# Module Information

- Dr Andrew Hazel (Reader - University of Manchester)

#### Course Delivery

Delivery Type | Delivery length / details |
---|---|

Lecture | 33 x 1 Hour Lectures |

#### Assessment

Assessment Type | Assessment length / details | Proportion |
---|---|---|

Semester Exam | 2 Hours Written Exam | 100% |

Supplementary Exam | 2 Hours Written Exam | 100% |

### Learning Outcomes

On successful completion of this module students should be able to:

1. Construct various integral transforms and be able to define Fourier, Laplace and Mellin transforms.

2. State the main properties of these three transforms and evaluate their inverses.

3. Make use of the transforms to solve linear ODEs and PDEs.

4. Describe situations in which it is beneficial to use Green's functions and Fredholm and Volterra integral equations.

5. Solve representative problems in applied mathematics using integral transforms and integral equation methods, for example ordinary and partial differential equations.

6. Demonstrate an ability to derive and solve linear integral equations

### Brief description

In their original formulation, many problems in mathematics appear intractable. The application of an integral transform to an equation often acts to simplify its solution. This module will explain the basic ideas behind integral transforms, generalizing the idea of Fourier transforms to include Laplace and Mellin transforms. Their use will be demonstrated in the solution of both ordinary and partial differential equations. The idea of generalized functions will be introduced, and their properties and applications described. The module will cover Green's functions and Fredholm theory, their link to integral equations, and their practical uses. Reference will be made to applications in which ordinary and partial differential equations describe problems such as potential flow and wave propagation.

### Content

• Distributions and the Dirac delta function.

• The Fourier transform. Linearity, dilation, translation, convolution, derivatives, the inversion formula.

• The Laplace transform. Linearity, dilation, convolution, derivatives, the inversion formula.

• The Mellin transform. Linearity, dilation, convolution, derivatives, the inversion formula.

• Applications of integral transforms to (partial) differential equations, integral equations, difference equations, solid and fluid mechanics.

• Integral equations. Degenerate (separable) kernels and solution method. General integral equation types. Neumann series and iterated kernels. Volterra and Fredholm equations, Fredholm theory

• Greens functions in 1D. Construction for constant coefficient ODEs and Sturm Liouville problems. Applications to the steady state heat equation and wave equation.

• Greens functions in 2D and 3D. Steady state heat equation and Potential flow problems (Laplace) and time-harmonic wave equation (Helmholtz).

### Module Skills

Skills Type | Skills details |
---|---|

Application of Number | Throughout |

Communication | Written answers to questions must be clear and well-structured, and should communicate students’ understanding |

Improving own Learning and Performance | Students are expected to develop their own approach to time-management regarding the completion of Example sheets on time, assimilation of feedback, and preparation between lectures. |

Information Technology | Use of blackboard |

Problem solving | Throughout |

Research skills | Students will be encouraged to independently find and assimilate useful resources. |

Subject Specific Skills | Students will be able to apply transforms to solve problems in many areas of mathematics and engineering. |

Team work | Students will be encouraged to work together on questions in workshops and on Example sheets. |

### Notes

This module is at CQFW Level 7