Inputs
Select a kernel, enter parameters, and compute the Green function. Optional tools include dG/dx, forcing integration, and a 1D kernel plot.
Example Data Table
Sample inputs and expected kernel values (rounded). Use them to sanity-check your configuration.
| Case | Inputs | Expected output | Notes |
|---|---|---|---|
| 1D Poisson (Dirichlet) | L=1, x=0.30, ξ=0.70 | G≈0.0900000000 | Uses G=x(L-ξ)/L for x<ξ. |
| 2D Laplace (free space) | (x,y)=(1,0), (ξx,ξy)=(0,0), ε=1e-9 | G≈-(1/(2π)) ln(1)=0 | At r=1, ln(1)=0, so G≈0. |
| 3D Laplace (free space) | (x,y,z)=(1,0,0), (ξx,ξy,ξz)=(0,0,0) | G≈0.0795774715 | G=1/(4πr) with r=1. |
Formula Used
How to Use This Calculator
- Pick a problem type that matches your operator and domain.
- Enter the field point and source point coordinates. For interval kernels, keep x and ξ inside [0, L].
- For Helmholtz, choose k away from resonant values where sin(kL) is near zero.
- Enable dG/dx to see the derivative when the kernel is differentiable away from the source.
- Enable u(x) to integrate u(x)=∫Gf (available for 1D interval kernels). Choose a forcing model and set N for accuracy.
- Use Download CSV or Download PDF after computing to export your data.
Technical Article
1) Green functions as impulse responses
A Green function is the field from a unit point source for a chosen linear operator. Once G(x,ξ) is known, many boundary-value problems reduce to integration or convolution. This calculator reports kernels and can numerically evaluate u(x)=∫Gf for 1D interval cases, for fast comparison across multiple scenarios.
2) Kernel families provided
The tool includes 1D Poisson and 1D Helmholtz kernels on [0,L] with Dirichlet boundaries, 2D and 3D Laplace free-space kernels, and the 1D diffusion (heat) fundamental solution for t>0. These cover electrostatics checks, modal wave problems, scattering benchmarks, and transient diffusion estimates.
3) Boundary conditions and interpretation
Dirichlet boundaries enforce zero field at x=0 and x=L, modeling grounded conductors or fixed constraints. Free-space Laplace kernels assume decay at infinity. If your geometry has mixed boundaries, treat these kernels as reference building blocks rather than final models.
4) Singularities and the role of ε
In free space, Laplace kernels diverge as distance r→0. The calculator replaces r with max(r,ε) (default 1e-9) to keep output finite for reporting and export. Choose ε smaller than your shortest meaningful length scale.
5) Helmholtz conditioning and resonant points
The Dirichlet Helmholtz kernel scales like 1/(k sin(kL)). When sin(kL) is near zero, the kernel becomes highly sensitive, indicating proximity to interval eigenmodes. For parameter sweeps, avoid kL≈nπ unless resonance behavior is the goal.
6) Forcing integration and accuracy data
With integration enabled, u(x)=∫₀⁼G(x,ξ)f(ξ)dξ is computed by a trapezoidal rule using N samples (default 800). For smooth inputs, error often decreases roughly as O(h²), where h=L/(N-1). For sharp Gaussians or high modes, increase N until u(x) converges.
7) Units and normalization checks
Kernel units invert the operator: the 3D Laplace kernel scales as 1/r, while the 2D kernel varies as -ln(r). The diffusion kernel integrates to 1 over space at fixed t. A quick check is that larger D or t broadens the curve and lowers its peak.
8) Validation and reporting workflow
Start with the example rows, then test symmetry and trends. For 1D Poisson, verify G(x,ξ)=G(ξ,x) and the slope jump in dG/dx at x=ξ. For Laplace kernels, confirm 1/r or -ln(r) behavior. Export CSV/PDF to document inputs, outputs, and any chosen regularization.
FAQs
1) Why does the Laplace kernel blow up at the source?
Because the Green function solves ∇²G=δ, a point source creates a singularity. The ε option limits reporting near r=0, but it does not replace a finite-size physical model.
2) What does the 1D Poisson kernel represent physically?
It is the response of a grounded 1D interval to a unit point load. Dirichlet boundaries force the field to zero at both ends, so the kernel peaks between them and vanishes at x=0 and x=L.
3) When should I increase the integration steps N?
Increase N when f(ξ) is sharp, oscillatory, or when u(x) changes as you vary N. A practical check is doubling N until u(x) stabilizes.
4) Why is dG/dx sometimes shown as NaN?
For 1D kernels, the derivative has a jump at the source location x=ξ. The calculator reports NaN exactly at that point to avoid implying a single-valued derivative.
5) What does the Helmholtz resonance warning mean?
When sin(kL) is near zero, the Dirichlet kernel denominator is tiny, amplifying sensitivity. This corresponds to interval eigenmode resonance, so unusually large values often indicate near-resonant settings.
6) Can I use these kernels for arbitrary geometries?
They are canonical kernels for simple domains. For complex boundaries, combine kernels with images, eigenfunction expansions, or numerical solvers such as boundary elements. Use this tool for benchmarking and quick analytic estimates.
7) How should I choose the regularization ε?
Pick ε smaller than the shortest meaningful length scale in your model. If you export values away from the source, 1e-9 is fine. If you probe very near the source, reduce ε carefully.