Calculator Inputs
Choose a solver mode, fill the inputs, and submit. Results appear above this form.
Example Data Table
| Mode | Key inputs | Main check | Sample output |
|---|---|---|---|
| 1D Heat | α=0.25, L=1, T=0.2, Nx=20, Nt=40 | r ≤ 0.5 | Center temperature decay over time |
| 1D Wave | c=1, L=1, T=0.5, Nx=20, Nt=80 | λ ≤ 1 | Oscillatory midpoint displacement |
| 2D Laplace | 16×16 grid, tol=0.0001, mixed boundaries | Residual below tolerance | Center potential and convergence log |
| Classification | A=1, B=0, C=1 | B²−4AC sign | Elliptic classification |
Formula Used
1D Heat equation
Model: u_t = αu_xx
Explicit step: u_i^{n+1} = u_i^n + r(u_{i+1}^n - 2u_i^n + u_{i-1}^n), where r = αΔt/Δx².
1D Wave equation
Model: u_tt = c²u_xx
Explicit step: u_i^{n+1} = 2u_i^n - u_i^{n-1} + λ²(u_{i+1}^n - 2u_i^n + u_{i-1}^n), where λ = cΔt/Δx.
2D Laplace equation
Model: u_xx + u_yy = 0
Jacobi update: u_{i,j}^{new} = 0.25(u_{i+1,j} + u_{i-1,j} + u_{i,j+1} + u_{i,j-1}).
PDE classification
For Au_xx + Bu_xy + Cu_yy + Du_x + Eu_y + Fu = G, compute B² - 4AC.
Positive means hyperbolic, zero means parabolic, and negative means elliptic.
How to Use This Calculator
1. Select the PDE mode
Choose heat, wave, Laplace, or classification based on the equation or numerical task you want to analyze.
2. Enter model inputs
Provide coefficients, domain size, boundaries, grid counts, or tolerance values. Smaller steps often improve accuracy but increase runtime.
3. Solve and review checks
Submit the form and review stability ratios, residuals, or discriminants to confirm whether the setup is numerically reasonable.
4. Export your results
Use the CSV or PDF buttons after calculation to save the result summary and computed data table for reporting or revision.
Frequently Asked Questions
1. What can this solver handle?
It handles four common tasks: solving the 1D heat equation, solving the 1D wave equation, iterating a 2D Laplace field, and classifying second-order PDEs.
2. Is this a symbolic PDE solver?
No. It focuses on practical numerical methods and equation classification. That makes it useful for study, quick checks, and small computational experiments.
3. Why does the heat solver show a stability warning?
The explicit heat scheme depends on r = αΔt/Δx². Values above 0.5 can produce unstable growth instead of physical diffusion.
4. Why does the wave solver use the Courant number?
The Courant number measures how far information travels in one time step. Values above one often break stability for the explicit update.
5. What does the Laplace residual mean?
The residual is the maximum change between two iterations. Smaller residuals mean the relaxation process is approaching a steady interior solution.
6. How are boundary conditions applied here?
The heat and wave solvers use fixed end values. The Laplace solver fixes all four edges while iterating the interior nodes.
7. Can I use larger grids for better accuracy?
Yes, but larger grids increase computation time. They may also require smaller time steps for explicit heat and wave calculations.
8. What does PDE classification tell me?
Classification reveals the qualitative character of the equation. Hyperbolic equations behave like waves, parabolic equations diffuse, and elliptic equations smooth toward equilibrium.