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## MTH3008 - Partial Differential Equations (2019)

MODULE TITLE | Partial Differential Equations | CREDIT VALUE | 15 |
---|---|---|---|

MODULE CODE | MTH3008 | MODULE CONVENER | Prof Vadim N Biktashev (Coordinator) |

DURATION: TERM | 1 | 2 | 3 |
---|---|---|---|

DURATION: WEEKS | 0 | 11 | 0 |

Number of Students Taking Module (anticipated) | 88 |
---|

A PDE is a differential equation in which the unknown function is a function of multiple independent variables and the equation involves its partial derivatives. The order is defined similarly to the case of ordinary differential equations, but further classification into elliptic, hyperbolic, and parabolic equations, especially for second order linear equations, is of utmost importance. Some partial differential equations do not fall into any of these categories over the whole domain of the independent variables and they are said to be of mixed type.

In this module, you will learn how PDEs can be well-posed or ill-posed, and will find out about a range of analytical techniques used to solve PDEs. The module will strengthen your ability to interpret theoretical mathematical concepts, and acquire a deeper understanding of how mathematics relates to real world problems. The module builds on material in the Differential Equation module MTH2003, in particular separation of variables and Fourier series. The module Vector Calculus and Applications MTH2004 is also a prerequisite, in particular definitions and use of gradient, divergence, and the divergence theorem.

Partial differential equations (PDEs) form a central part of mathematics. The laws of physics are formulated in terms of PDEs, so the subject is of great practical importance. However, the range of application of PDEs goes beyond the physical world into the modelling of subjects as diverse as ecology and economics. This leads to interesting connections between subjects that at first seem unrelated. The purpose of this module is to develop some of the main analytical and numerical techniques used to solve PDEs, building on the work done in MTH2003 and MTH2004. We will illustrate the topic using a range of real world examples.

On successful completion of this module, **you should be able to**:

**Module Specific Skills and Knowledge:**

1 demonstrate understanding of the classification of linear partial differential equations (PDEs) of first and second order;

2 apply a range of analytical techniques and a wider knowledge and appreciation of applications of PDEs in mathematics;

3 exhibit detailed knowledge of specific parabolic, elliptic and hyperbolic second order PDEs.

**Discipline Specific Skills and Knowledge:**

4 complete extended multi-step calculations using a variety of mathematical techniques;

5 translate unfamiliar problems into ones that can be tackled by familiar techniques;

6 show a knowledge of the relevance of PDEs in applications.

**Personal and Key Transferable/ Employment Skills and Knowledge:**

7 illustrate self-management and time management skills;

8 express complex abstract arguments in a logical and coherent manner;

9 use learning resources, including e-learning resources to extend their knowledge.

- Introduction. Examples of PDE models. First order PDEs. Linear, quasilinear and nonlinear cases;

- Second-order linear PDEs and their classification into elliptic, hyperbolic and parabolic classes. Domains, boundary conditions and well-posedness;

- Hyperbolic equations: method of characteristics, canonical form, wave equation in one and three dimensions, conservation of energy, D'Alembert and Kirchhoff formulas;

- Parabolic equations: canonical form, uniqueness and stability theorems, diffusion equation on finite and infinite domains, solution by transform methods;

- Elliptic equations: canonical form, uniqueness theorem, Laplace equation in finite and infinite domains, solution by transform methods;

- Green's function methods for solving non-homogeneous linear equations;

- Selected examples of solutions to nonlinear PDEs.

Scheduled Learning & Teaching Activities | 33.00 | Guided Independent Study | 117.00 | Placement / Study Abroad | 0.00 |
---|

Category | Hours of study time | Description |

Scheduled learning and teaching activities | 33 | Lectures/example classes |

Guided independent study | 30 | Assessment preparation |

Guided independent study | 57 | Study of notes and formative examples |

Guided independent study | 30 | Wider reading |

Form of Assessment | Size of Assessment (e.g. duration/length) | ILOs Assessed | Feedback Method |
---|---|---|---|

Examples Sheets | 5 during semester | All | Oral at lecture sessions. Solutions posted on ELE |

Coursework | 20 | Written Exams | 80 | Practical Exams | 0 |
---|

Form of Assessment | % of Credit | Size of Assessment (e.g. duration/length) | ILOs Assessed | Feedback Method |
---|---|---|---|---|

Coursework – based on questions submitted for assessment | 20 | 2 assignments, 30 hours total | All | Annotated script and written/verbal feedback |

Written Exam – closed book | 80 | 2 hours | All | Written/verbal on request, SRS |

Original Form of Assessment | Form of Re-assessment | ILOs Re-assessed | Time Scale for Re-reassessment |
---|---|---|---|

All above | Written Exam (100%) | All | August Ref/Def Period |

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.

If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

information that you are expected to consult. Further guidance will be provided by the Module Convener

ELE – http://vle.exeter.ac.uk

Reading list for this module:

Type | Author | Title | Edition | Publisher | Year | ISBN | Search |
---|---|---|---|---|---|---|---|

Set | Ockendon, J., Howison, S. , Lacey, A. & Movchan, A. | Applied Partial Differential Equations | Oxford University Press | 2003 | 978-0198527718 | [Library] | |

Set | Sneddon I.M. | Elements of Partial Differential Equations | McGraw-Hill | 1957 | [Library] | ||

Set | Smith, G.D. | Numerical Solution of Partial Differential Equations: Finite Difference Methods | 3rd | Oxford University Press | 1985 | 978-0198596509 | [Library] |

Set | Williams, W.E. | Partial Differential Equations | Clarendon Press | 1980 | 978-0198596332 | [Library] | |

Set | Tveito, A, Winther, R | Introduction to partial differential equations: A computational approach | 2008 | Springer-Verlag | 2009 | 978-3540887041 | [Library] |

Set | Logan, D.J. | Applied Partial Differential Equations | 2nd | Springer | 2004 | 978-0387209531 | [Library] |

CREDIT VALUE | 15 | ECTS VALUE | 7.5 |
---|---|---|---|

PRE-REQUISITE MODULES | MTH2003, MTH2004 |
---|---|

CO-REQUISITE MODULES |

NQF LEVEL (FHEQ) | 6 | AVAILABLE AS DISTANCE LEARNING | No |
---|---|---|---|

ORIGIN DATE | Tuesday 10 July 2018 | LAST REVISION DATE | Friday 30 August 2019 |

KEY WORDS SEARCH | Partial differential equations; parabolic equations; elliptic equations; hyperbolic equations; boundary value; initial value. |
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