hw9Marsh2008

hw9Marsh2008 - v ( x, 0). (c) Use this initial condition to...

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CAAM 336 · DIFFERENTIAL EQUATIONS Problem Set 9 Posted Friday 14 March 2008. Due Friday 21 March 2008 in class. 1. [70 points] Use the Fourier series method to solve the following initial boundary value problem ∂u ∂t - 2 u ∂x 2 = - 4 x, 0 x 1 , t 0 ∂u ∂x (0 , t ) = 1 ∂u ∂x (1 , t ) = 3 u ( x, 0) = x 2 by doing the following: (a) De±ne a function p ( x ) so that v ( x, t ) = u ( x, t ) - p ( x ) has homogeneous boundary conditions. Then write down the resulting partial di²erential equation for v . There are two things to keep in mind here: i. While p does not depend on t , it cannot be linear. You must do something else. ii. The initial condition and the right hand side for v will be di²erent from those for u . (b) Find the Fourier series for the new initial condition
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Unformatted text preview: v ( x, 0). (c) Use this initial condition to ±nd the Fourier series for v ( x, t ) for all positive t . (d) With the Fourier series for v , write down the Fourier series for u . (e) Does this solution stay bounded as you let t → ∞ ? (From last week’s homework, you can answer this question just by looking at the right hand side of the di²erential equation for v .) 2. [30 points] Use the Fourier series method to solve the periodic problem ∂u ∂t-∂ 2 u ∂x 2 = 0 ,-1 ≤ x ≤ 1 , t ≥ u (-1 , t ) = u (1 , t ) ∂u ∂x (-1 , t ) = ∂u ∂x (1 , t ) u ( x, 0) =-x 3 + x...
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This note was uploaded on 01/21/2012 for the course CAAM 330 taught by Professor Tompson during the Fall '09 term at UVA.

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