[Faller] 03FALL - HWK10 SOLN

[Faller] 03FALL - HWK10 SOLN - ECH 159 Fall 2003 Homework...

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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 < 0 , 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 )) = ( ) n U ( ω, t ) . This leads to t U ( ω, t ) = k ( ) 3 U ( ω, t ) and the solution is U ( ω, t ) = C ( ω ) exp( k ( ) 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 ( ) 3 t ) At t = 0 we find using the initial condition f ( x ) = 1 2 π Z -∞ exp( - iωx ) C ( ω )
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