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
It is nice. Thanks.
Comment by TTT — March 18, 2010 @ 17:36