Problem #3
UCI. MAE200B, winter 2006. by nasser Abbasi.
Problem: Harmonic function in a spherical shell. Find the
solution to the Laplace equation in a spherical shell bounded between
and
with the following boundary conditions
,
where
,
and
Solution
The laplacian in spherical coordinates is
Since the boundary conditions depends on
only, then the solution will not depend on
and hence the PDE simplifies to
Assume
,
and substitute in the above PDE, we obtain
divide by
and rearrange
The LHS depends on
only and the RHS depends on
and hence we apply separation of variables by setting both side equal to same
constant. Call this constant
for now (Later we will see that
for integer
).
Hence
hence we obtain the following 2 differential equations to solve
Looking at equation (2) for now, which we can write as
Now make the substitution
,
hence
But since
, the above can be written as
equation (4) is the Legendre differential equation when has solution only for
integral values of
,
and these values must be successive, hence we write
Now the
equation, equation (1), can be written by replacing
in terms of the eigenvalues
as
Equation (5) is the Euler differential equation. This equation has the
solution
and
Hence
Hence the overall solution
is
so
Applying B.C. at
and
we obtain the following 2 equations
Hence apply the inner product on each equation to evaluate the coefficients,
we obtain the following 2 equations (note that the weight is
when expressed in terms of
Lets call the integral
, and given that
hence
we write the above as
And for the BC at
we obtain
Hence
Now we need to solve eq (6) and eq (7) for
From (6)
Substitute into (7) we obtain
Hence
Substitute the above value for
in (6) to find
Hence finally we have
The following script prints few values of
and
We see that
and all other
and that
and all other
Hence the solution is
so given the above coefficients, we obtain the final solution as
To verify, pick some
and let
and
and verify the boundary conditions. This scripts verifies that the solution is
correct.