Green's Functions

A Green's function is the impulse response of a differential operator — the solution when the source term is a Dirac delta. If $\mathcal{L}$ is a linear differential operator, then:

$\mathcal{L}\, G(\mathbf{r}, \mathbf{r}') = \delta(\mathbf{r} - \mathbf{r}')$

Once we have G, the solution to $\mathcal{L}\, u = f$ is:

$u(\mathbf{r}) = \int G(\mathbf{r}, \mathbf{r}')\, f(\mathbf{r}')\, d\mathbf{r}'$

1D Poisson Equation

For $-d^2u/dx^2 = f(x)$ on $[0, L]$ with $u(0) = u(L) = 0$:

$G(x, x') = \frac{1}{L} \min(x, x')\bigl(L - \max(x, x')\bigr)$

This has a characteristic "tent" shape — it is continuous with a kink at $x = x'$.

The solution is then:

$u(x) = \int_0^L G(x, x')\, f(x')\, dx'$

For example, if $f(x) = 1$ (constant forcing), the solution is $u(x) = x(L-x)/2$ — a parabolic profile that satisfies the boundary conditions and the equation.

Free-Space Green's Function (1D Laplacian)

In 1D free space (no boundary conditions), the Green's function for $-d^2/dx^2$ is:

$G(x, x') = -\frac{|x - x'|}{2}$

Numerical Solution via Green's Function

We discretise $x'$ into $N$ points and approximate the integral as a sum:

$u(x) \approx \sum_{i=0}^{N-1} G(x, x_i')\, f(x_i')\, \Delta x'$

where $x_i' = (i + 0.5) \cdot L/N$ and $\Delta x' = L/N$.

Implementation

• greens_1d_poisson(x, x_prime, L) — returns $G(x, x')$ for the 1D Poisson problem
• greens_free_space_1d(x, x_prime) — returns $-|x - x'|/2$
• poisson_solution(x, f_values, L, N=100) — evaluates $u(x)$ by summing $G \cdot f \cdot \Delta x$

import math

def greens_1d_poisson(x, x_prime, L):
    return (1.0 / L) * min(x, x_prime) * (L - max(x, x_prime))