辅导MA3PD2解析R设计、数据结构解析

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