MATH3271

FW02

Solutions to Assignment 5                                                                                        

1. Legendre polynomials

(a)  Use Rodrigues’ formula to find the Legendre polynomials P0, P1 and P2.

(b)  Using Bonnet’s recurrence relation and the results from part (a) find the Legendre polynomials P3, P4 and P5.

            From the recurrence relation

so we get

(c)  Show that P2 and P4 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 x4 in a Legendre series in Pn(x).

            We have

Since x4 is an even function then A2k+1 = 0.  Since x4 is a polynomial of degree 4 then An = 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 Pn(cosθ).

We have

Note that cos(2θ) = 2cos(θ)2 – 1 which is an even function of cos(θ) so that A2k+1 = 0.  Similarly since cos(2θ) is a polynomial of degree 2 in cos(θ) An = 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 cos4θ.  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 A0 = 2/5, A2 = 8/7, A4 = 16/35 with the rest of the An = 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 |P2(cosθ)| and |P3(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 j0 and j1 over the interval [0, 20] on the same graph.  What can you say about the zeros of j1 relative to the zeros of j0?

            See MAPLE output.

(b) Find the first ten zeros of j0.  Note that j0 has the same zeros as J1/2.  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 An.

 

            We can write where Rn 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 Rn = Arn + B/rn+1.  In order for this solution to satisfy the boundary condition u(b, θ) = 0 we must have Abn + B/bn+1 = 0 which requires B = -Ab2n+1.  If we let An = Abn then we get the general solution

 

Applying the boundary condition we get

so that