Mathematical modeling
Unknowns, units, conservation, material laws, and boundaries: learn how to turn a physical question into equations you can explain.
Learn
A body warms up. A structure bends. A wake sheds vortices. A circuit divides a voltage. Start with the phenomenon that makes you curious, build its equations, and use Eqiora to explore what happens next.
Browse by topic
Section titled “Browse by topic”Each path explains the ideas it needs and links to the mathematics along the way, so you can start with any of them.
- Heat transferWhere does the heat go?Build an energy balance, derive conduction, and follow a body as it warms up.Fourier law · steady conduction · thermal storage
- Fluid mechanicsHow does a flow become a wake?Connect pressure and viscous stress to channel flow, cylinder forces, and alternating vortices.Conservation · Stokes flow · vortex streets
- Structural mechanicsHow does a body carry a load?Follow forces into displacement, strain, stress, and the work done by a load.Equilibrium · elasticity · finite elements
- Circuits and dynamicsWhat determines voltage and motion?Derive network balances, build a voltage divider, and see how stored energy produces dynamics.Circuit laws · power · exponential decay
- Inverse problemsWhat can a measurement tell us?Work backward from observations to a heat source, and learn which parameters the data can distinguish.Parameter recovery · sensitivity · uncertainty
Start a learning path
Section titled “Start a learning path”For a connected introduction, follow mathematical modeling → heat transfer → numerical simulation. First learn to express a physical question, then build and run a thermal model, then discover how the numerical approximation affects the answer.
Numerical simulation
Time steps, meshes, finite elements, finite volumes, and error: learn what a computer does with your equations and how to make a useful comparison.
From an equation to a reusable part
Section titled “From an equation to a reusable part”Each investigation starts with a question and a prediction you can make by
hand. Derive the equations, express the model in .eqi, and read the result.
Then open the reusable components: their equations explain how the same model
can be written more concisely. Change a material, a boundary, or a parameter,
and predict the consequence before running again.
Keep physical assumptions and numerical choices separate. A change in thermal conductivity changes the model; a finer mesh changes how you approximate it. The exercises help you explore both.
Follow an idea into another subject
Section titled “Follow an idea into another subject”- Conservation: follow energy through heat transfer, power through circuits, and momentum through fluid mechanics.
- Stress and work: connect loads in a solid with forces in a fluid.
- Prediction and measurement: solve a thermal problem, then ask what its temperature reveals in inverse problems.
- Approximation: use numerical simulation to understand time steps, spatial resolution, and the difference between a small residual and an accurate solution.
Connect the ideas to a result
Section titled “Connect the ideas to a result”See the cylinder-flow investigation or mixed-boundary elasticity for worked models and plots. When you reach your first computation, Get started sets up the shared environment. Guides explain practical tasks; Reference lets you look up a definition without leaving the subject.