Revision as of 18:16, 8 July 2013

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 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 $S$

Example

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}