Unit name | Applied Partial Differential Equations 2 |
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Unit code | MATH20402 |

Credit points | 20 |

Level of study | I/5 |

Teaching block(s) |
Teaching Block 2 (weeks 13 - 24) |

Unit director | Professor. Porter |

Open unit status | Not open |

Pre-requisites |
MATH 20900 Calculus 2 (MATH 11009 Mechancs 1 is preferable, but not required) |

Co-requisites |
None |

School/department | School of Mathematics |

Faculty | Faculty of Science |

Unit aims

To provide the student with the necessary mathematical tools in order to model a wide variety of different physical problems, ranging from waves on strings, the propagation of signals, the diffusion of heat in solids and chemicals in solution, traffic flow and the vibrations of membranes and surfaces.

General Description of the Unit

Partial differential equations (PDEs) are differential equations involving partial derivatives of functions of several variables. They are essential for understanding many physical processes including the behaviour of ocean waves, the flow of rivers, the diffusion of pollutants, aerodynamics, the operation of musical instruments, atomic physics, and many other branches of science. This unit will give an introduction to simple PDEs and how they arise in physical problems; it will develop techniques for solving them and understanding the behaviour of the solutions.

The unit will develop students' understanding of first year multivariable calculus and linear algebra. It will introduce Fourier series, the Fourier integral, the delta function and other methods for solving linear and nonlinear PDEs, (such as the method of characteristics) and will show how eigenvalues play a central role in applied mathematics. The course emphasises techniques and broad understanding rather than proofs.

Relation to Other Units

This unit will be a prerequisite for Mathematical Methods, Fluid Dynamics, Quantum Mechanics and other applied mathematics units. It gives applications of the vector calculus, complex variable methods and other material in Calculus 2 (or from 2014/15: Multivariable Calculus and Methods of Complex Functions), and includes material (Sturm-Liouville theory) relevant to Ordinary Differential Equations 2, though that course is not a prerequisite.

Further information is available on the School of Mathematics website: http://www.maths.bris.ac.uk/study/undergrad/

At the end of the course the student should should be able to:

- Understand the physical models and derive PDE's representing diffusion and wave propagation;
- identify appropriate boundary conditions for simple linear PDEs;
- solve linear two-dimensional PDEs on bounded spatial domains by separation of variables using Sturm-Liouville Theory and Fourier series;
- calculate and manipulate Fourier transforms, and use them to solve simple linear PDEs on unbounded spatial domains;
- transform to dimensionless variables and identify dimensionless parameters;
- use the method of characteristics to solve simple linear and nonlinear first order PDEs;
- describe some differences between linear and nonlinear PDEs;
- solve multi-dimensional linear PDE's using separation of variables in a variety of coordinate systems

Transferable Skills:

- Clear thinking; mathematical modelling of physical situations; skill in mathematical manipulation.

Three lectures and one problems class per week. Regular problem sheets will be distributed which will test the students' understanding of the material through a variety of problems ranging from elementary to difficult. Set questions will be marked promptly and returned with comments. Full solutions of all problems will be distributed.Problems classes will go through examples that compliment both the lectures and the worksheets.

100% Examination

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

If you fail this unit and are required to resit, reassessment is by a written examination in the August/September Resit and Supplementary exam period.

There are many books covering the subject. Reecommended texts to compliment the course are

- S. J. Farlow, Partial Differential Equations for Scientists and Engineers, Dover Publications 1993, QA377 FAR (Queen's Building) approx. £14.
- R. Haberman, Applied Partial Differential Equations, Pearson/Prentice-Hall 2004. QA377 HAB (Queen's Building) approx £60

Farlow is cheap, written in a straightforward and clear style, and covers most ot the course, rather briefly. Haberman has much more detail, and includes more up to date and advanced material beyond the scope of this unit, but would be useful to students planning to take more advanced courses in applied mathematics.