Green's Function
Description
Here Green's Function is the response to a unit impulse function at of the one dimensional heat equation . It can be used to solve the heat equation with an arbitrary initial condition as a convolution integral .
Two Impulses
Because the boundary conditions are zero at and the heat equation solution results in zero for , solutions can be added to obtain a new solution. By using Greens Function an initial condition plus a forcing function solution can be obtained through multiple integrations. To illustrate, the applet below shows an impulse response at and followed by another impulse at and . The final solution is the sum of the other two solutions. In this manor a general solution to the heat equation with forcing is