The University of Sydney
School of Mathematics and Statistics
Assignment
MATH3078/MATH3978/MATH4078: PDEs and WavesSemester 2, 2024
Web Page:https://canvas.sydney.edu.au/courses/60714
Lecturer: Robert Marangell
•This assignment is due by23:59 Sunday 29th September 2024. Late assignments
will not be accepted.
•Please submit yourtypesetAssignment as a single PDF document using TurnItIn
in Canvas. Handwritten assignments will not be accepted.
•Please show all working and present your arguments clearly.
•To ensure compliance with our anonymous marking obligations, please do not include
your name in any area of your assignment; only your SID should be present.
Copyright©2024 The University of Sydney1
1.(a) Solve Laplace’s equation on the interior of the half disk
H:=
{
(x,y)|x
2
+y
2
≤1,y≥0
}
with boundary conditions
u(x,0) = 0 for−1≤x≤1u(x,y) = Im
(
z
4
)
= 4x
3
y−4xy
3
forx
2
+y
2
= 1y >0.
(b) Plot your solution using your favourite software package.
(c) What isu(0,
1
2
)?
(d) What are the maximum and minimum values that the solution takes on the half
disk, and at what points do they occur?
2.Consider the following functionf(x) on the interval [0,7] (plotted below).
f(x) =
3x−
15
2
5
2
≤x≤3
6−
3
2
x3≤x≤
9
2
3
2
x−
15
2
9
2
≤x≤5
0,otherwise
(a) Compute the Fourier sine series of the periodic extension off(x) with period 14.
Compute the exact expression for the coeffieicients
(b) Write the solution to the boundary value problem (wave equation)
u
tt
=u
xx
u(0,t) =u(7,t) = 0
on the interval [0,7] with initial profilesu(x,0) =f(x) andu
t
(x,0) = 0.
(c) Use your favourite software package to make a plot of the solution from (b) at times
t= 3,4,andt= 5. Be sure to say which program you used, and what the code
you used to produce the plots.
3.Consider the PDE initial boundary value problem on the interval [0,π].
u
t
=u
xx
+ 2u
x
+u u(0,t) =u(π,t) = 0 withu(x,0) =f(x).(1)
(a) Solve the equation using separation of variables.
(b) Now solve the equation by making an ansatze
ax
w(x,t) =u(x,t), whereais a
real number to be determined. Findaso that ifu(x,t) solve eq. (1), thenw(x,t)
will solve the heat equation, with appropriately modified initial and boundary
conditions. Solve this heat equation and then undo your transformation to produce
the solutionuto eq. (1).
(c) Now either use separation of variables, or generalise the method from part (b)
by making the substitutionu=e
ax+bt
wfor unknown constantsaandbto solve
2
the following general, constant coefficient parabolic equation with homogeneous
Dirichlet boundary conditions:
u
t
=u
xx
+ 2Au
x
+Bu u(0,t) =u(π,t) = 0 withu(x,0) =f(x).(2)
Briefly discuss the the long term behaviour of the solution. Justify your answer.
4.Please list your sources for what you used to solve these problems. You can use what
you like to help you with figuring out a problem, but what you write up should be your
own understanding, and youneed to tell me what you used to get to this understanding.
This includes each other. So for example if you found an example of a certain type on a
given webpage, and used Mathematica to help you compute something, and you worked
together with another student to complete the assignment, you need to document all of
these facts on the sources page. You don’t need to adhere to a particular style of sourcing
To maintain anonymity, please only include the SID of the students you worked on. Your
assignment will be compared with the other students’ as well as what is on the web via
the turnitin program. Should you have too great of a match without properly citing your
sources, you may be in violation of the university’s academic honesty policy.
Solution:I used Mathematica for my graphs and some of the questions, as well as the
textbook by Olver, and the book Intro to PDEs by Strauss and some Chat GPT to draw
inspiration for some of the questions.
3
