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SCHOOL OF MATHEMATICS AND STATISTICS
Assignment 2
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MATH3078/3978: PDES AND WAVES Semester 2, 2018: 2024 – Write My Essay For Me | Essay Writing Service For Your Papers Online
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Due 5pm, Friday, 19 October 2017.
Upload a PDF of your assignment with the Turnitin feature on Canvas.
• You must use LaTeX to type your solutions.
• You must submit both a PDF of your solutions, and the LaTeX file.
• The assignment contains 6 primary question with multiple subparts each.
• MATH 3978 students must complete all 6 question for full marks.
• MATH 3078 students must complete any 5 question for full marks.
1D Linear Waves
We learned early in the course that
∂
2
t u = ∂
2
xu =⇒ u(t, x) = f(x − t) + g(x + t) (1)
Define the initial conditions
u(t = 0, x) = u0(x), ∂tu(t = 0, x) = v0(x). (2)
The initial conditions determine the functions f(x) and g(x).
Question 1(a): Write formulae for f(x) and g(x) in terms of u0(x) and v0(x). Hint: You answer
will involve integrals.
Question 1(b): Solve the wave equation (1) on the real line for the the following initial condition,
u(t = 0, x) ≡ u0(x) =
0 if x ≤ −1
1 + x if − 1 ≤ x ≤ 0
1 − x if 0 ≤ x ≤ 1
0 if 1 ≤ x
(3)
and initial velocity ∂tu(t = 0, x) = 0. For each finite time t < ∞, assume the solution vanishes as
|x| → ∞.
Question 1(c): Find a general solution to the dispersive wave equation, for x ∈ R.
∂tψ = ∂
3
xψ, ψ(t = 0, x) = ψ0(x) (4)
Hint: The solution will be in terms of integrals. This is very similar to solving the heat equation on
the real line.
Copyright
c 2018: 2024 - Write My Essay For Me | Essay Writing Service For Your Papers Online The University of Sydney
1
Circular Drum
The two-dimensional wave equation
1
c
2
∂
2
t u =
1
r
∂r (r∂ru) + 1
r
2
∂
2
θu (5)
governs small displacements, u = u(r, θ, t) of a circular drum. The coordinate r represents the radial
direction, and θ represents the angular direction. We want clamped boundary conditions on the outer
radius,
u(r = a, θ, t) = 0. (6)
Question 2(a): Solve for the time-periodic characteristic modes of equation (8) using separation of
variables. That is, find
u(r, θ, t) = R(r)Θ(θ)e
iωt
, (7)
where ω represents a characteristic frequency.
Question 2(b): Apply boundary conditions at the outer rim to determine a formula for the frequencies
ω.
Question 2(c): What are the frequencies (in Hz) of the 4 slowest modes for a drum with a 60 cm
diameter and wave speed c =200m/s.
Question 2(d): Solve the following forced-drum problem
1
c
2
∂
2
t u −
1
r
∂r (r∂ru) + 1
r
2
∂
2
θu
= A0 sin(Ω t) sin(ky) where y ≡ r sin θ (8)
Assume the boundary conditions u(r = a, θ, t) = 0, and the initial conditions u(r, t = 0) = 0. Assume
A0, Ω, and k are amplitude, frequency, and scale parameters respectively.
Question 2(e): For the drum parameters in part 2(c), what happens when Ω ≈ 1600 Hz?
2
Compact orthogonal polynomials
Consider the generating function
1
1 − 2tx + t
2
=
X∞
n=0
yn(x)t
n
. (9)
Question 3(a): Using equation (9), find a second-order differential equation of the form,
p(x)y
00
n
(x) + q(x)y
0
n
(x) + λn yn(x) = 0. (10)
Where p(x) and q(x) are polynomials in x (at most 2nd degree), and λn is an eigenvalue that depends
on n. What are p(x), q(x), and λn?
Question 3(b): Prove that
Z 1
−1
ym(x)yn(x)w(x) dx = 0 if m 6= n. (11)
for some weight function w(x). Define the weight such that
Z 1
−1
w(x) dx = 1. (12)
What is w(x)?
Question 3(c): Find
Γn =
Z 1
−1
yn(x)
2w(x) dx (13)
Hint: You need the generating function definition to solve this. You should be able to do the integrals
with the right change of variable.
Question 3(d): Use the generating function to find a three-term recurrence relation of the form,
anyn+1(x) + bnyn(x) + cnyn−1(x) = x yn(x) (14)
Find an, bn, and cn.
Question 3(e): Prove that
yn(x) = Xn
`=0
P`(x)Pn−`(x) (15)
where P`(x) are the Legendre polynomials. Hint: this question is easy if you use a specific property
of generating functions.
3
Burgers Shock
The goal in this question is to solve the nonlinear Burgers equation
∂tu + u∂xu = 0 (16)
for the same initial condition as in Question 1, i.e., equation (3).
x
u
Figure 1: Hint: The (symmetric) dotted triangle shows the shape of the initial condition; the (asymmetric)
dashed triangle shows the shape of the solution for some time after t = 0 and before a shock
forms; the solid (right) triangle shows the shape of the solution for some time after a shock forms.
Question 4(a): Find the characteristic curves, x = x(t, a), of the solution and invert for the initial
position of a curve a = a(x, t).
Question 4(b): Find the solution u(t, x) before a shock forms.
Question 4(c): Find the time, tc, and location, xc, where the inversion (of characteristic curves)
becomes impossible, and a shock forms.
Question 4(d): Assuming ξ(t) represents the shock location, use the following global condition
d
dt Z ∞
−∞
u(t, x) dx = 0, (both before and after the shock) (17)
to find ξ(t), and hence the solution u(t, x) after the shock forms.
Question 4(e): Compute the energy
E(t) ≡
1
2
Z ∞
−∞
u(t, x)
2 dx (18)
both before and after the shock forms. Make a plot of E(t).
Question 4(f): Now find the solution to Burgers equation with the following initial condition:
u0(x) = (
1 if − L < x < 0
0 otherwise.
(19)
Make plots of the solution during each qualitatively distinct phase of the dynamics.
Question 4(g): Make a plot of the energy in the solution for part 4(f).
4
A Burning Candle
The Fisher–Kolmogorov equation is a common model in chemistry and biology.
∂tT − κ ∂2
xT = r T
1 −
T
Θ0
. (20)
We can use this as a model for a burning candle. The left-hand side represents the linear diffusion of
heat within the candle. The right-hand side represents a reaction term that burns the wax at a given
rate as a nonlinear function of the temperature, T(x, t).
Question 5(a): Assume x is position along the candle, t is time, and T is temperature. What are
the units of the parameters in equation (20)? In addition, what are the units of p
κ/r and √
κr?
Question 5(b): We want to look for travelling wave solutions of the form
T = Θ0 F
x − λc0 t
`0
, (21)
where F(z) is a non-dimensional function of a non-dimensional variable; c0 is some characteristic
speed, `0 is some characteristic length, and λ is unknown. Transform equation (20) into an ODE of
the form
F
00(z) + λ F0
(z) + F(z) (1 − F(z)) = 0. (22)
Question 5(c): Look for solutions of the form
F(z) = 1
(1 + e
α z)
2
(23)
Find λ and α such that equation (23) solves equation (22).
Question 5(d): Make a plot of F(z).
5
The number of people
The population of people as a function of time and age obeys the following rules,
(I): At time t the number of people in a given age bin is:
Number of people who’s age is ∈ (a, a + ∆a) = n(t, a) = f(t, a) ∆a, (24)
where f(t, a) is the number density of people in a give age catagory.
(II): Each year, everyone gets a year older. In fact, every day everyone gets a day older, and so on.
(III): Each year, the probability of dying at a given age is
probability of dying ∈ (a, a + ∆a) = rD(t, a)∆a, (25)
where rD(t, a) is the death rate.
(IV): Each year, the probability of having a baby is
probability of a baby ∈ (a, a + ∆a) = rB(t, a)∆a, (26)
where rB(t, a) is the birth rate.
Question 6(a): Given the above rules, make an equation accounting of the number of people at
time t + ∆t and age a, given the population at time t. For this step, assume a > 0.
Question 6(b): Take the limit as ∆t → 0 and ∆a → 0. Derive a partial differential equation
governing the population dynamics as a function of time and age. For this part, assume a > 0.
Question 6(c): Given rule (IV), make an accounting of the number of people at age a = 0.
Question 6(d): Take the limit as ∆a → 0 and derive an integral condition for the number of people
starting out at age a = 0.
Question 6(e): Now make an equation that considers males and females separately. Assume that
a is the age range of females, and b is the age range of males. You should produce an equation for the
joint distribution of both populations f(t, a, b). You should assume different birth and death rates for
males and females, and all the rates should depend on t, a, b.
Question 6(f): Write a page paper – Describe plausible functions rB(t, a, b) and rD(t, a, b) from your experience with
people. For example, make simple model functions based on standard mathematical functions (e.g.,
exp(x), x
n
, log(x), tanh(x), etc.). If these functions depend on parameters say what the parameters
represent and what would be realistic values. Should your functions rB(t, a, b) and rD(t, a, b) depend
on f(t, a, b) in some particular way? Why or why not? Should your birth and death rate functions
depend on external factors that are exterior to normal population dynamics? For example, what might
be the response to a large war? Or what about the response to the invention of a new birth-control
technology? Or even just to the discovery of a new food source? How should the birth and death
rates for males and females depend on t, a, b?
6
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