# Science:Math Exam Resources/Courses/MATH257/December 2011/Question 05 (a)

MATH257 December 2011
Other MATH257 Exams

### Question 05 (a)

Consider the following problem involving Laplace's equation in an annular region:

$u_{rr}+{\frac {1}{r}}u_{r}+{\frac {1}{r^{2}}}u_{\theta \theta }=0,\quad 1 $u(1,\theta )=0,\quad u(2,\theta )=0,\quad u(r,0)=0,\quad u(r,\pi /2)=f(r).$ Use the method of separation of variables to solve the problem when $f(r)=\sin \left({\frac {2\pi }{\ln(2)}}\ln(r)\right).$ Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

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