Heat Transfer
2D Steady-State Conduction
Set the temperature on four walls β or insulate them β add internal heat, and watch a plate relax to steady state by the same relaxation the computer uses. Find out when the problem has no answer at all.
Transient Conduction
Quench a hot slab and watch the temperature ride inward over time. Lumped capacitance, the Biot number, and the exact moment the simple model starts to lie.
Convection & Boundary Layers
Blow fluid over a hot surface and grow the thermal boundary layer. Reynolds, Prandtl, Nusselt β and the moment laminar trips into turbulence.
Thermal Radiation
Heat a surface and watch its spectrum slide from infrared toward visible. Planck, Wien, StefanβBoltzmann β then trade photons between two gray surfaces.
Solar Heat Exchanger Design
A real published flat-plate solar collector problem: find the transcendental eigenvalues Ξ² tan Ξ² = Nu live, then compare the exact eigenfunction series to the lumped shortcut β and see exactly when the shortcut stops being safe.
Heated Copper Rod
An electrically-heated rod loses energy by convection and radiation at once β a nonlinear ODE with no closed form. Solve it live with RK4, find steady state by root-finding, and see which loss mechanism actually wins.
Circular Fin β Finite Difference vs. Exact
The steady temperature along a pin fin, solved analytically and on a finite-difference grid at once. Toggle insulated vs. convective tip, then refine the mesh and watch the discretization error collapse.
Math Methods
Fourier Series Builder
Stack sines until a square wave appears. Drag the number of terms, watch the Gibbs overshoot refuse to die β then flip to the spinning-circle phasor view.
Phase Portraits & Direction Fields
Click anywhere to drop a trajectory onto a vector field and watch it flow. Eigenvalue classification, saddles & spirals, plus nonlinear classics with limit cycles.
The Heat & Wave Equations
Separation of variables, made visible. Pluck a string or heat a rod and watch the modes evolve β one equation forgets, the other remembers.
Custom Fourier & PDE Solver
Bring your own function. Type any f(x), pick a sine/cosine/full series, read off the coefficients, then send it into the heat or wave equation. Built for checking real PDE homework.
The Auto-Tuner
Drive a damped oscillator with a square or sawtooth wave decomposed into Fourier harmonics. Each one passes through the transfer function β and one landing near resonance can dominate the rest.
Bessel Functions
Solve the wave or heat equation on a circle and sin/cos stop working β Bessel functions take over. Sweep the order, watch the zeros move, and see why they're a drumhead's, a cylinder's, and a planet's secret shared language.
Legendre Polynomials
The natural basis on a sphere β and, by coincidence that isn't a coincidence, exactly the optimal sample points for Gaussian quadrature. Sweep the order and watch the roots spread.
The Gamma Function
Factorial, smoothed out and extended past zero. Click anywhere on the curve to evaluate Ξ(x), watch it blow up at every non-positive integer, and flip sign across each pole.
Conformal Mapping: the Joukowski Airfoil
Flow around an airfoil is hard. Flow around a circle is easy. Solve the easy one, then watch every streamline bend through one transformation into flow around a real airfoil β lift included, verified against thin-airfoil theory.
Vibrations & Dynamics
The SpringβMassβDamper
The one system to rule them all. Pluck it and watch it ring down, or drive it with a shaker, tune mass, stiffness and damping, and cross into resonance.
Frequency Response & Bode Plots
Sweep the forcing frequency and trace the Bode magnitude and phase. Watch input and output sinusoids scale and slip out of step β where machines shake themselves apart.
Numerical Methods
Root Finding & Fixed-Point Iteration
Newton, secant, and fixed-point iteration hunting a root. Watch the cobweb staircase converge β or fly off, when the same equation is rearranged the wrong way.
Heat Transfer Through Concentric Cylinders
A real transcendental heat-conduction equation, solved five different ways. See a naive fixed-point rearrangement diverge, a working one crawl, and Steffensen's method turn that crawl into 3 iterations flat.
ODE Solver β Slope Fields & RK4
Type yβ² = f(x, y), see the direction field, click to drop an initial condition, and race Euler against RungeβKutta β with a mini ODE textbook alongside.
GaussβSeidel & Jacobi
Solve a linear system by iteration β watch it converge sweep by sweep, or diverge when the matrix isn't diagonally dominant. Includes a heated-fin and a circuit system to play with.
Potential Flow Over an Irregular Conduit
Laplace's equation on a domain with a slanted-wall cutout β the mesh, the finite-difference equations, and the matrix are all generated automatically from a list of boundary points, then solved live.
Bernoulli Equations
Nonlinear by exactly one term β a power of y. Strip it out with the right substitution and watch a direct numerical solve land right on top of the linearized one, even as it races toward a blow-up.
Explicit vs. Implicit: When a Time Step Blows Up
The Circular Fin, now heating up over time. Push the explicit time step past a hard speed limit and watch it diverge in a handful of steps β switch to implicit at the same step size and watch it calmly settle instead.
ADI: 2D Transient Conduction Without the Penalty
A hot plate, quenched on all four edges. 2D explicit's stability limit is even tighter than 1D's β push past it and watch it diverge. Switch to ADI and stay perfectly stable at 5Γ that same limit.
Ritz-Galerkin Method
Skip solving the differential equation exactly β pick a few trial functions, force the leftover error to be orthogonal to each one, and solve a tiny linear system instead. Watch it converge toward the exact Bessel-function solution as you add terms.
Ritz-Galerkin in 2D: a Heated Square Plate
The same trial-function trick, now on a plate. A second term improves the center β checked against an independent finite-difference solve β but watch the heatmap lose its symmetry, a real consequence of the trial space, not a bug.
Capstones
Design a Cooling Fin Array
Not a "verify the known answer" lab β a design brief. Hit a 50 W heat-rejection target on a fixed footprint by choosing fin diameter, length, and material, built on the verified single-fin physics from the Circular Fin lab.
Vibration Isolator Tuning
Builds on Spring-Mass-Damper and Frequency Response β design an isolator that keeps a forcing frequency away from resonance. On the way.
Choose Your Numerical Method
A physical problem, no method specified β pick the right tool from across the whole site and justify it. On the way.
Games
Match the Mystery Wave
A secret wave hides on screen, built from pure harmonics. Dial in the sliders to land on top of it, climb the match meter, and earn your stars. Fourier decomposition, by feel.
Classify the Fixed Point
A direction field flashes on screen β name the fixed point before the timer runs out. Saddle, node, spiral, or center? Build a streak as the matrices creep closer to the boundary.
Make It Converge
A linear-algebra puzzle where you swap rows to restore diagonal dominance and make the system converge. On the way.