defined by 
such that
for 
problem:
Find fundamental solution associated with operator defined by such that
for
answer:
The fundamental solution can be written as
And our goal is to determine 
. In the above 
are the 2 independent solution to the
homogenous equation 
We start by finding 
We try solution 
and substitute this into the above homogenous equation,
we obtain the characteristic equation 
, hence 
the 2 solution are 
and 
. Hence
our fundamental solution now looks like
Now consider the test function 
, hence
Where 
Hence expanding the differentiation in the above and simplifying we obtain
Now take 
, i.e. put a point source as input, then we are looking for 
from the
properties of delta function. In other words, we are looking for
Hence
Looking at the first integral, and perform integration by parts. In these calculations we note that
Hence
Now do integration by part on the last integral above
Now looking at (1) above, we now do integration by part on 
Consider the first integral above in the RHS, we write
Now do integration by part on the last term in the above line
Now we do integration by parts on 
Hence, from (2),(3),(4),(5), we have
or
![[ ]
ϕ (ξ )= ϕ (ξ )(2B)+ ϕ′(ξ) − ξ3+ Aξ3+ B ξ](ma35x.png) | (6) |
By looking at the coefficients on 
, and compare, we see that 
or
We can now use the continuity condition at 
and write
at 
, hence
Hence
Therefor the fundamental solution is
Here is a plot for few values of 
