M257_316_Midterm2_sec103W2009

M257_316_Midterm2_sec103W2009 - u t = 2 u xx , <...

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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 = 2 u xx , &lt; x &lt; 2 , t &gt; u x (0 ,t ) = 0 , u x ( 2 ,t ) = 2 t u ( x, 0) = 5 by using an appropriate eigenfunction expansion. Hint: You might nd it helpful to know that Z / 2 x 2 cos(2 nx ) dx = (-1) n 4 n 2 [50 marks]...
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