Please read the instructions below before you start the exam.
April/May 2020 MA3PD2 2019/0 A 800
UNIVERSITY OF READING
PARTIAL DIFFERENTIAL EQUATIONS II (MA3PD2)
Two hours
Answer ALL questions in section A and at least ONE question from section
B. (If more than one question from section B is attempted then marks from the
BETTER section B question will be used. If the exam mark calculated in this
way is less than 40%, then marks from the other section B question which has
been attempted will be added to the exam mark until 40% is reached).
vc2
Page 2
SECTION A
1. Let (r, ✓) denote polar coordinates, and suppose that u(r, ✓) satisfies the
Neumann Laplace problem⇢
urr + r�1ur + r�2u✓✓ = 0 in D = {0 < r < 1, 0 ✓ < 2⇡},
@u/@r = F (✓) on C = {r = 1, 0 ✓ < 2⇡},
in which the boundary forcing function F (✓) satisfies
R 2⇡
0 F (✓) d✓ = 0.
The solution u must be bounded in D and 2⇡-periodic in ✓.
(a) Use separation of variables to show that the most general solution
u(r, ✓) which is bounded and satisfies Laplace’s equation in D, and is
2⇡-periodic in ✓, can be written as