EXERCISES 2.3

2.3.1. For the following partial differential equations, what ordinary differential
equations are implied by the method of separation of variables?

* (a) du/dt = (k/r)(d/dr)(r du/dr)
  (b) du/dt =  k (d/dx)^2 u - v0 du/dx
* (c) (d/dx)^2 u + (d/dy)^2 u = 0
   (d) du/dt = (r/r^2) (d/dr)(r^2 du/dr)
* (e) du/dt = k (d/dx)^4 u
* (f) (d/dt)^2 u = c^2 (d/dx)^2 u


2.3.2. Consider the differential equation X'' + lambda X = 0.

Determine the eigenvalues lambda and corresponding eigenfunctions if X(x) satisfies
the following boundary conditions. Analyze the three cases for lambda positive, zero and negative.
You may assume that the eigenvalues are real.

  (a) X(0)=0, X(Pi)=0
* (b) X(0)=0, X(1)=0
  (c) X'(0)=0, X'(L)=0
* (d) X(0)=0, X'(L)=0
  (e) X'(0)=0, X(L)=0
* (f) X(a)=0, X(b)=0 (assume lambda positive)
  (g) X(0)=0, X'(L)+X(L)=0

2.3.3. Consider the heat equation
du/dt = k (d/dx)^2 u
subject to the boundary conditions
u(0,t) = 0 and u(L,t) = 0.
Solve the initial value problem if the temperature is initially

  (a) u(x,0) = 6 sin(9 Pi x/L)
  (b) u(x,0) = 3 sin(Pi x/L) - sin(3 Pi x/L)
* (c) u(x,0) = 2 cos(3 Pi x/L); Leave difficult integrals unevaluated.
  (d) u(x,0) = 1 on 0 < x < L/2, u(x,0) = 2 on L/2 < x < L

2.3.4. Consider
du/dt = k (d/dx)^2 u
subject to u(0, t) = 0, u(L, t) = 0, and u(x, 0) = f (x).
*(a) What is the total heat energy in the rod as a function of time?
 (b) What is the flow of heat energy out of the rod at x = 0? at x = L?
*(c) What relationship should exist between parts (a) and (b)?

2.3.5. Evaluate (be careful if n = m) the integral over 0<x<L of the product
of sin(n Pi x/L) and sin(m Pi x/L) for n>0,m>0.
Use the trigonometric identity sin(a) sin(b) = (1/2)(cos(a-b)-cos(a+b)), being careful if a+b=0 or a-b=0.

*2.3.6. Evaluate the integral over 0<x<L of the product
of cos(n Pi x/L) and cos(m Pi x/L) for n>0,m>0.
Use the trigonometric identity cos(a)cos(b) = (1/2)(cos(a+b)+cos(a-b)), being careful if a+b=0 or a-b=0.

2.3.7. Consider the following boundary value problem (if necessary, see Sec. 2.4.1):
du/dt = k (d/dx)^2 u with du/dx=0 at x=0 and x=L, and u(x,0)=f(x).

(a) Give a one-sentence physical interpretation of this problem.
(b) Solve by the method of separation of variables. First show that there
are no separated solutions which exponentially grow in time. [Hint:
The answer is
u(x, t) = A0 + sum of terms A[n] exp(-lambda[n] k t) cos(n Pi x/L) for
n=1 to infinity.
What is lambda[n]?

(c) Show that the initial condition, u(x, 0) = f (x), is satisfied if
f(x) = A0 + sum of terms A[n] cos(n Pi x/L) for n=1 to infinity.

(d) Using Exercise 2.3.6, solve for A0 and A[n] (n > 1).

(e) What happens to the temperature distribution as t approaches infinity? Show that
it approaches the steady-state temperature distribution (see Sec. 1.4).

*2.3.8. Consider
du/dt = k (d/dx)^2 u - alpha u
This corresponds to a one-dimensional rod either with heat loss through the
lateral sides with outside temperature 0 Celsius (alpha > 0, see Exercise 1.2.4) or with
insulated lateral sides with a heat sink proportional to the temperature.
Suppose that the boundary conditions are
u(0,t) = 0 and u(L,t) = 0.

(a) What are the possible equilibrium temperature distributions if a > 0?
(b) Solve the time-dependent problem [u(x, 0) = f (x)] if a > 0. Analyze
the temperature for large time (t --+ oo) and compare to part (a).

*2.3.9. Redo Exercise 2.3.8 if alpha < 0. [Be especially careful if -alpha/k = (n Pi/L)^2.]

2.3.10. For two- and three-dimensional vectors, the fundamental property of dot
products, A.B = |A||B| cos(theta), implies that
|A.B| < |A| |B|. (2.3.44)
In this exercise we generalize this to n-dimensional vectors and functions,
in which case (2.3.44) is known as Schwarz's inequality. [The names of
Cauchy and Buniakovsky are also associated with (2.3.44).]
(a) Show that |A - gamma B|^2 > 0 implies (2.3.44), where gamma = A.B/B.B.
(b) Express the inequality using both [complicated summations]
*(c) Generalize (2.3.44) to functions. [Hint: Let A. mean the integral of
A(x)B(x) over x=0 to x=L]

2.3.11. Solve Laplace's equation inside a rectangle:
(d/dx)^2 u + (d/dy)^2 u = 0
subject to the boundary conditions
u(0,y)=g(y), u(x,0)=0,
u(L,y)=0, u(x,H)=0.
(Hint: If necessary, see Sec. 2.5.1.)