Examples that actually compile.

Lorenz Attractor

Chaotic three-state ODE solved with DoPri5, plus uncertainty propagation across initial conditions.

Bouncing Ball (Event Detection)

Hybrid system showing event detection at ground contact, with restitution-based velocity reset.

• events
• dense output

ODE Solver Zoo & Auto-Selection

Tour of every numra ODE solver, plus the auto-selection logic that picks the right one from problem hints.

• stiff
• dense output

Van der Pol Oscillator

Stiff ODE benchmark comparing explicit DoPri5 against implicit Radau5 across stiffness levels.

• stiff

1D Heat Equation

Linear parabolic PDE solved by method of lines with Dirichlet boundaries.

• stiff

Geometric Brownian Motion (Monte Carlo)

SDE option-pricing example with ensemble statistics across Euler-Maruyama, Milstein, and SRA1.

Mackey-Glass DDE

Classic chaotic delay differential equation solved by method of steps.

Pendulum DAE

Pendulum solved both as an ODE in polar coordinates and as a DAE with a holonomic constraint.

Mackey-Glass — Parameter Sweep

Delay-parameter study showing the periodic → chaotic → hyperchaotic transition in Mackey-Glass.

Fractional Relaxation

Caputo-derivative FDE solved by L1 scheme, with the Mittag-Leffler exact solution as reference.

Viscoelastic Material (Prony Series)

Integro-differential equation for stress relaxation, evaluated with O(1)-memory Prony recursion.

Stefan Problem (Moving Boundary)

One-phase melting/solidification PDE on a fixed domain via coordinate transformation.

• stiff

Stochastic Heat Equation

Linear parabolic SPDE with additive space-time white noise; deterministic limit recovered as σ → 0.

• stiff

SPDE Heat — Adaptive Time Stepping

Stochastic heat equation comparing fixed vs. adaptive time stepping; convergence to deterministic limit.

• stiff