MA147 Mathematical Methods and Modelling 1
Lecturer: Emma Davis
Term(s): Term 1
Status for Mathematics students: This module is not available to Maths students
Commitment: 20h lectures, 10h videos and/or handouts, problem sheets
Assessment: 15% from assignments and 85% from exam
Formal registration prerequisites: None
Assumed knowledge: None (standard entry criteria for Maths-related subjects suffice)
Useful background: Modelling with differential equations, solution techniques for linear differential equations of first and second order, eigenvalues and eigenvectors of 2x2 matrices, python and jupyter notebooks
Synergies: MA124 Maths by Computer (python programming, problem solving on the computer)
Leads to: The following modules have this module listed as assumed knowledge or useful background:
- MA144 Methods of Mathematical Modelling 2
- MA265 Methods of Mathematical Modelling 3
- MA250 Introduction to Partial Differential Equations
- MA254 Theory of ODEs
- MA256 Introduction to Mathematical Biology
- MA261 Numerical Methods and Computing
- MA269 Asymptotics and Integral Transforms
Content:
- Introduction to mathematical modelling with differential equations: Modelling cycle, principles and observations, types of problems, scaling and dimensional analysis, simplification and reduction, perturbation methods.
- Intro to differential equations: Classification, general first order equations, autonomous equations, stability, integrating factors for linear equations, separation and substitution methods for nonlinear equations.
- Systems and higher order equations: Relation between higher order equations as systems, general 2x2 systems, autonomous systems, phase portraits, linearisation and linear stability, general theory for linear systems, eigenspace analysis in case of constant coefficients.
- Further problems and techniques: a selection from discretisation principles and difference equations, control problems, dynamical systems, attractors and linearisation.
Learning Outcomes: By the end of the module students should be able:
- To understand the modelling cycle in science and engineering, to formulate mathematical models and problems using differential equations, and to use a variety of methods to reveal their main underlying dynamics.
- To apply a range of techniques to solve simple ordinary differential equations (first order, second order, first order systems), and to gain insight into the qualitative behaviour of solutions.
- To confidently deploy computational methods and software to validate results, to approximate solutions of more challenging problems, and to further investigate them.
Books:
Robinson, James C. An Introduction to Ordinary Differential Equations. Cambridge University Press, 2004.
Witelski, B. and Bowen, M., Methods of Mathematical Modelling: Continuous Systems and Differential Equations. Springer, 2015.
Logan, David. A First Course in Differential Equations. Springer, 2015.
Holmes, Mark H. Introduction to the Foundations of Applied Mathematics. Springer, 2019.