Asymptotics
Sheet 8
1. For the first-order nonlinear differential equation
df
− f = εf 2 e−t
dt
with ε 1 and initial condition f (0) = 1 determine an approximation by using the
multiple-scale method. Show that the resulting expression is the exact solution.
2. For the lightly damped linear oscillator
ẍ + 2εẋ + ω 2 x = 0
(a) use multiple-scale analysis to the find the solution in second order, i.e. go one order
beyond what was done in the lecture.
(b) compare your result with the exact solution.
3. Consider the oscillator
y 00 (t) + ω 2 (t)y(t) = 0.
(1)
(a) Show that the change of variables
Z
t 7→ T = f (t) =
t
ω(x)dx
transforms equation (1) to
d2 y
ω 0 (τ ) dy
+
y
+
= 0,
dT 2
ω 2 (τ ) dT
(2)
where τ = t is the slow time.
(b) Solve equation (2) by multiple scale thus recovering the WKB approximation to (1).
(Hint: Unlike the other examples we considered, here dτ /dT = /ω(τ ) (check!). See
Example 11.3.3.)
4. Consider the boundary layer problem in the inner scale,
d2 y
dy
+ a + by = 0,
2
dt
dt
y(0) = A, y(1/) = B, a > 0.
Perform multiple scale expansion thus recovering the boundary-layer solution
y(t) = Beb/a e−bx/a + (A − Beb/a )e−ax/ + O().
(Hint: See Example 11.3.4)
(3)