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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 
 
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