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ACM41000: Uncertainty Quantification

创建日期：2024-04-26 14:12:41

所属分类：会计Accounting

标签： ACM41000

Uncertainty Quantification

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ACM41000: Uncertainty Quantification

Assignment 1
This is a graded assignment. The due date is Wednesday 16th February at 5pm.
Upload your solutions to Brightspace.
1. (a) i. Solve the first-order differential equation
dy
dx −
3y
x+ 1 − (x+ 1)
4 = 0. (1)
ii. Solve the inhomogeneous second-order differential equation
d2y
dt2 − 3
dy
dt + 2y = (1 + t) exp(3t). (2)
(b) Consider the system
du
dt =
1
3(u− v)(1 − u− v), (3a)
dv
dt = u(2 − v) (3b)
i. Determine the four fixed points.
ii. Calculate the eigenvalues of the Jacobian at these points.
iii. Hence, characterise each fixed point.
Page 1 of 5
2. In the high-end art world, forgeries still exist. Over the years, many have
been officially certified by historians and traded by reputable dealers. This
is often in spite of clear evidence that the painting is a forgery. One such
painting is ‘The Supper at Emmaus’ (Figure 1), supposedly by celebrated
17th Century Dutch artist Vermeer, but actually by renowned art forger van
Meegeren in 1936, inspired by Caravaggio’s piece of the same name. This
painting was sold to the Rembrandt Society for $300,000 (or $5,600,000
in today’s money) and donated to the Museum Boijmans Van Beuningen
in Rotterdam. Later, in 1947 van Meegeren was tried and found guilty
of fraud and forgery. The evidence used was based on water and ethyl
alcohol resistivity, evidence on the production of “age cracks” in the paint,
the presence of the colour cobalt blue (unknown in the 17th Century), and
an artificial resin invented in the 19th Century. However, some notable
art experts continued to claim several paintings including ‘The Supper at
Emmaus’ to be genuine Vermeers. In 1967 scientists at Carnegie Mellon
University produced thorough and conclusive proof that ‘The Supper at
Emmaus’ was a forgery. We will look at the basis of their method.
Figure 1: ‘The Supper at Emmaus’ by Henricus Antonius van Meegeren.
White lead (Pb210) is a radioactive substance (half life 22 years) and a
pigment of major importance in paintings. It is manufactured from ores
which contain uranium and elements to which uranium decays. One of
these is Radium 226 (Ra226) (half life 1600 years). While still part of the
ore, the amount of Ra226 decaying to Pb210 is equal to the amount of Pb210
disintegrating per unit time, i.e., Pb210 and Ra226 are in a ‘radioactive
equilibrium’.
Page 2 of 5
In the manufacture of the pigment though, the radium and most of its
descendants are removed. The Pb210 begins to decay and continues to do
so until it reaches equilibrium with the smaller amount of Ra that survived
the chemical process.
Let’s define the following variables
• y(t) – amount of Pb210 per gram of ordinary lead at time t
• y(t0) = y0 – the initial amount of Pb210 at manufacture time, t0
• r(t) – number of disintegrations of Ra226 per minute per gram of or-
dinary lead
We use the following differential equation model for radioactive decay
dy
dt = −λy + r(t), (4a)
y(t0) = y0, (4b)
where λ = 3.151 × 10−2 is the decay constant for Pb210. We may take r(t)
to be a constant since the painting is at most 400 years old and the half-life
of Ra226 is 1600 years.
For the painting to be real, the original quantity of Pb210 must be in equilib-
rium with the larger quantity of radium in the ore from which the pigment
was manufactured. This translates to requiring
λy0 to be in the range of 0 − 200. (5)
Consider the current time to be the time of the investigation, whereby
values from the ‘The Supper at Emmaus’ painting are
λy = 8.5, (6a)
r = 0.8. (6b)
We will reverse this calculation to see if these values correspond to a λy0
for a supposedly 300 years old painting.
(a) Solve the differential equation (4) using the integrating factor tech-
nique.
(b) Assume the painting is not a forgery. Take t − t0 = 300 years along
with current data values (6) to estimate λy0 from your ODE solution.
(c) Is your number for λy0 so far out of range as to conclude that the ‘The
Supper at Emmaus’ is a forgery?
Page 3 of 5
3. Manufacturers, retailers, and advertising agencies understandably want to
predict the decision process of an individual consumer. Consider the fol-
lowing model formulated for predicting consumer behaviour towards a par-
ticular product, say brand X. The variables for the model are
• B(t) – buying level for brand X
• M(t) – motivation/attitude towards brand X
• C(t) – level of communication (e.g., advertising) of brand X
where t is time, and these variables are related by
dB
dt = b(M − βB), (7a)
dM
dt = a(B − αM) + γC. (7b)
where a, b, α, β, and γ are constants which are typically (but not always)
positive.
(a) Rearrange (7) by isolating M in terms of B and substituting to form
a single second-order differential equation for B (explicitly containing
C).
(b) Assume that a constant advertising campaign takes place such that
C(t) = C¯, a constant. Solve the second-order differential equation
obtained in (a).
(c) Interpret the long-term dynamics of your solution in (b) for
i. αβ > 1
ii. αβ < 1
Which makes more physical sense?
(d) Suppose that the advertising campaign is not constant, but slows down
over time. Solve your differential equation obtained in (a) with
C(t) = C¯ exp(−t). (8)
(e) Solve the system of differential equations (7) in R with initial condi-
tions
B(0) = 0, M(0) = 0, (9)
for each of the two sets of parameters:
a = 1, b = 1, α = 2, β = 1, γ = 1, C = 1, (10)
Page 4 of 5
and
a = 1, b = 1, α = 2, β = 1, γ = 1, C(t) = exp(−t), (11)
and plot B(t) in each case until t = 20.

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