Exact Solution form Fundamental Solution PDE

Question

Let $f$ be a function on $\mathbb{R}_+\times\mathbb{R}^d$. Let $L$ be some differential operator, like $L=\frac{\partial}{\partial t}+\frac{\partial^2}{\partial x^2}$. Consider for some function $g$ the PDE

$$ L[f]=g$$

equipped with the initial condition that $f(0,x)=h(x)$. If we are told the fundamental solution ( i.e the solution when $f(0,x)=\delta(x_0)$ ) how do I from this get the solution for when $f(0,x)=h(x)$ ?

As an example I am thinking of eq (12.27) the Langevin equation from the book Elements of Nonequilibrium Statistical Mechanics by V. Balakrishnan.

Im confused when comparing it to the wikipedia https://en.wikipedia.org/wiki/Fundamental_solution#Proof_that_the_convolution_is_a_solution because one is relating fundamental solutions about the RHS of the PDE and the other is relating them to initial conditions.

Answer 1

Because of linearity, one can convert an initial condition into a forcing by superposition. In particular, with $(Lf)(t,x)=0,f(0,x)=g(x)$, one can write $f(t,x)=g(x)+h(t,x)$, then $(Lf)(t,x)=(Lg)(t,x)+(Lh)(t,x)=0,h(0,x)=0$. Then the now-inhomogeneous equation to be solved is $(Lh)(t,x)=-(Lg)(t,x)$, where that right side is just a given function now. This is amenable to the type of analysis you were seeing at this point.

This is assuming that the initial condition is regular enough to be in the domain of $L$. If it isn't then you really need to work with the weak formulation anyway just to have everything make sense.