
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

• Integrated by Justin Marshall.