Mathematics 174 – Applied Mathematics 2 Midterm Problem Set February 24, 2000

Groundrules and Directions

This is an open-book, open-notes, open-library, open Swords 335, unlimited time assignment. If you use something you find in a book other than Haberman in one of your solutions, please footnote it. You must work independently, and you may not consult other faculty members in working on these problems. When you have completed the problem set, please write a brief statement saying that you have abided by these ground rules, sign it, and turn it in with your solutions.

In all parts of problems where solution of a BVP is asked for, give your answer in series form, together with formulas for the coefficients.

Don’t kill trees unnecessarily for the Maple output you submit: delete irrelevant work and resize your plots before printing!!

Due: No later than 5:00 pm, Thursday, March 2.

Standing Assumptions: In all parts of both problems, m is an integer and m ≥ 0.

I. For the 2D heat equation on a circular disk of radius a, if the outer edge is insulated, we are led to consider the problem

\{ \begin{matrix}u_{t} = k∇^{2}u \\ u_{r}(a, θ, t) = 0 \\ u(r, θ, 0) = α(r, θ) \end{matrix}

A) If we look for product solutions u(r, θ, t) = f(r)g(θ)h(t), what ODE’s must h and g

satisfy, and what boundary value problem for f(r) results? B) Show that for each Bessel function of the first kind J_{m}(z) there is an infinite sequence

of numbers w_{mn} ≥ 0 where J′_{m}(w_{mn}) = 0. (Note: I’m looking for an actual proof here, given the facts we know about J_{m}(z), its zeroes, etc. What theorem from 1-variable calculus guarantees the existence of places where the derivative of a function is zero?)

C) What are the eigenvalues and eigenfunctions of the boundary value problem for f(r)? D) Solve the problem:

\{ \begin{matrix}u_{t} = k∇^{2}u \\ u_{r}(a, θ, t) = 0 \\ u(r, θ, 0) = r sin(3θ) \end{matrix}

II. In this problem, you will study the eigenvalues and eigenfunctions of the Sturm-Liouville problem:

(1)  \{ \begin{matrix}\frac{d}{dr} (r \frac{df}{dr} ) − \frac{m^{2}}{r} f = −λrf \\ f(a) = 0 & where a > 0 \\ f(b) = 0 \end{matrix}

arising from problems on annular (ring-shaped) regions, and use them to solve several boundary value problems.

In class on 2/22, we saw that the change of variables z^{2} = λr^{2} takes this to a new type of boundary value problem for the Bessel equation of order m:

(2)  \{ \begin{matrix}z^{2} \frac{d^{2}f}{dz^{2}} + z \frac{df}{dz} + (z^{2} − m^{2})f = 0 \\ f(\sqrt{λ}a) = 0 \\ f(\sqrt{λ}b) = 0 \end{matrix}

(Note: the square roots are justified here since all eigenvalues satisfy λ ≥ 0, as we will see shortly.)

The general solution of Bessel’s equation of order m is a linear combination of the Bessel function of the first kind, J_{m}(z), and the Bessel function of the second kind, which is called Y_{m}(z):

f(z) = c_{1}J_{m}(z) + c_{2}Y_{m}(z)

For m ≥ 1, Y_{m}(z) ∼ z^{−m} as z → 0, so it is not defined for z = 0. (Similarly, Y_{0}(z) ∼ c ln(z) for a constant c, so it is not defined for z = 0 either.) That is why we have not used (or needed) Y_{m}(z) before – in all of our work on circular domains, we have wanted solutions f of the Bessel equation with |f(0)| < ∞. Now, however, our boundary conditions are at a, b > 0 and we are interested in functions on the interval [a, b]. Their behavior at z = 0 is not relevant!

A) Show that for each integer m ≥ 0, we will get an infinite sequence of eigenvalues λ_{mn} of (1), n = 1, 2, 3, …, and corresponding eigenfunctions ϕ_{mn}(r) determined uniquely up to a constant multiple.

B) Using the Rayleigh quotient for (1), show that all eigenvalues satisfy λ_{mn} ≥ 0. C) Is there a smallest eigenvalue overall? If so, for which m is the minimum achieved? If

not, say why not. D) Show that if f(z) = c_{1}J_{m}(z) + c_{2}Y_{m}(z) is a nonzero function satisfying (2), then

(3)  J_{m}(\sqrt{λ}a)Y_{m}(\sqrt{λ}b) − J_{m}(\sqrt{λ}b)Y_{m}(\sqrt{λ}a) = 0

E) The relation (3) in part D is actually an equation for λ. It cannot be solved exactly by any usual technique, but we can get approximations to solutions by graphical and numerical methods in Maple. Taking a = 1 and b = 2, by plotting, estimate the locations of the first 4 solutions of (3) for m = 0, 1, 2. Then use the fsolve command in Maple to generate accurate approximations of each of your roots. Notes: The Bessel function of the second kind of order m is called BesselY(m,z) in Maple. The command fsolve(equation(var),var=c); will use a numerical method similar to Newton’s Method to solve the equation equation(var) = 0 for the variable var, starting from the initial approximation var = c.

F) Once you have an approximation for λ_{mn}, how do you get an approximation to the corresponding eigenfunction ϕ_{mn}(r)?

G) Studying the vibrations of an annular (ring-shaped) membrane with inner (r = a) and outer (r = b) edges fixed at zero displacement leads to consideration of the boundary value/initial value problems

(4)  \{ \begin{matrix}u_{tt} = c^{2}∇^{2}u \\ u(a, θ, t) = 0 \\ u(b, θ, t) = 0 \\ u(r, θ, 0) = α(r, θ) \\ u_{t}(r, θ, 0) = β(r, θ) \end{matrix}

As always, the product solutions u(r, θ, t) = f(r)g(θ)h(t) are called the normal modes of vibration. There are four types of normal modes (“cosine-cosine, cosine-sine, etc.”) of (4) for each pair of integers (m, n) with m ≥ 0, n ≥ 1. Incorporating your work from parts A through F, use Maple to plot (approximations to) the “cosine-cosine” normal modes with m = 0, 1, 2, n = 1, 2, 3, 4 for a ring with a = 1 and b = 2. (Hand in plots with t = π/4; you can try animations too if you like!) Discuss the results, giving numbers and locations of nodal curves, etc.

H) Using your work from parts A through F again, solve the following BVP for the heat equation on an annular region as simply as possible:

\{ \begin{matrix}u_{t} = k∇^{2}u \\ u(a, θ, t) = 0 \\ u(b, θ, t) = 0 \\ u(r, θ, 0) = (r − a)(b − r) sin(2θ) \end{matrix}