PDE Solver

Solve partial differential equations numerically using finite difference methods. Visualize heat diffusion, wave propagation, and steady-state solutions.

How It Works

Heat Equation: ∂u/∂t = α ∂²u/∂x²

The heat equation models how temperature diffuses through a material over time. The parameter α (thermal diffusivity) controls how fast heat spreads. Larger α means faster diffusion.

Method: Explicit forward-time central-space (FTCS) finite difference scheme. At each time step: u_iⁿ⁺¹ = u_iⁿ + r(u_i+1ⁿ - 2u_iⁿ + u_i-1ⁿ), where r = αΔt/Δx². Stability requires r ≤ 0.5.

Wave Equation: ∂²u/∂t² = c² ∂²u/∂x²

The wave equation describes vibrations and wave propagation. The parameter c is the wave speed. Solutions exhibit propagation, reflection, and superposition.

Method: Leapfrog (central difference in both time and space) scheme. u_iⁿ⁺¹ = 2u_iⁿ - u_i^n-1 + r²(u_i+1ⁿ - 2u_iⁿ + u_i-1ⁿ), where r = cΔt/Δx. The CFL condition requires r ≤ 1.

Laplace Equation: ∂²u/∂x² + ∂²u/∂y² = 0

The Laplace equation describes steady-state temperature distribution (or electrostatic potential) in a region, given fixed values on the boundary. The solution is the equilibrium state with no internal sources.

Method: Gauss-Seidel iterative relaxation (or SOR for faster convergence). Each interior point is updated to the average of its four neighbours: u_i,j = (u_i+1,j + u_i-1,j + u_i,j+1 + u_i,j-1) / 4. Iteration continues until the maximum change is below the specified tolerance.

Boundary Conditions

• Dirichlet: Fixed value at the boundary (e.g. u = 0). Models a surface held at constant temperature.
• Neumann: Fixed derivative at the boundary (du/dx = 0). Models an insulated surface with no heat flux.
• Periodic: The solution wraps around: u(0) = u(L). Models an infinite or cyclic domain.