- #1

- 20

- 0

## Homework Statement

A the density of a gas [tex]\rho[/tex] obeys the modified diffusion equation

[tex]\frac{\partial \rho(x,t)}{\partial t}-D\frac{\partial^2 \rho(x,t)}{\partial x^2}=K\delta(x)\delta(t)[/tex]

**A)**Express [tex]\rho[/tex] in terms of its 2D fourier transform [tex]\widetilde{\rho}(p,\omega)[/tex] and express the right hand side as a Fourier type integral.

**B)**Find the function [tex]\widetilde{\rho}(p,\omega)[/tex]

[

**Note:**It might occur to you that this function could contain a term of the form [tex]f(p)\delta(Dp^2-i\omega )[/tex], where [tex]f(p)[/tex] is an arbitrary function. If so, you should assume that [tex]f(p)=0[/tex]. 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:

[tex]\rho(x,t)=\frac{1}{2\pi}\iint_{-\infty}^{\infty}dp\; d\omega\; \left [e^{i(px-\omega t)}\widetilde{\rho}(p,\omega) \right ][/tex]

and

[tex]\delta(x)\delta(t)=\frac{1}{4\pi^2}\iint_{-\infty}^{\infty}dp\; d\omega\; \left [e^{i(px-\omega t)} \right][/tex]

**B)**Sticking the results from A) into the diffusion equation, taking the derivatives and rearranging:

[tex]\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][/tex]

...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.

Last edited: