homework9

# homework9 - cos cαt Z ∞-∞ f τ)cos α τ-x dτdα...

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Problem 1 Solve the initial value problem for the inhomogeneous wave equation: u tt = u xx + cos(2 x ) sin( ωt ) , u x (0 ,t ) = u x ( π,t ) = 0 , 0 < x < π, u ( x, 0) = 0 , u t ( x, 0) = 0 , where ω is a real valued parameter. Problem 2 By using Fourier Integral Formula derive the solution of the wave equation u tt = c 2 u xx , x ( -∞ , + ) , t > 0 , which satis es the initial conditions u ( x, 0) = f ( x ) and u t ( x, 0) = 0 when x ( -∞ , + ) . Transform this solution to the d'Alembert's form. Hints for the transformation: ( δ ( x ) is Dirac Delta function ): δ ( x - x 0 ) = 1 π Z 0 cos( k ( x - x 0 )) dk Z + -∞ f ( x ) δ ( x - x 0 ) dx = f ( x 0 ) Check the answer: u ( x,t ) = 1 Z +
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Unformatted text preview: cos cαt Z + ∞-∞ f ( τ )cos( α ( τ-x )) dτdα Problem 3 Let F denote the periodic function, of period l , where F ( x ) = ‰ l 4-x, when ≤ x ≤ l/ 2 x-3 l 4 , when l/ 2 < x ≤ l. [a] Describe the function F ( x ) graphically. and show that it is, in fact, the even periodic extension, with period l , of the function f ( x ) = l 4-x, ≤ x ≤ l/ 2 . [b] Find the Cosine Fourier Series of the function f ( x ) . 1...
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