Title: Methods of Mathematical Modelling – 1
Syllabus: 
1.	Mathematical modelling:  
(focus on modelling with differential equations) 
modelling cycle 
ad-hoc vs first principles, law of mass action 
mathematical problem types 
model validation 
dimensional analysis, nondimensionalization 
scaling, simplification and reduction
perturbations (time permitting); 
Applications and examples: 
logistic population growth, SEIR, examples for chemical reactions, examples for nondimensionalization 
2.	First order equations: 
ordinary differential equations, classification 
explicit and implicit solutions 
trivial case, using fundamental theorem of calculus 
autonomous equations, stationary points and stability 
comments on well-posedness 
linear equations, integrating factor 
separable equations 
substitution method;
Applications and examples: 
landing spacecraft, logistic population growth with hunting, mixing problems, Newton’s cooling law, economic models (for advertising) 
3.	Higher order equations: 
(focus on second order equations) 
linear second order equations, complementary function, particular integral, Wronskian 
constant coefficient case, auxiliary function (complete discussion) 
perturbations and asymptotic expansions (time permitting); 
Applications and examples: 
oscillating RLC circuits, projectiles, mass-spring system, vibrations 
4.	Systems: 
(focus on 2x2 sytems) 
relation to higher-order equations / reformulating second order equations 
autonomous systems 
direction fields and phase portraits 
linearization, linear stability 
discussion of homogeneous systems with constant coefficients; 
Applications and examples: 
Lotka-Volterra / predator-prey 

Covered in assignments and question sheets: 
difference equations, numerical solvers for differential equations (usage only), reduction of order, energy preservation / decay 
Plans for 2023/24: 
Students struggled with the fairly abstract dimensional analysis at the beginning. Instead, first illustrating its benefits first some differential equations, and then discussing it more generally. After the second order equations seems a good place. 
The pendulum hasn’t been discussed but is an important example. I plan to discuss the nonlinear one with a perturbation analysis following Holmes’ book. The current projectile example can go on a question sheet, possibly into an assignment. 
Linear first order equations and second order equations with constant coefficients seemed to have been quite easy, probably well covered in schools. I can go faster there. 
In turn, I plan to do a bit more on discrete (in time) examples and difference equations fairly early in the course. For model validation or inverse problems, optimisation problems are also easier to formulate.