Followed by a question posted here we are now interested in how can we find the Green function for the following problem

where
a constant and with boundary condition
.
Note that, for heat equation, in one variable, the Green’s function is a solution of the initial value problem

with boundary condition

where
is the Dirac delta function and this Green function is already known as
.
Note that, the Green’s function for heat equation in the whole line 

is
.
However, this is not true once we restrict ourselves to the half-plan

with boundary condition
.
In order to find the correct one, we need to use the so-called reflection principle as follows: For each
, we define
, called the inverse point of
. It can be readily verified that the function

which implies
.
Now let us go back to the viscous Burgers type equation. By the following change of variable

one easily sees that

which gives
.
In order to eliminate the term
, we use the following change of variable

we obtain

which yields
.
Hence we have the following summary
.
We are now in a position to talk about the most interesting point of this entry: How to find the Green function for backward once we already have the Green function for pushforward? It is obvious to see that the boundary condition doesn’t change, i.e.
.
The Green function for the last equation is already known, called
.
The Green function for the equation for
can be constructed by looking at the substitution used. Precisely,
.
The Green function for the equation for
is constructed a little bit crazy, what we need is the following
.
The most important property one should check is the boundary condition, roughly speaking, the Green function must satisfy
.
See also: Green’s function and differential equations