M257_316_Midterm2_sec101n2W2009

M257_316_Midterm2_se - u t = α 2 u xx< x< π 2 t> u(0,t = 0 u x π 2,t = t 2 2 u x 0 = 0 by using an appropriate expansion in terms of the

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Math 257/316, Midterm 2, Section 101/102 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 for the heat equation: u t = u xx - u, 0 < x < 1 , t > 0 u x (0 ,t ) = 0 , u x (1 ,t ) = sinh(1) u ( x, 0) = cosh( x ) - x Hint: You might find it helpful to know that Z 1 0 - x cos( nπx ) dx = ( 2 /n 2 π 2 , n odd 0 , n even . [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 (0 ,t ) = 0 , u x ( π/ 2 ,t ) = t 2 / 2 u ( x, 0) = 0 by using an appropriate expansion in terms of the eigenfunctions sin(2 n +1) x corresponding to the eigenvalues λ n = (2 n + 1) where n = 0 , 1 ,... Hint: You might find it helpful to know that Z π/ 2-x sin((2 n + 1) x ) dx = (-1) n +1 / (2 n + 1) 2 and that Z te γt dt = te γt /γ-e γt /γ 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 The University of British Columbia.

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