# MA146-10 Methods of Mathematical Modelling 1

Department
Warwick Mathematics Institute
Level
Bjorn Stinner
Credit value
10
Module duration
10 weeks
Assessment
Multiple
Study location
University of Warwick main campus, Coventry

##### Introductory description

The module introduces the fundamentals of mathematical modelling and scaling analysis, before discussing and analysing difference and differential equation models in the context of physics, chemistry, engineering as well as the life and social sciences.

##### Module aims

To introduce the basic concepts of ODEs and difference equations and their solutions.

##### Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

1. Introduction to mathematical modelling: Mathematical modelling cycle, types of models (sto-chastic, deterministic, discrete, continuous, ….).
2. Scaling and dimensional analysis: Buckingham’s Pi Theorem, examples from chemical reac-tions and projectile motion, perturbation methods (time-permitting).
3. First order linear equations: first order linear equations, examples of existence and unique-ness, integration techniques (integrating factors, ..).
4. Second order equations: general homogeneous equations and linear second order equations with constant coefficients, reduction to 2x2 systems, sketching the flow under a vector field (2d only, phase diagrams).
5. Nonlinear equations and 2x2 systems: linear stability such as predator and prey models.
6. Difference equation: discrete population models such as the logistic model/fishery manage-ment, stability and instability of solutions (6+7 could be moved before 4).
7. Discretisation techniques: explicit and implicit Euler, connection to difference equations, sta-bility.
##### Learning outcomes

By the end of the module, students should be able to:

• understand the modelling cycle and be able to formulate and analyse simple models themselves
• use scaling, non-dimensionalisation and linear stability techniques to reveal and understand the main underlying dynamics/driving factors
• solve simple ODEs (first order and second order) and interpret their qualitative behavior
• solve simple difference equations and understand their connection to continuous ODEs
• understand the basic concepts of numerical approximation

Logan, David. A first course in differential equations. Springer, 2006.
Robinson, James C. An introduction to ordinary differential equations. Cambridge University Press, 2004.
Holmes, Mark H. Introduction to the foundations of applied mathematics. Springer, 2009.
Hermann, Martin, and Masoud Saravi. Nonlinear ordinary differential equations. Springer India, 2016.
Witelski, B. and Bowen, M., Methods of Mathematical Modelling: Continuous Systems and Differential Equations, Springer, 2015

##### Subject specific skills

The module introduces the fundamentals of mathematical modelling and scaling analysis, before discussing and analysing difference and differential equation models in the context of physics, chemistry, engineering as well as the life and social sciences.

##### Transferable skills

Students will acquire key modelling and problem solving skills which will empower them to address problems in a large range of scientific fields with confidence.

## Study time

Type Required
Lectures 20 sessions of 1 hour (20%)
Online learning (independent) 9 sessions of 1 hour (9%)
Private study 13 hours (13%)
Assessment 58 hours (58%)
Total 100 hours
##### Private study description

Working on assignments, going over lecture notes, text books, exam revision.

## Costs

No further costs have been identified for this module.

You do not need to pass all assessment components to pass the module.

##### Assessment group D
Weighting Study time
Assignments 15% 20 hours
In-person Examination 85% 38 hours
##### Assessment group R
Weighting Study time
In-person Examination - Resit 100%
##### Feedback on assessment

Marked homework (both assessed and formative) is returned and discussed in smaller classes. Exam feedback is given.

## Courses

This module is Core for:

• Year 1 of UMAA-G105 Undergraduate Master of Mathematics (with Intercalated Year)