## MAS6446 Mathematical methods and modelling of natural systems

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 Semester 2, 2019/20 20 Credits Lecturer: Dr Alex Best Timetable Aims Outcomes Assessment

Part 1: This part of the course introduces methods which are useful in many areas of mathematics. The emphasis will mainly be on obtaining approximate solutions to problems which involve a small parameter and cannot easily be solved exactly. These problems will include the evaluation of integrals and the solution of differential equations. Examples of possible applications are: oscillating motions with small nonlinear damping, the effect of other planets on the Earth's orbit around the Sun, boundary layers in fluid flows, electrical capacitance of long thin bodies, central limit theorem correction terms for finite sample size.

Part 2: Mathematical modelling enables insight into a wide range of scientific problems. This part will provide a practical introduction to techniques for modelling natural systems. Students will learn how to construct, analyse and interpret mathematical models, using a combination of differential equations, scientific computing and mathematical reasoning. Students will learn the art of mathematical modelling: translating a scientific problem into a mathematical model, identifying and using appropriate mathematical tools to analyse the model, and finally relating the significance of the mathematical results back to the original problem. Study systems will be drawn from throughout the environmental and life sciences.

There are no prerequisites for this module.
No other modules have this module as a prerequisite.

## Outline syllabus

Part 1:
• Integral methods and differential equations: Dirac δ-function, Fourier and Laplace transforms, applications to differential equations, Green functions.
• Asymptotic expansions: algebraic equations with small parameter, asymptotic expansion of functions defined by integrals.
• Differential equations with a small parameter.
Part 2:
• Evolution within ecological populations.
• From individual movement decisions to spatial patterns in animal populations.
• Individual and collective behaviour of cells.

## Aims

Part 1:
• To develop methods for solving differential equations using integral transforms and representations.
• To introduce asymptotic methods for evaluating integrals.
• To introduce asymptotic methods for solving differential equations.
Part 2:
• develop students' skills in comprehending problems, formulating them mathematically and obtaining solutions by appropriate methods;
• provide practical demonstrations of how mathematical modelling may be used to gain insight into the dynamics of natural systems;
• build on mathematical methods (ordinary/partial differential equations, linear stability analysis, scientific computing in Python) learned at earlier levels, and expose students to how they can be used to model natural systems.

## Learning outcomes

By the end of the unit, a candidate will 1. be able to solve differential equations using integral transforms; 2. be familiar with the ideas of mathematical modelling and understand the utility of mathematical modelling to understand natural phenomena (QAA benchmark for mathematics, 5.4v); 3. be able to formulate a mathematical model that represents the dynamics of a natural system and apply appropriate methods to obtain solutions (QAA benchmark for mathematics, 5.4v); 4. be able to gain insight in to a natural system through the application of mathematical modelling techniques; 5. recognise how mathematical methods learned at earlier levels may be used to model natural systems.
32 lectures, no tutorials, 9 practicals

## Assessment

Part 1: One formal 2 hour written examination. Format: 4 questions from 5.

Part 2: 3 pieces of coursework and 1 oral presentation.

## Timetable

 Tue 12:00 - 12:50 lecture Hicks Lecture Theatre 5 Tue 16:00 - 16:50 lecture Diamond Computer Room 3 / Room 207 Thu 14:00 - 14:50 lecture Arts Tower Lecture Theatre 3 Fri 14:00 - 14:50 lecture Hicks Lecture Theatre 3