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.