Math 6420 • PDE

The flux \(q\) of dilute chemicals in a solute is usually assumed
to be governed by Fick's law, \(q=-D\nabla c\), where \(c\) is the concentration. In an electric
field, if the chemical species is ionic, then the flux is given by the Nernst-Planck
equation \(q=-D\nabla c-\mu c\nabla\phi\) where \(\phi\) is the potential of the electric field.
\begin{enumerate}
\item[a)] Assuming the field \(\nabla\phi\) is constant, and is independent of the
concentration of the ionic species, find an evolution equation governing the concentration
\(c\) as a function of time.
\item[b)] Suppose the concentrations at the two ends of a one dimensional domain are held
fixed. Find the concentration as a function of time, starting from arbitrary initial
conditions.
\end{enumerate}

Caution: The spatial operator is not self-adjoint! A transformation of the form
\(c(x,t)=e^{\alpha x}u(x,t)\) is helpful.
\end{enumerate}