P384_Assignment_8 - 1 Physics 384 2016 Assignment 8 Due in class Friday November 25 Problem I Consider the inhomogeneous ODE d2 y(x k 2 y(x = f(x dx2

P384_Assignment_8 - 1 Physics 384 2016 Assignment 8 Due in...

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1 Physics 384 2016 Assignment 8 Due in class Friday November 25 Problem I Consider the inhomogeneous ODE d 2 y ( x ) dx 2 - k 2 y ( x ) = f ( x ) where the boundary conditions are that y ( ±∞ ) = finite. Use the Fourier transform method to show that the Green’s function G ( x ; x 0 ) for this problem, which satisfies d 2 G ( x ; x 0 ) dx 2 - k 2 G ( x ; x 0 ) = δ ( x - x 0 ) can be expressed as G ( x ; x 0 ) = - 1 2 π ˆ -∞ e ik 0 ( x - x 0 ) k 0 2 + k 2 dk 0 Evaluate the integral and show that G ( x ; x 0 ) = - 1 2 k e - k | x - x 0 | x Problem II Use Fourier transform methods to show that the Green’s function for Poisson’s equation ~ 2 G ( ~ r ; ~ r 0 ) = 4 πδ 3 ( ~ r - ~ r 0 ) is given by G ( ~ r ; ~ r 0 ) = 1 | ~ r - ~ r 0 | Note: This requires a bit more care than some of the examples we’ve already treated. In particular, the Fourier transform of G does not actually exist, without modification, since the existence of a transform requires that the function be absolutely integrable. In this case,
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