Math 550 Fall 2003
Methods of Asymptotic Analysis
Review Exercises:
You should be able to do these problems without much trouble, perhaps with the assistance
of the course notes, one of the texts on reserve (Haberman or Zauderer), and/or your favorite
texts on ODE’s and PDE’s. If you find any of these questions particularly difficult, you should
spend time reviewing the appropriate material IN THE FIRST WEEK!
1. Find the solution to the following:
ut − a(t)u = b(t)
utt + c1 ut + c2 u = 0
(rur )r = 0
(r 2 ur )r = 0
(rur )r + k 2 ru = 0
What extra information is needed to completely specify the solution?
2. Find the solution of
xux + tut − cu = 0
for c = constant
Where is (are) the singular point(s)?
3. Find the solution to the following problems:
a) Heat (Diffusion) equation on an infinite domain (in 1D, 2D, or 3D):
ut = D∇2 u
u(x, 0) = h(x)
−∞ < x < ∞
b) Wave Equation on an infinite domain, using d’Alembert’s solution of
utt = c2 uxx
Initial displacement:
u(x, 0) = f (x)
Initial velocity:
ut (x, 0) = g(x)
Contrast the behavior of the two solutions from parts a) and b) as t → ∞.
4. a) Use the Fourier transform to solve 3 a,b) again.
b) How does the solution change if the problem is defined on a semi-infinite domain,
0 ≤ x < ∞ with u(0, t) = 0?
c) How does the solution change for ux (0, t) = 0?
5.a)What method would you use to solve 3 a) and b) on a finite domain, e.g. with boundary
conditions at x = 0 and x = 1?
b) What is the form of the solution if the boundary conditions are u(0, t) = 0 and u(1, t) = 0?
c) What if the conditions are ux (0, t) = 0 and u(1, t) = 0?