the module

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2022/3 academic year.