6.4: Initial-Boundary Value Problems

Consider the initial-boundary value problem for $c=c(x,t)$

$$\begin{eqnarray} \label{sol1} c_t&=&D\triangle c\ \ \mbox{in}\ \Omega\times (0,\infty)\\ \label{sol2} c(x,0)&=&c_0(x)\ \ x\in\overline{\Omega}\\ \label{sol3} \frac{\partial c}{\partial n}&=& 0\ \ \mbox{on}\ \partial\Omega\times (0,\infty). \end{eqnarray}$$

Here is $\Omega\subset\mathbb{R}^n$, $n$ the exterior unit normal at the smooth parts of $\partial\Omega$, $D$ a positive constant and $c_0(x)$ a given function.

Remark. In application to diffusion problems, $c(x,t)$ is the concentration of a substance in a solution, $c_0(x)$ its initial concentration and $D$ the coefficient of diffusion.
The first Fick's rule says that

$$w=D\partial c/\partial n,$$

where $w$ is the flow of the substance through the boundary $\partial\Omega$. Thus according to the Neumann boundary condition ($\ref{sol3}$), we assume that there is no flow through the boundary.

Contributors and Attributions

• Prof. Dr. Erich Miersemann (Universität Leipzig)

• Integrated by Justin Marshall.