Poisson's Equation

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In this article we discuss the solution of inhomogenious Laplace's Equation, which is called Poisson's equation. \begin{align} \nabla ^2 u(\mathbf{r})=\rho(\mathbf{r}) \end{align} In a related article we found that the solution to an inhomogenious linear PDE is \begin{align} u(\mathbf{r})=\int_V G(\mathbf{r},\mathbf{r}_0)\rho(\mathbf{r}_0)dV(\mathbf{r}_0) \tag{1}=G(\mathbf{r},\mathbf{r}_0) * \rho(\mathbf{r}_0) \end{align}

To avoid confusion I want to make clear explicitly that we need to distinguish between in/homogenious PDE's and in/homogenious boundary conditions. Inhomogenious PDE's are given by $\mathcal{L}u=\rho$, whereas homogenious PDE's have $\rho=0$. Homogenious boundary conditions mean, that there are essentially no boundary conditions, in electrostatics these are problems with e.g. charges in free space. Inhomogenious boundary conditions are those where additionally to a given charge distribution in space, the solution is constrained to have a certain potential on some region $S$ in space. For example a charge distribution is kept in some metal sphere, which itself is held at a certain potential. We distinguish two boundary-value problems

  • Dirichlet boundary conditions, where $u(\mathbf{r})$ is specified on some surface $S$ (e.g. the metal sphere)
  • Neumann boundary conditions, where $\partial u(\mathbf{r})/ \partial n$ is given on some surface $S$


Find the solution $G(\mathbf{r},\mathbf{r}_0)$ for Poisson's equation (inhomogenious Laplace's equation) with homogenious boundary conditions in three dimensions that tends to zero as $|\mathbf{r}|\rightarrow \infty$. \begin{align} \nabla ^2 G(\mathbf{r},\mathbf{r}_0)=\delta(\mathbf{r}-\mathbf{r}_0) \end{align} In other words, find the Green's function for Poisson's Equation. First notice that the problem is spherically symmetric about $\mathbf{r}_0$. To find $G(\mathbf{r},\mathbf{r}_0)$ we integrate over the sphere with Volume $V$ and Surface $S$ centered at $\mathbf{r}_0$. \begin{align} \int_V \nabla ^2 G(\mathbf{r},\mathbf{r}_0) \, dV= \int_V \delta(\mathbf{r}-\mathbf{r}_0) \,dV =1 \tag{2} \end{align} and following Gauß's theorem \begin{align} \int_V \nabla \cdot \nabla G(\mathbf{r},\mathbf{r}_0) \, dV= \oint_S \nabla G(\mathbf{r},\mathbf{r}_0) \, d\mathbf{S} \end{align} We expect the solution to be spherically symmetric about $\mathbf{r}_0$, this means the solution is constant on spheres with a particular radius, thus $G(\mathbf{r},\mathbf{r}_0)$ must be a function of $|\mathbf{r}-\mathbf{r}_0|$ \begin{align} G(\mathbf{r},\mathbf{r}_0)=G(|\mathbf{r}-\mathbf{r}_0|)=G(r) \end{align} where $r$ is the radial distance from $\mathbf{r}_0$. Equation (2) becomes \begin{align} \oint_S \nabla_r G(r) \, d\mathbf{S}=4\pi r^2 \nabla_r G(r)=1 \end{align} Integrating we find the solution \begin{align} \int \nabla_r G(r) dr=\int \frac{1}{4\pi r^2} dr=-\frac{1}{4\pi r} \end{align} which is

Green's function for homogenious boundary conditions.

\begin{align} G(\mathbf{r},\mathbf{r}_0)=G(|\mathbf{r}-\mathbf{r}_0|)=-\frac{1}{4\pi}\frac{1}{|\mathbf{r}-\mathbf{r}_0|} \end{align}