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The University of British Columbia Sessional Examinations - April 2007
Mathematics 401 Green’s Functions and Variational Methods
Closed book examination Time: 2.5 hours
Special Instructions:
Do any 8 of 10 questions. If more than 8 questions are attempted, the best 8 marks will be taken. Each question is out of 10. No notes, calculators, or books.
Rules governing examinations
- Each candidate should be prepared to produce his or her library/AMS card upon request.
- Read and observe the following rules: No candidate shall be permitted to enter the examination room after the expiration of one half hour, or to leave during the first half hour of the examination. Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities in examination questions. CAUTION - Candidates guilty of any of the following or similar practices shall be immediately dismissed from the examination and shall be liable to disciplinary action. (a) Making use of any books, papers or memoranda, other than those authorized by the examiners. (b) Speaking or communicating with other candidates. (c) Purposely exposing written papers to the view of other candidates. The plea of accident or forgetfulness shall not be received.
- Smoking is not permitted during examinations.
April 2007 Mathematics 401 Name Page 2 of 4 pages
[10] 1. Consider the ODE BVP for u(x), x ∈ [0, 1] given below:
u′′^ + xu′^ − u = f (x), u(0) = 0, u(1) = 0.
(a) [5 marks] Find the adjoint of this problem.
(b) [5] Let G(s, x) be the Green’s function for the original problem above, i.e.
u(x) =
0
G(s, x)f (s)ds
Write the conditions that G(s, x) must satisfy. It is not necessary to find G explicitly.
[10] 2. Determine which of the following problems for u(x, t), x ∈ R, t ≥ 0 are well posed. Justify.
(a) [5] utt = uxxxx with u(x, 0) and ut(x, 0) given.
(b) [5] ut = −uxx − uxxxx with u(x, 0) given.
[10] 3. Consider u(x, y) that solves ∆u = f (x, y)
with f given with compact support Ω contained in the unit disk centred at the origin. Recall that the solution can be written
u(x, y) =
2 π
Ω
ln |(x, y) − (s, t)|dsdt
(a) [5] Consider x = (x, y) far from the origin, i.e. = 1/|x| is small. Reproduce the multipole expansion derived in class for the solution u, showing the terms of O(1) and O().
(b) [5] Consider the specific case of Ω the unit disk and f ≡ 1 in Ω. Evaluate the terms you found in part (a) above. Simplify.
[10] 4. Consider u(x, y) in the domain Ω = {(x, y) : 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1 } (a square).
(a) [8] Consider the functional F [u] =
Ω
u^2 dA
for u with zero values on the boundary of Ω, subject to the constraint ∫
Ω
(u^2 x + u^2 y)dA = 1.
Find the maximum of F and the maximizing function.
(b) [2] Discuss what occurs when a minimum of F above with its constraint is sought.
[10] 5. Consider the ODE eigenvalue problem below for u(x), x ∈ [0, 1]:
−u′′^ = λa(x)u, u′(0) = 0, u′(1) = 0
with a(x) > 0 given.
Continued on page 3
