Mathematics 373 – Applied Mathematics (PDE) Problem Set 4 March 27, 2001

I. For each of the following vector fields and surfaces, verify the Divergence Theorem by direct computation of the surface integral and the triple integral: A) F = (z, y, x), S the sphere of radius 1 centered at (0, 0, 0). B) F = (x + y, y + z, x + z), S the surface of the tetrahedron bounded by the coordi-

nate planes and the plane x + y + z = 1. (For the surface integral, you will need to parametrize each of the four faces of the tetrahedron separately, and add the corresponding surface integrals.) Answer: 1/2 (both values).

II. Recall the definitions of the hyperbolic cosine and sine functions:

cosh x = \frac{e^{x} + e^{−x}}{2} sinh x = \frac{e^{x} − e^{−x}}{2}

A) Show that for all x ∈ R, cosh^{2} x − sinh^{2} x = 1. Consider the parametric curve α(t) = (cosh t, sinh t) in the plane. How does this curve help explain the names hyperbolic cosine and sine?

B) Express the derivatives of the hyperbolic functions cosh x, sinh x, and the functions tanh x = \frac{sinh x}{cosh x}, coth x = \frac{cosh x}{sinh x}, sechx = \frac{1}{cosh x}, cschx = \frac{1}{sinh x} analogous to the other trig functions in terms of the same hyperbolic functions.

C) Show that any function g(x) = Ae^{x} + Be^{−x} can be written as a linear combination of cosh(x) and sinh(x).

II. Consider the heat equation

u_{t} = k(u_{xx} + u_{yy})

on the rectangle [0, a] × [0, b] in the (x, y)-plane. (Think of a thin metal plate in this shape and heat conduction within the plate.) An equilibrium temperature distribution is a solution u(x, y, t) of the heat equation that does not depend on t – in other words, a solution that has “settled down” and is not changing any longer. When u is independent of t, the heat equation reduces to

(1)  u_{xx} + u_{yy} = 0

which is called Laplace’s Equation. In this problem, we will apply separation of variables and Fourier series techniques to solve a boundary value problem for Laplace’s Equation on the rectangle.

A) Say u(x, y) = X(x)Y (y) solves (1). What 2 ODE’s are satisfied by X and Y? B) A typical problem here would be to solve (1) together with boundary conditions such

as

u_{y}(x, 0) = u_{y}(x, b) = 0,

all x (horizontal edges of the plate insulated), and

u(0, y) = 0 u(a, y) = g(y)

for some given function g (vertical edges held at given temperatures). What are the eigenfunctions and eigenvalues for the boundary value problem you get for Y (y)?

C) Show how to develop a Fourier series solution of Laplace’s Equation satisfying the boundary conditions from B and these initial conditions. (Note: you will probably want to use the hyperbolic functions from problem II here to deal with the factors X(x) in the product solutions.)

IV. Suppose we have a thin metal ring with constant c, ρ, K_{0} in the shape of D = {(r, θ) : a ≤ r ≤ b, 0 ≤ θ ≤ 2π} in the polar coordinate plane. The ring has a radially symmetric equilibrium temperature function u = u(r) that satisfies

\frac{∂u}{∂r}(a) = β, \frac{∂u}{∂r}(b) = 1.

Show that β = b/a and interpret this result physically. (Note: As for the case of rectangular coordinates, \frac{∂u}{∂r}(a) is proportional to the heat flux across the inner boundary of the ring. Similarly at r = b.)

Assignment

Individual writeups due in class, Tuesday, April 3.