Algebraic problems
1. Find the rescalings for the roots of
2 x3 − (1 − + 32 )x2 + (3 − 3 + 22 − 3 )x − 2 + 3 − 3 = 0,
and thence find two (non-trivial) terms in the approximation for each root, using (a)
iteration and (b) expansion.
2. Develop two terms of the perturbation solutions to
δx3 − (1 + δ + 2δ 2 − δ 3 )x2 + (2 − δ 2 − δ 3 )x + 2δ − 3δ 2 − δ 4 = 0,
for δ 1 and δ 1.
3. Develop perturbation solutions to
x3 − (6 + + 2 )x2 + (12 + 3 + 32 + 23 )x − 8 − 2 − 32 − 23 − 4 = 0
finding the three terms in the approximation for each root, x = x0 + α xα + 2α x2α , and
determining α along the way.
4. Develop three terms of the perturbation solutions to the real roots of
2
ex−x = x2 .
Eigenproblems and regularly perturbed differential equations
1. Find the corrections to the leading-order eigenvalues of the matrix problem Ax =
λx + Bx, for
6 −3
4 2
(b)
(a)
3 0
2 4
and
B=
1
1
1
2
2. Find the corrections to the leading-order eigenvalues of the differential equation,
y 00 + λy + xy = 0,
with y(−1) = y(1) = 0.
3. Normal modes of a slightly mis-shapen membrane
Consider the two-dimensional wave equation,
φxx + φyy = φtt ,
where subscripts denote partial derivatives, on the region,
0 ≤ x ≤ 1,
x(1 − x) ≤ y ≤ 1,
with φ = 0 on the boundary. Consider normal-mode √
solutions for which φ(x, y, t) =
Φ(x, y) cos(ωt). Show that Φ = sin πx sin 3πy and ω = π 10 is a leading-order eigensolution. Calculate the O() corrections to this solution.
Integrals
1. Use the method of repeated integration by parts or rescaling to obtain four terms in
the asymptotic approximation to the integral,
Z
for x → 0. Note that
∞
t−3 e−4t dt,
x
γ=−
Z
∞
e−t log t dt
0
is Euler’s constant.
2. Find the leading-order behaviour for x 1 of
(a)
Z
∞
e
0
xt(4−t3 )
dt
(1 + 2t2 )
(b)
Z
∞
e
νt−x cosh2 t
dt
−∞
(c)
Z
π
√
4
sinh t e−x sin t dt
0
3. Evaluate the first two terms as → 0 of
√
Z ∞
xdx
3/2
( + x) (1 + x)
0
4. Evaluate the first two terms as m approaches unity from below of the elliptic integral,
Z
π/2
0
(1 − m2 sin2 θ)1/2 dθ
Matched asymptotic expansions
1. Consider
y 00 + (1 + 22 )y 0 + y = 0,
in 0 ≤ x ≤ 1,
with y(0) = 0 and y(1) = e−1 . Find three terms of the outer solution, applying only the
boundary condition at x = 1. Next find three terms in an inner approximation for the
boundary layer near x = 0 applying the boundary condition at x = 0. Determine the
constants of integration by matching (a) over an intermediate region, and (b) using van
Dyke’s rule with P = Q = 2. Compute the composite approximation, C2,2 y.
2. The function y(x) satisfies
y 00 + x3/4 y 0 + y = 0,
in 0 ≤ x ≤ 1,
and y(0) = 0 and y(1) = 1. First find the rescaling for the boundary layer near x = 0,
and obtain the leading order inner approximation. Then find the leading order outer
approximation and match the two approximations.
3. Calculate two terms of the outer solution of
(1 + )x2 y 0 = [(1 − )xy 2 − (1 + )x + y 3 + 2y 2 ]
in 0 ≤ x ≤ 1,
with y(1) = 1. Locate the non-uniformity of the asymptoticness and hence the rescaling
for an inner region. Thence find two terms for this inner solution.
Reading exercise
Read chapter 5.2 of Hinch. Then solve the following problem.
4. The function f (r) satisfies
3
frr + fr + 4f fr = 0,
r
in r > 1,
and is subject to the boundary conditions, f (1) = 0 and f → 1 as r → ∞. Obtain
an asymptotic expansion for f at fixed r and → 0 in the asymptotic sequence, 1, ,
2 log(1/), 2 . Then find an expansion for f at fixed ρ = r as → 0 in the sequence 1, 2 .
Match these expansions. You might want to use the solution of question 1 of the Integrals
problems.
Multiple Scales
1. Obtain an asymptotic approximation for x to order one, which is valid for t = O( −1 ),
when
ẍ + [(ẋ2 − 1)ẋ + x3 ] + x = 0
with x(0) = 1 and ẋ(0) = 0.
2. Find the leading-order approximation for times of order −1 to
ẍ + y ẋ + x = y 2
ẏ = (x − 2y 2 )
with ẋ(0) = y(0) = 1 and x(0) = 0.
3. Obtain an asymptotic approximation for x to order one, which is valid for t = O( −1 ),
when
ẍ + x = ex , x(0) = 1, ẋ(0) = 0.
4. Mathieu problem. The equation:
ÿ + (a + 2 cos t)y = 0
with
1 2
n + a1 + 2 a2
4
We are interested in discovering when solutions grow exponentially; the initial conditions
are not particularly relevant. Let’s solve the problem with multiple scales:
a=
y = y0 (τ, T, T2 ) + y1 (τ, T, T2 ) + 2 y2 (τ, T, T2 ) + ...
τ = t, T = t, T2 = 2 t.
Verify that the leading-order solution can be written as
y0 = A(T, T2 ) cos
nτ
nτ
+ B(T, T2 ) sin
2
2
For n = 1, show that some of the inhomogeneous terms of the y1 equation are resonant.
Eliminate the baddies, write a single equation for A, and show that the solution grows
exponentially (signifying parametric instability) over a certain range of a 1 ; translate this
back to a sector of the (a, )−plane.
For n = 2, convince yourself that we should drop the first slow timescale T , and write
y0 = A(T2 ) cos τ + B(T2 ) sin τ
Now press on to O(2 ) and eliminate the bad guys again, to write the equation for A. Once
more, determine the sector of the (a, )−plane where there is exponential growth.
Finally, repeat the exercise for n = 0.