M257_316_Midterm2_sec103W2009

# M257_316_Midterm2_sec103W2009 - u t = α 2 u xx< x< π...

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Math 257/316, Midterm 2, Section 103 November 20, 2009 Instructions. The duration of the exam is 55 minutes. Answer all questions. Calculators are not allowed. Maximum score 100. 1. Solve the following inhomogeneous initial boundary value problem: u t = u xx - u, 0 < x < 1 , t > 0 u x (0 ,t ) = 0 , u (1 ,t ) = cosh(1) u ( x, 0) = cosh( x ) + πx 2 Hint: You might ﬁnd it helpful to know that Z 1 0 x cos (( n + 1 2 ) πx ) dx = ( - 1) n ( n + 1 2 ) π - 1 ( n + 1 2 ) 2 π 2 [50 marks] 2. Solve the following inhomogeneous initial boundary value problem for the heat equation:
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Unformatted text preview: u t = α 2 u xx , < x < π 2 , t > u x (0 ,t ) = 0 , u x ( π 2 ,t ) = 2 t u ( x, 0) = 5 by using an appropriate eigenfunction expansion. Hint: You might ﬁnd it helpful to know that Z π/ 2 x 2 cos(2 nx ) dx = π (-1) n 4 n 2 [50 marks]...
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## This note was uploaded on 04/06/2012 for the course MATH 257 taught by Professor Peirce during the Fall '08 term at UBC.

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