Lecturer: Johanna Knapp Second Semester, 2021
School of Mathematics and Statistics University of Melbourne
MAST20030: Differential Equations
Assignment 3
Due: 9am on Monday, October 11
• This assignment has 50 marks.
• Make sure that you show all your working.
• Marks will be awarded for clear explanations, correct notation and legible writing, not just for arriving at the
correct solution.
• All questions will be marked.
Online submission:
1. Write your answers by hand on blank paper. Write on one side of the paper only. Start each question on a
new page. Write the question number at the top of each page.
2. Scan your submission to a single PDF file with a mobile phone or a scanner. Scan from directly above to
avoid any excessive keystone effect. Check that all pages are clearly readable and cropped to the A4 borders
of the original page. Poorly scanned submissions may be impossible to mark.
3. Upload the PDF file via the Canvas Assignments menu and submit the PDF to the Gradescope tool by first
selecting your PDF file and then clicking on Upload PDF. Gradescope will then ask you to identify on which
of the uploaded pages your answers to each question are located.
Question 1. (10 marks)
(a) Determine the Fourier series of the function
푓 (푥) = cos푥 0 < 푥 < 휋
2
extended to an odd function on −휋2 < 푥 < 휋2 and then periodically to R. (3 marks)
(b) Plot the graphs of 푓 and its odd extension to −휋2 < 푥 < 휋2 . Furthermore produce three plots of the Fourier
series, containing the first 10, 20, and 50 terms, respectively. The plots should have domain −휋 ≤ 푥 ≤ 휋 and
range [−1.5, 1.5]. If you do not have a computer, handmade sketches are OK. (2 marks)
(c) What does this Fourier series converge to? Explain your answer by quoting an appropriate result from the slides.
(3 marks)
(d) Use your result to compute
∞∑
푘=0
(2푘 + 1) (−1)푘
16푘2 + 16푘 + 3
(2 marks)
continued on next page
MAST20030, Assignment 3 2
Question 2. (16 marks)
Consider the inhomogeneous one-dimensional heat equation
휕푢
휕푡
= 퐷
휕2푢
휕푥2
+ (1 − 푥2) cos 푡, 0 < 푥 < 1, 푡 > 0 (1)
with homogeneous boundary conditions
휕푥푢 (0, 푡) = 0, 푢 (1, 푡) = 0, 푡 > 0.
and the initial condition
푢 (푥, 0) = 0.
(a) Show that separation does not work for (1). (1 marks)
(b) Expand the 푡-independent factor of the inhomogeneous term in (1), i.e. (1 − 푥2), in terms of eigenfunctions of
the boundary value problem. (4 marks)
(c) Expand 푢 (푥, 푡) in terms of the eigenfunctions of the boundary value problem with 푡-dependent coefficients.
Using the result of (b), derive first-oder ODEs for these coefficients. (4 marks)
(d) Solve the first-oder ODEs of (c). (5 marks)
(e) Impose the initial condition to arrive at the (formal) solution for 푢 (푥, 푡). (2 marks)
Question 3. (24 marks)
Consider the one-dimensional wave equation
휕2푢
휕푡2
=
휕2푢
휕푥2
, 0 ≤ 푥 ≤ 1, 푡 ≥ 0
with boundary conditions
푢 (0, 푡) = 0, 푢 (1, 푡) = 0, 푡 > 0.
and initial conditions
푢 (푥, 0) = 푥4 − 2푥3 + 푥, 휕푡푢 (푥, 0) = 12휋 sin 3휋푥 cos 3휋푥
(a) Solve this IBVP. (11 marks)
(b) Show that the solution is genuine for 푥 ∈ [0, 1] and 푡 ≥ 0. (6 marks)
(c) Prove that the solution to the IBVP is unique by employing the “energy method” for an appropriate function 푣 ,
where the energy functional is defined by
퐸 [푣] (푡) = 1
2
∫ 1
0
[ (
휕푡푣 (푥, 푡)
)2 + (휕푥푣 (푥, 푡))2] d푥 .
You will need to demonstrate that this functional is 0 for all 푡 > 0 and then show why this implies that any
solution to is unique. (7 marks)
End of assignment.

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