# LINEAR PDES - 2020/1

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

Module Leader

LLOYD David (Maths)

Number of Credits: 15

ECTS Credits: 7.5

Framework: FHEQ Level 5

JACs code: G100

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

Module Availability

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 |
---|---|---|

School-timetabled exam/test | IN-SEMESTER TEST (50 MINS) | 20 |

Examination | EXAMINATION | 80 |

Alternative Assessment

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 two hour examination (three best answers contribute to exam mark) at the end of Semester 1; worth 80% 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

Overall student workload

Independent Study Hours: 117

Lecture Hours: 33

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.

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 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 |

Mathematics MMath | 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 |

Financial Mathematics 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 2020/1 academic year.