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knowlewj01
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Homework Statement
at t=0, x=0, a schoolboy sets off a stink bomb halfway down a corridor that is long enough to be considered infinite. The dispersion of the particles obey the modified diffusion formula:
[itex] \frac{\partial \rho (x,t)}{\partial t} - D\frac{\partial^2 \rho(x,t)}{\partialx^2}=K\delta(x)\delta(t)[/itex]
1) Express the density in terms of its Fourier transform and substitute into the above equation, also write out the dirac delta function as a product of the integral forms.
2) find the function [itex]\tilde{\rho}(p,\omega)[/itex]
there is another part but i need to understand this bit before i make n attempt on it
Homework Equations
Inverse Fourier Transform:
[itex] f(x,t) = \frac{1}{2\pi}\int^\infty_{-\infty} dp \int^\infty_{-\infty} d\omega e^{ipx - i\omega t} \tilde{f}(p,\omega) [/itex]
Dirac Delta Function:
[itex] \delta (x-x_0) = \frac{1}{2\pi}\int^\infty_{-\infty} e^{ip(x-x_0) dp[/itex]
The Attempt at a Solution
having made a substitution for [itex] \rho(x,t)[/itex] for the above equation and using the delta functions for x and t i get:
[itex] \int^\infty_{-\infty} dp \int^\infty_{-\infty} d\omega e^{ipx - i\omega t} \left(Dp^2 - i\omega\right) \tilde{\rho}(p,\omega) = \frac{K}{2\pi}\int^\infty_{-\infty} dp \int^\infty_{-\infty} d\omega e^{ipx - i\omega t}[/itex]
From here, I am not sure how to get the function [itex]\tlide{\rho}(p,\omega)[/itex]
is there some property of Fourier transforms or delta functions that i am forgetting?
Thanks
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