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## MTH2003 - Differential Equations (2022)

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

MODULE CODE | MTH2003 | MODULE CONVENER | Prof Vadim N Biktashev (Coordinator) |

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

DURATION: WEEKS | 11 weeks | 0 | 0 |

Number of Students Taking Module (anticipated) | 300 |
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Differential equations are at the heart of nearly all modern applications of mathematics to natural phenomena. Computerised applications play a vital role in many areas of modern technology. Mathematically, all rates of change and acceleration can be described by derivative functions. These include the growth of populations, the spread of diseases, movement of physical objects in response to forces acting on them, or even the fluctuations of the stock market. You will learn the basic principles of differential equations, and will apply that knowledge to some every day phenomena. Then you will learn about methods of finding solutions for some classes of differential equations. In particular you will develop methods to solve the wave and heat equations which are fundamental to modelling many physical processes.

This course will enable you to demonstrate an understanding of, and competence in, a range of analytical tools for posing and solving differential equations, and their application to situations in science and technology.

Prerequisite modules: MTH1002 or NSC1002 (Natural Science Students) or equivalent.

The aim of this module is to introduce you to some representative types of ordinary and partial differential equations and to introduce a number of analytical techniques used to solve them exactly or approximately.

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

**Module Specific Skills and Knowledge:**

1 demonstrate a working knowledge of how to identify, classify and solve a range of types of ordinary and partial differential equations;

2 reveal an insight into their application and derivation;

3 show some knowledge of a selection of special functions and series methods used for solution of these differential equations.

**Discipline Specific Skills and Knowledge:**

4 exhibit an understanding of range of analytical tools for posing and solving differential equations;

5 display competence in applying these tools;

6 prove an understanding of the concept of using differential equations for mathematical modelling of natural phenomena and engineering applicatoins.

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

7 demonstrate an ability to monitor your own progress and to manage time;

8 show an ability to formulate and solve complex problems.

- Review of simple methods for solving first order ordinary differential equations (ODEs). Sufficient conditions to guarantee existence and uniqueness to such ODEs.

- The general linear second order ODE. Equations with constant coefficients, Euler-Cauchy equations.

- Power series methods: Leibniz-Maclauren method and method of Frobenius (selected examples).

- Boundary value problems, eigenfunctions and eigenvalues. Orthogonality of eigenfunctions of Sturm-Liouville problems.

- Selected orthogonal systems of functions: trigonometric functions, Legendre polynomials, Bessel functions. Series in orthogonal functions, including Fourier series.

- Examples of linear partial differential equations (PDEs) and their solutions, including initial/boundary-value problems for one-dimensional wave equation and one-dimensional heat equation, on the whole real line and on a finite interval.

- Solution of PDEs in two and three spatial dimensions using separation of variables, a.k.a. normal modes or (generalised) Fourier series, in Cartesian, polar and spherical coordinates.

- Systems of first-order differential equations. Linear Systems. Diagonalization for constant coefficients case. Variation of parameters.

- Selected applications of ODEs and PDEs.

Scheduled Learning & Teaching Activities | 38.00 | Guided Independent Study | 112.00 | Placement / Study Abroad |
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Category | Hours of study time | Description |

Scheduled learning and teaching activities | 33 | Lectures including examples classes |

Scheduled learning and teaching activities | 5 | Tutorials |

Guided independent study | 112 | Lecture and assessment preparation; wider reading |

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

Exercise sheets | 5 x 10 hours | All | Discussion at tutorials and solutions provided in ELE; tutor feedback on submitted solutions. |

Coursework | 10 | Written Exams | 90 | Practical Exams | 0 |
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Form of Assessment | % of Credit | Size of Assessment (e.g. duration/length) | ILOs Assessed | Feedback Method |
---|---|---|---|---|

Written exam – closed book | 90 | 2 hours (January) | All | Written/verbal on request, SRS |

Coursework exercises 1 | 5 | 15 hours | All | Annotated script and written/verbal feedback |

Coursework exercises 2 | 5 | 15 hours | All | Annotated script and written/verbal feedback |

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

Written Exam* | Written exam (2 hours) (90%) | All | August Ref/Def period |

Coursework Exercises 1* | Coursework exercises 1 (5%) | All | August Ref/Def period |

Coursework Exercises 2* | Coursework exercises 2 (5%) | All | August Ref/Def period |

*Please refer to reassessment notes for details on deferral vs. Referral reassessment

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

**Web based and Electronic Resources: ****ELE: **http://vle.exeter.ac.uk

Reading list for this module:

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

Set | Arfken, G.B. & Weber, H.J. | Mathematical Methods for Physicists | Electronic | Harcourt/ Academic Press | 2005 | 000-0-120-59825-6 | [Library] |

Set | O'Neil, P.V. | Advanced Engineering Mathematics | 2nd | Wadsworth | 1987 | 000-0-534-06792-1 | [Library] |

Set | Stephenson, G. & Radmore, P.M. | Advanced Mathematical Methods for Engineering and Science Students | Cambridge University Press | 1990 | 000-0-521-36860-X | [Library] | |

Set | Boyce, W E, Di Prima, R C | Elementary differential equations and boundary value problems | 9th edition | John Wiley and Sons | 2009 | 978-0-470-39873-9 | [Library] |

Set | Kreyszig, E. | Advanced Engineering Mathematics | 9th | Wiley | 2006 | 978-0471728979 | [Library] |

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

PRE-REQUISITE MODULES | MTH1002 |
---|---|

CO-REQUISITE MODULES |

NQF LEVEL (FHEQ) | 5 | AVAILABLE AS DISTANCE LEARNING | No |
---|---|---|---|

ORIGIN DATE | Tuesday 10 July 2018 | LAST REVISION DATE | Friday 05 August 2022 |

KEY WORDS SEARCH | Differential equations; orthogonal functions. |
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