Maths 361 Assignment 1
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
Department of Mathematics
Maths 361
Assignment 1, due 11:59pm on 21 March 2023
Include all your working.
Full marks: 61 marks.
Getting help
● No cheating You are allowed to discuss with others how to go about a question, but you must produce your own computations and write your answers by yourself.
● Course material The course book, your notes from lectures, and the tutorials all provide helpful information for answering the assignment questions.
● Office Hours You can get help from Graham Donovan in office 303.221 during office hours. I’m also available at other times – stop by, or email to make an appointment.
[SHOW YOUR WORK]
1) (6 marks) Classify each of the following PDEs by linearity, order and homogeneity.
(a) ut北 + u北北北 = x2
(b) ut + uu北 = 0
(c) utt + tanh(x)u北 = e-北2
2) (15 marks) Consider the function f (x) = x2 e x for x ∈ [0, 2].
(a) Construct the even extension of f and find its Fourier series.
(b) Construct the odd extension of f and find its Fourier series.
(c) Plot the partial series for N = 1, 2, 3, 4 and 10 for each of the extensions above and compare with the extensions of f . You may use the matlab live notebook from the tutorial as a starting point, or any other software of your choice.
3) (25 marks) Consider the following problem:
ut = u北北 , 0 < x < 5, t > 0
u(0, t) + u北 (0, t) = 2, u北 (5, t) = 0, t > 0
u(x,0) = 0, 0 < x < 5.
(a) Find the steady state solution us (x) of this problem.
(b) Write a new PDE, boundary conditions and initial conditions for U (x, t) = u(x, t) e us (x)
(c) Use separation of variables to find a solution to the PDE, boundary conditions and initial conditions. You must justify each step of your solution carefully to get full marks. (Hint: if you are unable to write the eigenvalues explicitly, leave them in implicit form.)
(d) Estimate the first three eigenvalues; explain how you have done this.
(e) Suppose you had tried to apply separation of variables directly to the original problem
without removing the steady state solution. At what point would this approach fail? Explain.
4) (15 marks) Consider the following eigenvalue problem:
y\\ + λy = 0, 0 < x < 5
y\ (0) = 0, y(5) = 0
(a) Is this a Sturm-Liouville problem? Explain.
(b) What conclusions can you draw from application of the theorem(s) to this problem?
(c) Find all eigenvalues and eigenfunctions.
2023-03-16