How to Use MATLAB for Mathematical Modelling

Use Experimental Data to Improve a Model

Not every modelling problem starts with a neat equation.

Sometimes you have measurements first.

Imagine that you measure the temperature of the coffee every ten minutes:

time = [0 10 20 30 40 50 60];
temp = [90 55 40 31 26 23 21];

plot(time,temp,'o')
xlabel('Time (minutes)')
ylabel('Temperature (°C)')
grid on

Now you can compare your theoretical model with the observations.

This is where curve fitting and parameter estimation become useful. Instead of simply choosing a value for (k), you can estimate a value that gives a better match to the measurements.

For more advanced work, MATLAB provides tools for fitting curves and analysing how well a fitted model represents the available data. Its mathematical-modelling workflow also supports data-driven, first-principles, and hybrid approaches.

There is an important warning here: a model that matches your existing data extremely well is not automatically a good model.

You should also look at residuals and, where possible, test the model against observations that were not used during fitting.

Experiment With Model Parameters

Once your basic model works, start changing things.

For example, what happens if the cooling constant is smaller?

kValues = [0.03 0.08 0.15];

figure
hold on

for k = kValues
    coolingModel = @(t,T) -k*(T - Ta);
    [t,T] = ode45(coolingModel,tspan,T0);
    plot(t,T,'LineWidth',2)
end

xlabel('Time (minutes)')
ylabel('Temperature (°C)')
legend('k = 0.03','k = 0.08','k = 0.15')
grid on

Now you have three possible scenarios on one graph.

This is much more informative than calculating one case and stopping.

Parameter experiments can help you understand questions such as:

• Which variable has the biggest effect?
• How sensitive is the result to measurement errors?
• What happens when a parameter is doubled?
• Which parameter values produce realistic behaviour?
• Does the model remain stable when conditions change?

This type of investigation is often where mathematical modelling becomes genuinely interesting.

MATLAB for Optimisation Problems

Prediction is only one reason to build a mathematical model.

Sometimes you want MATLAB to find the best solution.

For example, you might want to minimise:

• Manufacturing cost
• Energy consumption
• Material usage
• Travel time
• Production waste