**MATH3271**

**FW02**

**Solutions to Assignment 5 **

1. Legendre polynomials

(a) Use Rodrigues’ formula to find the Legendre
polynomials P_{0}, P_{1} and P_{2}.

_{}

(b) Using Bonnet’s recurrence relation and the
results from part (a) find the Legendre polynomials P_{3}, P_{4}
and P_{5}.

From the recurrence relation

_{}

so we get

_{}

(c) Show that P_{2} and P_{4} are
orthogonal over the interval [-1, 1].

_{}

(d) Use MAPLE to plot the first four Legendre
polynomials on the interval [-1, 1] on the same graph. Put a label on the various curves (by hand)
in order to identify each polynomial.

See MAPLE output

2. (a) Expand x^{4} in a Legendre series in
P_{n}(x).

We have

_{}

Since x^{4}
is an even function then A_{2k+1} = 0.
Since x^{4} is a polynomial of degree 4 then A_{n}
= 0 when n > 4. Thus we have

Thus

_{}

which you can verify using the results
from question 1.

(b) Expand cos(2θ)
in a Legendre series in P_{n}(cosθ).

We have

_{}

Note that cos(2θ) = 2cos(θ)^{2} – 1 which is an even
function of cos(θ) so that A_{2k+1} = 0. Similarly since cos(2θ)
is a polynomial of degree 2 in cos(θ) A_{n} = 0 for n > 2. Now

_{}

making the usual substitution that x =
cosθ. Thus

_{}

Thus

_{}

3. Dirichlet problem with symmetry

(a) Solve the following Dirichlet problem for the
sphere

_{}, 0 ≤ r ≤ 1, 0 ≤ θ ≤ π

subject to the boundary condition that u(1,
θ) = 2 cos^{4}θ. You
may use the results from

question 2(a) here.

The solution to the symmetric
Dirichlet problem for the interior of a sphere is

_{}

The boundary
condition is

_{}

Thus we must have
A_{0} = 2/5, A_{2} = 8/7, A_{4} = 16/35 with the rest
of the A_{n} = 0. Thus the solution is

_{}

(b) Use MAPLE to produce a three-dimensional plot
of the solution from part (a). The
simplest way to do this is to use cylindrical coordinates with ρ replaced
by r and φ replaced by θ. Then
u will be plotted as the z-coordinate. Note that the range of θ is [0, π].

See MAPLE output.

4. Use MAPLE to
produce a three-dimensional plot of |P_{2}(cosθ)|
and |P_{3}(cosθ)|.

See MAPLE
output.

5. MAPLE does not
have a specific function for the spherical Bessel functions but since

_{}

we can use BesselJ(n+1/2,x) to
evaluate these functions.

(a) Plot j_{0}
and j_{1} over the interval [0, 20] on the same graph. What can you say about the zeros of j_{1}
relative to the zeros of j_{0}?

See MAPLE
output.

(b) Find the first ten zeros of j_{0.} Note that j_{0} has the same zeros as
J_{1/2}. Identify these zeros.

See MAPLE
output.

6. Consider the symmetric Dirichlet problem
between two spheres, i.e.

_{}, 0 ≤ θ ≤ π

subject to the boundary conditions u(a,
θ) = f(θ) and u(b, θ) = 0.
By using the full solution to Euler’s equation show that u(r, θ) is
given by

_{}

Derive an explicit expression for the coefficients A_{n}.

We can write _{}where R_{n} satisfies the Euler equation

_{}

Thus the indicial equation is s(s-1) + 2s – n(n+1)=0 which has solutions s = n or s = -(n+1). Thus R_{n} = Ar^{n} + B/r^{n+1}. In order for this solution to satisfy the
boundary condition u(b, θ) = 0 we must have Ab^{n}
+ B/b^{n+1} = 0 which requires B = -Ab^{2n+1}. If we let A_{n}
= Ab^{n} then we get the general solution

_{}

Applying the boundary condition we get

_{}

so that

_{}