How to Use MATLAB for Mathematical Modelling

Mathematical modelling sounds more complicated than it needs to be. At its simplest, you take a real situation, describe it with mathematics, and then use a computer to see what those equations tell you.

MATLAB is well suited to this process because it lets you work with equations, numerical calculations, data, graphs, and simulations in the same environment. You can use it for anything from a small population-growth exercise to a detailed engineering model.

The trick is not to begin with MATLAB commands. Begin with the problem.

Once you understand what you are trying to represent mathematically, MATLAB becomes much easier to use.

What Does Mathematical Modelling Mean?

A mathematical model is a simplified description of something that happens in the real world.

Think about a tank filling with water. The amount of water changes over time, and the rate of change depends on the flow entering the tank. You can describe that relationship with an equation and then use MATLAB to calculate how the water level changes.

The same idea applies to much larger problems, including:

• Mechanical systems
• Electrical circuits
• Population growth
• Financial forecasting
• Heat transfer
• Chemical reactions
• Control systems
• Optimisation problems
• Biological processes
• Structural and physical systems

MATLAB supports both mathematical and data-driven approaches. You can build a model from scientific principles, work with measured data, or combine the two approaches. MathWorks specifically describes MATLAB as supporting symbolic and numerical modelling, curve fitting, statistics, optimisation, and differential-equation solving.

Start With the Mathematics, Not the Software

One of the easiest mistakes to make is opening MATLAB immediately and trying to figure out the equations afterward.

I prefer the opposite approach.

First, describe the real problem. Then identify the variables and parameters. After that, decide which mathematical relationships describe the system.

For example, imagine that you want to study how a cup of coffee cools down.

You could use Newton's law of cooling:

[
\frac{dT}{dt}=-k(T-T_a)
]

Here, (T) represents the coffee's temperature, (T_a) is the surrounding temperature, and (k) controls how quickly the coffee cools.

At this point, you already have a mathematical model.

MATLAB's job is to help you solve it, experiment with it, and understand what the solution means.

That distinction matters. A computer can solve an equation correctly even when the equation itself is a poor description of reality.

Build a Simple Model in MATLAB

Let's turn the cooling example into a small MATLAB model.

Suppose the coffee starts at 90°C, the room is at 20°C, and we choose a cooling constant of 0.08.