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 $r=1$ and $r=2$ with the following boundary conditions

MATH ,MATH where MATH, MATH and $z=r\cos\phi$

Solution


3d.bmp

The laplacian in spherical coordinates is

MATH

Since the boundary conditions depends on $\phi$ only, then the solution will not depend on $\theta$ and hence the PDE simplifies to MATH

Assume MATH, and substitute in the above PDE, we obtain MATH

divide by $R\ \Phi$ and rearrange

MATH

The LHS depends on $R\left( r\right) $ only and the RHS depends on MATH and hence we apply separation of variables by setting both side equal to same constant. Call this constant $k$ for now (Later we will see that MATH for integer $n$). Hence

MATH

hence we obtain the following 2 differential equations to solve MATH

Looking at equation (2) for now, which we can write as

MATH

Now make the substitution $x=\cos\phi$, hence MATH

MATH
and by chain ruleMATHhence
MATH
so substitute these into eq (3) we obtainMATH

But since $x=\cos\phi$ , the above can be written as

MATH

MATH

equation (4) is the Legendre differential equation when has solution only for integral values of $k$, and these values must be successive, hence we write

MATH
$\ $ for integer $n$, and the solution to (4) becomes the Legendre polynomials MATH for $n=0,1,\cdots $ but since $x=\cos \phi $ then the solution to equation (2) is MATH

Now the $R\left( r\right) $ equation, equation (1), can be written by replacing $k$ in terms of the eigenvalues $n$ asMATH

Equation (5) is the Euler differential equation. This equation has the solution MATH and MATH Hence MATH

Hence the overall solution isMATH

so MATH

Applying B.C. at $r=1$ and $r=2$ we obtain the following 2 equations

MATH

Hence apply the inner product on each equation to evaluate the coefficients, we obtain the following 2 equations (note that the weight is $\sin \phi $ when expressed in terms of $\phi $

MATH

Lets call the integral MATH , and given that MATHhence we write the above as

MATH

And for the BC at $r=2$ we obtain

MATH

Hence MATH

Now we need to solve eq (6) and eq (7) for $A_{n},B_{n}$

From (6) MATH

Substitute into (7) we obtain

MATH

Hence MATH

Substitute the above value for $B_{n}$ in (6) to find $A_{n}$

MATH

Hence finally we have

MATH

The following script prints few values of $A_{n}$ and $B_{n}$


Figure

We see that MATH and all other $A_{n}=0$ and that MATH and all other $B_{n}=0$

Hence the solution is

MATH

so given the above coefficients, we obtain the final solution as

MATH

To verify, pick some $\phi $ and let $r=1$ and $r=2$ and verify the boundary conditions. This scripts verifies that the solution is correct.


Figure