Model impulse responses for linear operators with confidence. Tune boundaries, potentials, damping, and grid resolution. Get G(x,x0) values, diagnostics, CSV tables, and PDFs instantly.
Pick an operator, boundary conditions, and a point source. The tool returns a discrete Green function G(x, x0) on a 1D grid.
Sample configuration for a Dirichlet Poisson case on L=1, N=201, x0=0.50.
| x | G(x, x0) | Note |
|---|---|---|
| 0.00 | 0.000000 | Left boundary |
| 0.25 | ~0.1875 | Rises toward source |
| 0.50 | peak | Nearest node to x0 |
| 0.75 | ~0.1875 | Symmetric for this case |
| 1.00 | 0.000000 | Right boundary |
A Green function is defined by solving a linear operator equation with a delta source:
This calculator discretizes the 1D operator on a uniform grid and solves:
For Poisson and Helmholtz families the interior stencil uses:
Boundary conditions are enforced by replacing the first and last matrix rows with Dirichlet, Neumann, or Robin constraints.
A Green function is the impulse response of a linear differential operator. Once G(x,x0) is known, any forcing f(x) can be mapped to a solution by superposition. In one dimension, the calculator returns a discrete N-point approximation on [0,L].
The Poisson family models diffusion, electrostatics, and steady transport through -p\,d^2/dx^2 + q. The Helmholtz option adds a screening scale using k^2, common in waveguides and Yukawa-type kernels. The Schrödinger-like form uses V-E to represent energy-shifted response.
Boundary choices encode how the system interacts with its environment. Dirichlet enforces fixed values at endpoints, Neumann enforces zero slope, and Robin blends both with \alpha G + \beta G' = 0. For example, large \beta favors small gradients, while \beta=0 with nonzero \alpha approximates a hard constraint.
The grid spacing is \Delta x = L/(N-1). Increasing N sharpens the impulse and improves spatial detail, but it can amplify conditioning issues when the operator is weakly constrained. As a practical target, start with N=201 and increase gradually while monitoring the residual and the min/max range.
The continuous source \delta(x-x0) is represented by a single-node impulse with b_j=1/\Delta x. This scaling keeps the discrete integral consistent so that \sum_i b_i\,\Delta x\approx 1. If you shift x0, the nearest index changes, which is expected for a grid-based method.
With G available, a linear boundary value problem L u=f can be approximated by a discrete convolution-like sum: u(x_i)\approx\sum_j G(x_i,x_j)\,f(x_j)\,\Delta x. The exported table supports quick post-processing in scripts for multiple right-hand sides without re-deriving formulas.
The tool reports an L2 residual for the linear system, which should decrease as constraints and resolution improve. Many applications also check qualitative structure: continuity away from x0, a slope jump consistent with the impulse, and an integral magnitude that stays stable under moderate N changes.
CSV export provides the full grid for plotting, fitting, or building response operators. The PDF export summarizes parameters, diagnostics, and sampled values for lab notes. For reproducibility, record L, N, boundary types, and coefficients, then verify that repeated runs produce consistent residuals and peak location.
It computes a discrete 1D Green function G(x,x0) for a chosen linear operator and boundary conditions by solving a tridiagonal finite-difference system.
Because \Delta x changes with N, affecting how the delta source and second derivative are approximated. Increasing N typically improves accuracy until conditioning limits appear.
Use stronger boundary constraints, avoid extreme Robin parameters, and try moderate N first. If Helmholtz, verify k and offsets produce a well-posed problem on [0,L].
p scales the second-derivative stiffness, q shifts the diagonal as a reaction/offset term, and k adds screening via k^2 in the Helmholtz family.
Yes. Robin boundaries can emulate partial absorption by tuning \alpha and \beta. The best parameters depend on your physical boundary law and the units used in your model.
Often yes for self-adjoint operators with symmetric boundary conditions, but not always. Non-symmetric conditions or operator choices can break reciprocity, so use the symmetry as a check, not a guarantee.
Use CSV for plots and numerical integration, and PDF for documentation. When publishing results, include L, N, boundary settings, and coefficients so others can reproduce the kernel.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.