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Unformatted text preview: ECH 159, Fall 2003, Homework 10, November, 2003 Problem 1 The linearized Korteveg de Vries Equation reads ∂ t u = k∂ 3 x u,-∞ < x < ∞ , u ( x, 0) = f ( x ) (a) Fourier transform the equation from x to ω and solve for U ( ω,t ) = F ( u ( x,t )) (b) Using the convolution theorem write the solution for u ( x,t ) in form of an integral over f ( x ). (c) Write down the final solution for f ( x ) = 0 ,x < ,f ( x ) = 1 ,x > 0. For the following functions find a Fourier series. If necessary define a suitable extension. Sketch the function and the Fourier series over the interval- 10 < x < 10. Solution (a) We do the forward transform of the KdV equation (we chose to put the 2 π factor in the back transform) Z ∞-∞ exp( iωx ) ∂ t udx = k Z ∞-∞ ∂ 3 x udx As we know the transform of the n th derivative reads F (d n x u ( x,t )) = ( iω ) n U ( ω,t ) . This leads to ∂ t U ( ω,t ) = k ( iω ) 3 U ( ω,t ) and the solution is U ( ω,t ) = C ( ω )exp( k ( iω ) 3 t ) (b) We know that u ( x,t ) is the backtransform of U ( ω,t ) u ( x,t ) = 1 2 π Z ∞-∞ exp(- iωx ) C ( ω )exp( k ( iω ) 3 t ) dω...
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This note was uploaded on 02/13/2011 for the course ECH 159 taught by Professor Gates during the Winter '08 term at UC Davis.
- Winter '08