- #1
mudkip9001
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Homework Statement
A the density of a gas \rho obeys the modified diffusion equation
\frac{\partial \rho(x,t)}{\partial t}-D\frac{\partial^2 \rho(x,t)}{\partial x^2}=K\delta(x)\delta(t)
A) Express \rho in terms of its 2D Fourier transform \widetilde{\rho}(p,\omega) and express the right hand side as a Fourier type integral.
B) Find the function \widetilde{\rho}(p,\omega)
[Note: It might occur to you that this function could contain a term of the form f(p)\delta(Dp^2-i\omega ), where f(p) is an arbitrary function. If so, you should assume that f(p)=0. If this doesn't occur to you, then don't worry about it!]
The Attempt at a Solution
A) I'm pretty sure I can do this:
\rho(x,t)=\frac{1}{2\pi}\iint_{-\infty}^{\infty}dp\; d\omega\; \left [e^{i(px-\omega t)}\widetilde{\rho}(p,\omega) \right ]
and
\delta(x)\delta(t)=\frac{1}{4\pi^2}\iint_{-\infty}^{\infty}dp\; d\omega\; \left [e^{i(px-\omega t)} \right]
B) Sticking the results from A) into the diffusion equation, taking the derivatives and rearranging:
\iint_{-\infty}^{\infty}dp\; d\omega\; \left [e^{i(px-\omega t)}\widetilde{\rho}(p,\omega) \right]=\frac{K}{2\pi(Dp^2-i\omega)}\iint_{-\infty}^{\infty}dp\; d\omega\; \left [e^{i(px-\omega t)} \right]
...and here i get stuck. how do i solve this? I have noticed that the two sides are conspicuously similar, but I'm not sure what conclusions I can make from that.
Edit: the choice of exponentials for the transform (negative for the time) is the convention that was recomended in lectures.
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