Here Green's Function is the response to a unit impulse function at [math]x=\xi[/math] of the one dimensional heat equation [math]u_t=\alpha^2u_{xx}[/math]. It can be used to solve the heat equation with an arbitrary initial condition as a convolution integral [math]u\left(x,t\right)=\frac{1}{2\alpha\sqrt{\pi t}}\int_{-\infty}^{\infty}f\left(x\right)e^{-\frac{\left(x-\xi\right)^2}{4\alpha^2t}}d\xi[/math].

Because the boundary conditions are zero at [math]\pm\infty[/math] and the heat equation solution results in zero for [math]u_t-\alpha^2u_{xx}=0[/math] , 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 [math]t=0[/math] and [math]x=\xi_1[/math] followed by another impulse at [math]t=t_2[/math] and [math]x=\xi_2[/math]. The final solution is the sum of the other two solutions. In this manor a general solution to the heat equation with forcing is[br][br]