- #1
jdstokes
- 520
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I want to solve the partial differential equation
[itex]\Delta f(r,z) = f(r,z) - e^{-(\alpha r^2 + \beta z^2)}[/itex]
where [itex]\Delta[/itex] is the laplacian operator and [itex]\alpha, \beta > 0[/itex]
In full cylindrical symmetry, this becomes
[itex]\frac{\partial_r f}{r} + \partial^2_rf + \partial^2_z f = f - e^{-(\alpha r^2 + \beta z^2)}[/itex]
Applying the Fourier transform along the cylindrical symmetry axis one obtains the following ODE
[itex]d^2_r\hat{f} + \frac{1}{r}d_r\hat{f} - (k_z^2 + 1) \hat{f} = \mathcal{F}\{e^{-(\alpha r^2 + \beta z^2)}\}[/itex]
where
[itex]\mathcal{F} \equiv \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}dz\, e^{i k_z z}[/itex].
The solution to the homogeneous part, according to Mathematica is
[itex]\hat{f} = C_1 J_0(ir\sqrt{k_z^2 + 1}) + C_2 Y_0(-ir\sqrt{k_z^2 + 1})[/itex]
for arbitrary constants C_1, C_2 and J_0 and Y_0 are Bessel functions of the first and second kind, respectively.
I tried to take the inverse Fourier transform of [itex]\hat{f}[/itex] using Mathematica, but it won't give a result. Is anyone aware of, or know of a book containting this integral transform? If not, how about the limiting case as [itex]k_z \rightarrow \infty[/itex]
[itex]\hat{f} = C_1 J_0(irk_z) + C_2 Y_0(-irk_z)[/itex]
Thanks.
James
[itex]\Delta f(r,z) = f(r,z) - e^{-(\alpha r^2 + \beta z^2)}[/itex]
where [itex]\Delta[/itex] is the laplacian operator and [itex]\alpha, \beta > 0[/itex]
In full cylindrical symmetry, this becomes
[itex]\frac{\partial_r f}{r} + \partial^2_rf + \partial^2_z f = f - e^{-(\alpha r^2 + \beta z^2)}[/itex]
Applying the Fourier transform along the cylindrical symmetry axis one obtains the following ODE
[itex]d^2_r\hat{f} + \frac{1}{r}d_r\hat{f} - (k_z^2 + 1) \hat{f} = \mathcal{F}\{e^{-(\alpha r^2 + \beta z^2)}\}[/itex]
where
[itex]\mathcal{F} \equiv \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}dz\, e^{i k_z z}[/itex].
The solution to the homogeneous part, according to Mathematica is
[itex]\hat{f} = C_1 J_0(ir\sqrt{k_z^2 + 1}) + C_2 Y_0(-ir\sqrt{k_z^2 + 1})[/itex]
for arbitrary constants C_1, C_2 and J_0 and Y_0 are Bessel functions of the first and second kind, respectively.
I tried to take the inverse Fourier transform of [itex]\hat{f}[/itex] using Mathematica, but it won't give a result. Is anyone aware of, or know of a book containting this integral transform? If not, how about the limiting case as [itex]k_z \rightarrow \infty[/itex]
[itex]\hat{f} = C_1 J_0(irk_z) + C_2 Y_0(-irk_z)[/itex]
Thanks.
James