How to Use MATLAB for Mathematical Modelling

Or you might want to maximise:

• Profit
• Efficiency
• Output
• Reliability

MATLAB includes optimisation tools for a range of problem types. The right method depends on whether your problem is linear or nonlinear, whether it has constraints, and what type of variables you are working with.

A simple objective function could look like this:

objective = @(x) (x(1)-3)^2 + (x(2)-2)^2;

x0 = [0 0];

x = fminunc(objective,x0);

This is only a basic example, but the principle scales to much more realistic optimisation problems.

The important thing is to understand the objective function before choosing a solver. MATLAB can perform the optimisation, but you still need to decide what “best” actually means.

When Should You Use Simulink?

For relatively small mathematical models, writing MATLAB code is often enough.

For larger dynamic systems, however, a block-diagram environment can make the structure easier to understand.

That's where Simulink comes in.

Simulink lets you construct models using blocks representing components and relationships. It can be useful for systems involving controllers, sensors, mechanical components, electrical systems, and other interacting elements.

For example, a basic control system might look conceptually like:

Reference → Controller → Physical System → Sensor → Feedback

Instead of writing every relationship manually, you can represent the system graphically and then simulate it.

Simulink is particularly useful when a project involves many interconnected components or when you need to repeatedly test a system under different conditions.

Modelling Problems That Involve Space

Some problems change not only over time but also across a physical area.

Heat moving through a metal plate is a simple example.

In that situation, you may need a partial differential equation rather than an ordinary differential equation.

MATLAB provides tools for PDE-based modelling and finite-element analysis, allowing you to define geometries, meshes, equations, conditions, and numerical solutions. The MathWorks modelling environment covers both first-principles approaches and finite-element workflows for problems described by PDEs.

The general process is something like:

1. Define the physical problem.
2. Write the governing PDE.
3. Describe the geometry.
4. Create an appropriate mesh.
5. Apply boundary and initial conditions.
6. Solve the model.
7. Visualise the results.
8. Compare the result with expected behaviour or experimental data.

The mathematics becomes more demanding at this stage, but the underlying workflow has not really changed.