PDE Solvers

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 \(u(x,t)\).