1D Green's Function for the Laplacian

What you are seeing: the Green's function $G(x, x_0)$ for $-u'' = f$ on $[0, L]$ with Dirichlet boundary conditions. Drag the source location $x_0$ ; the Green's function (orange tent) is plotted. Below, the solution $u(x)$ for the user-selected $f(x)$ , obtained via convolution with $G$ .

Figure 1. Green's function (top) and solution for a chosen source (bottom).

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f(x)

WHAT TO TRY

Drag the poke position x0 along the top string: the Green tent G(x, x0) follows, peaking under your finger at height x0(1 - x0). The response is largest for a poke at the centre and vanishes at the pinned ends.
Change the distributed load f(x) between constant, step, Gaussian and sine: the bottom string re-settles into u(x) = integral G(x,s) f(s) ds, the superposition of one tent per source point. A constant load sags into a parabola.
Watch the sweep build u(x) one poke at a time, left to right: each source adds its own scaled tent, and the diagnostic confirms u(L/2) = L squared over 8 for the constant load.