# 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.