PDE Solvers

Heat, Wave, and Laplace equations.

Parameters

Diffusivity α

Spatial N

Time steps

Initial condition

CFL number r=αΔt/Δx² —

Stable? (r ≤ 0.5) —

Max T —

Wave speed c

Spatial N

Time steps

Initial displacement

CFL = cΔt/Δx —

Stable? (CFL ≤ 1) —

Snapshots —

Grid size N×N

Iterations

BC: top value

Source f (Poisson)

Color: blue=0 → red=1 (temperature/potential field)

Select a PDE in the list to learn more.

Partial Differential Equations (PDE) Handbook

Mathematical Principles & Theorems

Simulates second-order linear PDEs in 1D space: (1) 1D Heat / Diffusion Equation (Parabolic): \(u_t = \alpha u_{xx}\), modeling thermal dissipation, solved via FTCS explicit finite differences with Courant-Friedrichs-Lewy (CFL) stability condition \(r = \frac{\alpha \Delta t}{\Delta x^2} \le \frac{1}{2}\); (2) 1D Wave / Vibration Equation (Hyperbolic): \(u_{tt} = c^2 u_{xx}\), modeling acoustic and string oscillations, with d'Alembert traveling wave solution \(u(x,t) = f(x-ct) + g(x+ct)\).

Operating Instructions

Select PDE model: 1D Heat Equation (Diffusion) or 1D Wave Equation (Oscillation).
Configure thermal diffusivity \(\alpha\) or wave speed \(c\), grid points, and step size using numeric inputs.
Select initial condition profile: Gaussian Pulse, Sinusoidal Mode, Square Step, or Plucked String.
Click Run to view 60fps time evolution canvas displaying spatial field amplitudes \(u(x,t)\).