MATH10001 Mathematical Workshop

Mathematical Modelling and Problem Solving

Traditional view of maths?

● Maths is useless

● The only jobs maths can be used for are accountancy and teaching

● Maths has no link with the real world

The truth is that mathematicians have changed the world,

Radio, digital revolution

Engineering

Computers

Security

Health

Telecommunications Fourier

Fourier Transforms

Joseph Fourier, 1768-1830

Mobile phones, digital radios, MP3 players etc. all use these ideas.

, MP3 players etc all use the sameideas

Internet shopping

Internet transfer protocols based on mathematics

Credit cards and online shopping

– Modern encryption algorithms like RSA use prime number theorems

Scanners

Security screening at the airport

MRI scanners in hospitals

Mathematical modelling in engineering:

eg. Finite element modelling to reduce vibration

Divide car into small

cuboids/tetrahedra.

Treat it a bit like masses

and springs in a network.

Vibrations modelled using eigenvalues of matrices.

Traffic Management

Variable speed signs “shock waves”

Many models of traffic flow Road design Traffic control Better throughput Better safety

Real WorldMathematical

Model

Explanations

& PredictionsSolution

modelling

problem

solving

interpretation

testing Modelling Cycle

Steps in the modelling cycle

1. Identify the problem

2. Define the variables and parameters

• Variables are quantities that can change in a problem

• Parameters are quantities that remain constant

3. Make assumptions

4. Write down a model - relationships between variables

5. Solve the model

6. Test the solution against the real life problem

7. Refine the model

Example – modelling the growth of bacteria

We start with 500 bacteria in a Petri dish. After one day we have 525 bacteria, after two days we have 551 bacteria.

1. The problem is to find a formula for the number of bacteria after n days.

2. Variables include the population, growth rate, time, temperature, amount of food, amount of space left in dish etc. Parameters include initial population, size of dish, initial amount of food.

3. To simplify the problem we make certain assumptions – ignore the amount of food, temperature and space in dish and assume that the growth rate is constant.

4. Define the model: first introduce some notation:

We write down the relationship

5. We have

Assuming a growth rate of 0.05 (from our observations) we get

population initial

days after population

rate growth

experiment of start from days in time

0a

na

k

n

n

.1)1( 111 nakkaaa nnnn any for

.)1(...)1()1( 022

1 akakaka nnnn

.0500)05.1()1( 0 naka nnn all for

6. Test the solution: we test our solution against observations.

We can see that the model works well at the start but after 6 days the model is not accurately predicting the population. This tells us that we need to modify our model.

Day Actual population Predicted population

0 500 500

1 525 525

2 551 551

3 575 579

4 598 608

5 610 638

6 620 670

7. The growth rate appears to be decreasing over time. This could be due to a change in food available or room to grow. We chose to ignore those variables in the original model. As the growth rate is changing we could replace our constant rate by one which is a function of n.

Why do we need mathematical models? Why don’t we simply make lots of observations?