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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.
- Fall '09