Lecturer: Ferran Brosa Planella

Term(s): Term 1

Status for Mathematics students: This module is not available to Maths students

Commitment: 30h lectures, plus handouts and 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

Leads to: The following modules have this module listed as assumed knowledge or useful background:

• MA145 Mathematical Methods and 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:

1. Introduction to mathematical modelling with differential equations: modelling cycle, principles and observations, types of problems, scaling and dimensional analysis, simplification and reduction.
2. Introduction to differential equations: classification, general first order equations, autonomous equations, stability, phase portraits, integrating factors for linear equations, separation and substitution methods for nonlinear equations.
3. Higher order differential equations: Linear second order equations, both homogeneous and inhomogeneous, linear second order equations with constant coefficients, auxiliary equations.
4. Difference equations: General difference equations, relation to the Euler's method, first order linear difference equations, second order linear difference equations with constant coefficients, autonomous equations, chaos.
   
5. Systems of equations: systems of difference and differential equations, relation with higher order equations, linear systems of differential equations, homogeneous linear systems with constant coefficients, phase portraits, autonomous systems, linearisation and linear stability.

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 interpret the results from the mathematical analysis in order to provide understanding about a physical system.

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.

Additional Resources