# 2022/23 Undergraduate Module Catalogue

## SOEE2661 Advanced Mathematics 4

### 10 creditsClass Size: 50

Module manager: Dr Alex Rap
Email: a.rap@leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2022/23

### Pre-requisites

 SOEE2041 Advanced Mathematics 3

Module replaces

SOEE2660 Mathematics for Geophysical and Environmental Sciences 4

This module is not approved as a discovery module

### Module summary

This module introduces students to the mathematical techniques required to solve differential equations arising in the Geophysical Sciences.

### Objectives

To provide students with sufficient Mathematical background for understanding their studies in Geophysical Sciences.

Learning outcomes
On completion of this module, students should be able to:

a) solve ordinary differential equations using series solutions, obtain properties of these solutions and relationships between them;
b) use separation of variables to solve the Laplace, diffusion and wave equation in Cartesian, spherical and cylindrical polar co-ordinates employing Fourier series or other appropriate eigenfunctions;
c) obtain solutions to standard partial differential equations using Fourier transforms;
d) understand how these equations apply to relevant geophysical phenomena.

### Syllabus

- Partial Differential Equations: Laplace's equation, diffusion equation, wave equation.
- Separation of variables in cylindrical and spherical co-ordinates.
- Differential Equations: Introduction to Sturm-Liouville theory-eigenvalues, eigenfunctions, differential operators, orthogonality.
- Series solutions of ODEs - Frobenius' method.
- Legendre's equations and Legendre polynomials.
- Bessel's equations.
- Generating functions and recurrence relations.
- Fourier Transforms: Fourier sine and cosine transforms and their relationship to Fourier series.
- Complex Fourier transforms.
- Inverse transform and transform pair.
- Applications to the solution of differential equations.
- Properties - transforms of derivatives, convolution theorem.

### Teaching methods

 Delivery type Number Length hours Student hours Lecture 22 1.00 22.00 Tutorial 10 1.00 10.00 Private study hours 68.00 Total Contact hours 32.00 Total hours (100hr per 10 credits) 100.00

### Private study

Studying and revising of course material.
Completing of assignments and assessments.

### Opportunities for Formative Feedback

Example sheets are provided for students to work on independently. Assistance with these and regular informal feedback to students is given in the weekly tutorial class to continuously monitor progress. Solutions are provided after each topic is completed

### Methods of assessment

Coursework
 Assessment type Notes % of formal assessment Assignment Problem sheet - One assignment 15.00 Total percentage (Assessment Coursework) 15.00

Re-sits will be assessed by examination only.

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) (S1) 2 hr 00 mins 85.00 Total percentage (Assessment Exams) 85.00

Re-sits will be assessed by examination only.