Mathematical modeling
How long does a warm object take to cool? What temperature lies between two layers of insulation? Why does changing a boundary condition change the answer? These questions lead from physical assumptions to equations you can explain, solve by hand in simple cases, and express in Eqiora.
This path assumes algebra and introductory calculus. You can read its mathematics without running code. If you already have a physical subject in mind, start with heat transfer, fluid mechanics, structures, or circuits and dynamics, and come back to these chapters when you need a particular modeling idea.
A connected investigation
Section titled “A connected investigation”Begin with a bead whose whole temperature changes together. Derive its energy balance, normalize it into scalar decay, and compare a run with the exponential solution. Then ask what happens inside a thick wall, where one temperature is no longer enough. Fields, conservation, material laws, and boundary conditions turn that question into a spatial model.
Along the way you write the small equations yourself before opening the library components that package the same relations, so you know which assumptions an import brings with it. Each chapter ends with a deliberate mistake and exercises with guidance.
Build a model
Section titled “Build a model”- Models are not simulations — derive a cooling balance and distinguish changing the physical assumptions from changing how a computer solves it.
- Quantities, dimensions, and units — check power and time scales, distinguish temperature from a temperature difference, and normalize a model without losing its meaning.
- Algebraic relations and networks — derive a two-layer thermal network, count independent relations, and recognize conservation and material laws in imported electrical parts.
- Ordinary differential equations — derive and run exponential decay, interpret its characteristic time, and ask what changes when the environment moves.
Extend it through space
Section titled “Extend it through space”- Fields and spatial domains — replace one temperature with a field, calculate a gradient, and read the supports and boundary names of a flow model.
- Conservation laws — account for storage, outward flux, and generation; derive a local heat balance and recover the lumped model by integration.
- Constitutive laws — derive slab resistance from Fourier conduction, test stress laws in simple motions, and inspect reusable material components.
- Boundary and interface conditions — connect the environment to the equations, join two materials, and recognize missing references or contradictory data.
Put it to use
Section titled “Put it to use”To run the decay example, follow Get started once; the package
and files it sets up are all you need. The .eqi file holds the equations, and
Python chooses the numerical method and observations.
Choose the next question that interests you:
| Your question | Continue with |
|---|---|
| How does heat spread through a body? | Heat transfer |
| How much of a discrepancy comes from numerical approximation? | Numerical simulation |
| How do pressure and viscosity determine a flow? | Fluid mechanics |
| How do loads produce strain and displacement? | Structural mechanics |
| How do connected components determine currents and evolving states? | Circuits and dynamics |
| What can observations tell us about an unknown parameter? | Inverse problems |
For procedural help, use run and inspect; for exact syntax, use the Language reference. The Gallery develops larger physical investigations.