Dimensional analysis
ci
1. Suppose we have a physical law a = C mr
where C is a constant, c is the speed of light, i
is some electrical current, m is a mass and r is a distance. The fundamental dimensions
are electric charge Q, mass M , length L and time T . What dimension must C have?
You may solve the problem either under the assumption that a is an acceleration or for
a general variable a.
2. Suppose some non-trivial physical law (dimensionally consistent of course) relates two
variables `1 and `2 in the following way f (`1 , `2 ) = 0, where both `1 and `2 represent
distances. Show that the physical law is equivalent to ``21 = k, for some constant k.
3. Suppose we have two basic dimension L1 and L2 and 3 variables q1 , q2 , q3 . The dimensional matrix is
1
1 1
2
A=
.
1
−1 2
2
Show that a dimensionally consistent physical law f (q1 , q2 , q3 ) = 0 can equivalently be
written g(π) = 0 for some dimensionless quantity π. Find a possible expression for π.
4. Suppose we are studying water waves that move under the influence of gravity. We
suppose that the variables of interest are the acceleration in free fall g, the velocity of
the wave v, the height of the wave h and the wave length `. We also suppose that they are
related by a dimensionally consistent equation f (g, v, h, `) = 0. Determine the minimum
number of dimensionless π-variables needed to describe this problem according to the
Buckingham Pi-theorem and give possible expressions for the πi . You will need to find
the dimensions yourself.
Introduction to PDE’s
5. Which of the following PDE’s are linear? Remember that motivation is required.
1. u0t u = 2
0
2. u000
xxx + sin(u)ux + u = 0
3. 2u + u00xx + u00tt = 0
6. Show that f (x, y) = ex (cos(y)x − sin(y)y) is a harmonic function.
7. Solve the problem
u0t − u00xx = 0, 0 < x < π, t > 0
u(x, 0) = 2 sin(x) cos(x), 0 < x < π,
u(0, t) = u(π, t) = 0, t > 0.
8. Consider the problem
u0t − u00xx = 0, 0 < x < π, t > 0
u(x, 0) = f (x), 0 < x < π,
u(0, t) = u(π, t) = 0, t > 0.
Suppose that f (x) can be written as a finite sine sum f (x) =
solution formula to prove that
lim u(x, t) = 0,
PN
n=1 bn
sin(nx). Use the
t→∞
for every x ∈ (0, π). Explain why you would expect this result on physical grounds.
Sturm-Liouville etc
9. Show that f1 (x) = 1 and f2 (x) = 1 − x are orthogonal on the interval [0, ∞) with the
weight function r(x) = e−x . Hint: One can use integration by parts.
10. Solve the problem
(e−2x y 0 )0 + e−2x y = 0, 0 < x < 1,
y(0) = 1,
y 0 (0) = 0.
11. Solve the problem
u00tt − u00xx = 0, 0 < x < 1, t > 0
u(x, 0) = f (x), 0 < x < 1,
u0t (x, 0) = 0, 0 < x < 1,
u(0, t) = u(1, t) = 0, t > 0,
where f (x) = x if 0 < x <
1
2
and f (x) = 1 − x if
1
2
< x < 1.
12. Solve the problem
u00tt − 2u0t = u00xx , 0 < x < π, t > 0
u(x, 0) = sin(x), 0 < x < π,
u0t (x, 0) = 0, 0 < x < π,
u(0, t) = u(1, t) = 0, t > 0,
by separation of variables.
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Transform theory
13. The Laguerre polynomials pn are a sequence of polynomials that are orthogonal on the
interval [0, ∞) with the weight function r(x) = e−x . The first two polynomials in the
sequence are p0 (x) = 1 and p1 (x) = 1 − x. Find the first two generalized Fourier
coefficients of q(x) = x2 with respect to the Laguerre polynomials.
14. Find the Laplace transform of f (t) = et θ(t − 1) + 1.
15. Find the Fourier transform of f (t) = θ(t)e−t sin(t).
16. Using the Laplace transform, find a function y such that y 00 − 3y 0 + y = 0, y(0) = 0 and
y 0 (0) = 1.
Dynamical systems and chaos
17. Set x0 = 2, f (x) = x2 − 1 and xn+1 = f (xn ) for n ≥ 1. Compute x2 .
18. Set f (x)
= x2 − 1 and consider the dynamical system defined by xn+1 = f (xn ) Show
√
that 1−2 5 is a fixed point of the system. Is it a stable fixed point?
19. Set x0 = 1 and xn+1 = sin(xn ) for n ≥ 1. Show that limn→∞ xn = 0. Hint: It may be
helpful to show that {xn } is a decreasing sequence.
20. Determine the Lyapunov exponent of the system in the previous question. You may use
the limit limn→∞ xn = 0 even if you have not proved it.
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