Green's functions and delta initial conditions

Question

The Green's function of the $Lu=u_t-u_{xx}=0$ differential equation is a Gaussian distribution with variance $t/2$ and mean $0$, which is also the solution to the differential equation itself, for a $\delta$ initial condition at $x=0$. The general solution here is a convolutional product $G*f$ for initial $u(0,x)=f(x)$. For the wave equation $Lu=u_{tt}-u_{xx}-u_{yy}-u_{zz}=0$, Green's function is a spherical wavefront with a $\delta$ distribution, exactly what you'd expect from a $\delta$ initial condition. The 1st differential equation here is a parabolic, the 2nd hyperbolic, so they're quite different. I understand how Green's function is used to solve inhomogeneous differential equations using a convolutional product; if $LG=\delta$, $L(f*G)=f*(LG)=f*\delta=f$, so $y=f*G$ solves $Ly=f$. But that is not what seems to be going on here. So I don't understand, what is the relationship between Green's function and $\delta$ initial conditions? To what extent does this $y=G*f$ trick for initial conditions $f$ work? Why does this look so similar to solving an inhomogeneous equation?

Answer 1

Note first that for the heat equation and for the wave equation we have a "special" coordinate $t$, which we associate with time. Therefore, for these problems it is natural to consider "initial value problems," whereas for your second example there is no such thing as the initial condition, so the analogy is not that straightforward as you describe.

Second (and to answer your question), for any inhomogeneous linear problem of the form $$ Lu=f(t,x),\quad u(0,x)=0,\tag{1} $$ with a special coordinate $t$, one can consider a family of the initial value problems $$ Lv=0,\quad v(\tau,x)=f(\tau,x),\quad \tau\geq 0.\tag{2} $$ If you can solve the second problem, then the solution to the original problem is simply given as $$ u(t,x)=\int_{0}^t v(t,x;\tau)d\tau, $$ where $v(t,x;\tau)$ is the solution of $(2)$ for $t_0=\tau$. That is, for the linear problems the initial value problem for the homogeneous equation and solution to an inhomogeneous equation are very much related (this is called Duhamel's principle).