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5.3 Resolution of poisson screened equation with the use of Green function

Poisson's screened equation uses a linear operator so its solutions are superposable and this suggests a general method for solving this equation. Suppose that we could construct all of the solutions generated by point sources, provided that they satisfy the appropriate boundary conditions. Any general source function can be built up out of a set of suitably weighted point sources, so the general solution of Poisson's equation must be expressible as a weighted sum over the point source solutions. Thus, once we know all of the point source solutions we can construct any other solution. In mathematical terminology, we require the solution to

$$\bigtriangleup G(\v{r}, \v{r}') = -\delta(\v{r} - \v{r}')$$ (5.15)

which goes to zero as $\vert{\v{r}}\vert \rightarrow\infty$. The function $G(\v{r}, \v{r}')$ is the solution generated by a unit point source located at position $\v{r}'$. This function is known as a Green's function and from now on we will consider it to depend only on the difference $\v{r}-\v{r}$. In other words

$$G(\v{r}, \v{r}') =G(\v{r}- \v{r}')$$ (5.16)

This reflects the translational invariance of the unbounded domain with the disturbance depending only on the relative separation from the source. The solution generated by a general source function $f(\v{ r})$ is simply the appropriately weighted sum of all of the Green's function solutions:

$$u({\v{r}}) = \int_\Omega G(\v{r}-\v{r}') f(\v{r}') d^3 r'.$$ (5.17)

The formalism of Green function and the proof of this solution is provided in annex C. As stated above, this is a superposition of screened $1/r$ functions, weighted by the source function $f$ and with $\lambda$ acting as the strength of the screening.

An analytical expression of the Green functions associated to Poisson's screened equation can be evaluated. The presence of the term $k^2$ suggests the use of spherical polar coordinates $(\rho,\theta,\phi)$, with the polar axis along $\v{R}=\v{r}-\v{r}'$, such that, $\v{p}=\v{\rho}$, so $\v{\rho} \cdot \v{R}=\rho R \cos \theta$ , and the elementary integration volume being $dV=d\rho\: \rho d\theta \: \rho\sin\theta d\phi$. Equation 5.15 becomes:

$$G(\v{R}) = \frac{1}{(2\pi)^{3}} \int_0^\infty \rho^2 \int_0^\pi \... ...ac{e^{i \rho R \cos\theta}}{\rho^2+\lambda^2} d\theta d\rho \int_0^{2\pi} d\phi$$ (5.18)

The angular integration present no difficulty, letting us with the $\rho$ integral:

$$G(\v{R}) = \frac{1}{2\pi^{2}R} \int_0^\infty \frac{\rho}{\rho^2+\lambda^2} \frac{e^{i\rho R}-e^{-i\rho R}}{2i} d\rho$$ (5.19)

And the beauty of holomorph analysis to gives us:

$$G(\v{R}) = \frac{1}{4\pi^{2}R} \int_{-\infty}^\infty \frac{i \rho... ...da)(\rho-i\lambda)} , i\lambda \right) \right] = \frac{e^{-\lambda R}}{4 \pi R}$$ (5.20)

with the use of contour techniques around the pole $\rho = i\lambda$ and the residue theorem.

Let's summarize the previous steps and write the final results for the particular case of interest where $\lambda =0$, corresponding to the solution of Poisson's equation:

$$G(\v{r}-\v{r}') = \frac{1}{(2 \pi)^3} \int \frac{1}{k^2} e^{ i \v{k} \cdot (\v{r}'-\v{r}) } d^3 k = \frac{1}{4\pi\vert\v{r}'-\v{r}\vert }$$ (5.21)

$$\phi(\v{r}) = \frac{1}{\epsilon_0} \int_\Omega \rho(\v{r'}) G(\v{r}-\v{r}') dr'... ...1}{4\pi \epsilon_0} \int_\Omega \frac{\rho(\v{r}')}{\vert\v{r}'-\v{r}\vert} dr'$$ (5.22)

The expression of $\phi$ as the convolution between $\rho$ and $G$ is an interesting results for implementing Poisson's solver algorithm.

Emmanuel Branlard
http://emmanuel.branlard.free.fr/