ISEGORIA / MATH ENCYCLOPEDIA
Partial differential equations: fields in motion
Heat, waves, and potential fields on a grid.
Before you begin: Calculus and vectors
Predict, manipulate, then check your reasoning against the example and question. Graphs illustrate the mathematics; they do not replace a proof.
1. Heat diffusion
A rod of length 10 starts with two warm spots. The heat equation is solved by finite differences, and the time slider (or Play) shows the temperature at any moment; the space–time picture below shows the whole history at once. Sharp peaks decay fastest because the flux is proportional to the gradient. Compare insulated ends, which keep the total heat fixed, with ends held at zero, which let it drain away.
Worked example. With insulated ends the total ∫u dx stays constant and the rod tends to its mean temperature; a Gaussian spot of width σ spreads to width √(σ² + 2κt).
Watch out. This explicit scheme is stable only when r = κΔt/Δx² ≤ 1/2; the time step here is chosen so that r = 0.4.
Why does the peak fall?
Heat flows down the gradient into neighbouring cells.
2. Wave equation
A string of length 10 is plucked near one end and released from rest. The solution is d’Alembert’s: the pulse splits into two halves moving left and right at speed c, and each end reflects them. Scrub the time or press Play, change the wave speed, and switch between fixed ends, which invert the reflected pulse, and free ends, which do not. The space–time picture shows the pulses zigzagging along the lines x ± ct.
Worked example. A fixed end reverses the sign of the reflected displacement; after a time 2L/c the string returns to its starting shape.
Watch out. F is the initial shape extended beyond the string, oddly about a fixed end and evenly about a free end. The model ignores damping and nonlinear stretching.
What controls wave speed?
The parameter c, set by tension and linear density.
3. Potential fields
Two small electrodes sit inside a grounded box. Drag them and set their potentials. The potential everywhere else solves Laplace’s equation, computed on a grid by relaxation until every value equals the average of its four neighbours. Equipotentials are drawn every 0.1, and field lines follow E = −∇V from the electrodes, crossing the equipotentials at right angles.
Worked example. Field lines cross equipotentials at right angles, and with both electrodes positive the potential between them has a saddle, not a dip.
Watch out. The solution is computed on an 81 × 61 grid with fixed walls at V = 0; the electrodes are small discs held at fixed potential.
Where is the field strongest?
Where equipotentials are packed most closely.