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### M257_316_Midterm2_sec101W2006

Course: MATH 257, Fall 2011
School: UBC
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Word Count: 153

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257/316, Math Midterm 2, Section 101 17 November 2006 Instructions. The duration of the exam is 55 minutes. Answer all questions. Calculators are not allowed. Maximum score 100. 1. Let f (x) = cos(2x) (a) Determine the sine series expansion for f (x) on the interval 0 &lt; x &lt; . (b) Is the Fourier series for f (x) on the interval &lt; x &lt; the same expansion as you found above in part (a)?...

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257/316, Math Midterm 2, Section 101 17 November 2006 Instructions. The duration of the exam is 55 minutes. Answer all questions. Calculators are not allowed. Maximum score 100. 1. Let f (x) = cos(2x) (a) Determine the sine series expansion for f (x) on the interval 0 < x < . (b) Is the Fourier series for f (x) on the interval < x < the same expansion as you found above in part (a)? Is there a way to answer this without working out the expansion? Hint: It may be useful know to that Z sin(nx) cos(2x)dx = 0 0 if n is even if n is odd 2n n2 4 [20 marks] 2. Solve the following inhomogeneous initial boundary value problem for the heat equation: ut = uxx + et sin x, 0 < x < , t > 0 u(0, t) = 0 and u(, t) = 0 u(x, 0) = 0 [40 marks] 3. Determine the solution to the following boundary value problem: 1 1 urr + ur + 2 u r r = 0, 0 < r < a, 0 < < u(r, 0) = 0 = u(r, 2 r0 ), u(r, ) < 2 u(a, ) = sin(4) [40 marks] 1
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