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 P_n(x).

We have

Since x⁴ is an even function then A_2k+1 = 0. Since x⁴ 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 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 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 |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 j_0. Note that j₀ 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 = 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 A_n = Abⁿ then we get the general solution

Applying the boundary condition we get

so that