# Mathematics BSc/MA/MSci/MA(SocSci)

## Mathematics 2D: Mathematical Methods and Modelling MATHS2033

• Academic Session: 2020-21
• School: School of Mathematics and Statistics
• Credits: 10
• Level: Level 2 (SCQF level 8)
• Typically Offered: Semester 2
• Available to Visiting Students: Yes
• Available to Erasmus Students: Yes

### Short Description

This course aims to introduce theory and methods used in mathematical modelling. Topics covered in the course include: population modelling, local minima and maxima of functions, Fourier series and integral transforms. It is an essential course for intending honours students. The emphasis is on methods and applications.

### Timetable

2 x lectures per week. Fortnightly tutorials.

### Co-requisites

Mathematics 2A: Multivariable Calculus, Mathematics 2B: Linear Algebra.

### Assessment

One degree examination (80%) (1 hour 30 mins); coursework (20%).

Main Assessment In: April/May

Are reassessment opportunities available for all summative assessments? No

Reassessment will not be available for continuous assessment. Immediate feedback will be given to students at the assignment deadline, and it is not practical to set multiple versions of each of these exercises.

In total 80% of the assessment can be repeated for students not reaching the threshold grade.

Note that students are required to submit 75% of the assessments in order to receive a grade for the course. Accordingly, students who have a substantial period of illness and have all eAssignments and tutorial group work MV'ed, will not be in danger of failing to receive credits.

### Course Aims

This course aims to introduce theory and methods used in mathematical modelling. The first part of the course aims to provide an introduction to the modelling of populations using difference and differential equations. Students will learn to construct the solution of the equations arising in such models and qualitatively analyse these solutions. The second part of the course aims to teach students important methods from calculus that are used in mathematical modelling. These include: analysis of local minima and maxima of functions, Fourier series and integral transforms.

### Intended Learning Outcomes of Course

By the end of this course, students will be able to:

■ Derive and analyse single-species dynamical systems, including the exponential and logistic models with predation.

■ Analyse the stability of equilibrium points in first order autonomous differential equations and construct simple bifurcation diagrams for single-species dynamics systems.

■ Diagonalise suitable matrices and apply this to find solutions to systems of ordinary differential equations and difference equations.

■ Determine the stationary points of functions of several variables; classify these stationary points using first principles and the Hessian criterion; apply the method of Lagrange multipliers to solve practical extreme value problems.

■ Determine Fourier series for given functions defined on finite intervals.

■ Compute the Fourier transform and Laplace transform of a given function; apply the properties of Beta and Gamma functions to evaluate certain integrals.

■ State the definitions, results and formulae presented in lectures; apply and adapt these to solve suitable problems.

### Minimum Requirement for Award of Credits

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.