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