2. Consider a drug delivery system in which the drug is delivered with a constant flow or flux
$\frac{\partial u}{\partial x}$
at the surface of the skin, $x = 0$. You can think of it as a drup pump at
$x = 0$. We assume $u(x, t)$ ($M/L$ in 1 space dim.) is the concentration satisfying a chemical
diffusion problem with a constant flow or flux at the boundary $x = 0$:
$\frac{\partial u}{\partial t} = D \frac{\partial^2 u}{\partial x^2}$, $x > 0$, $t > 0$
IC: $u(x, 0) = 0$, BC: $D \frac{\partial u}{\partial x}(0, t) = -A$, $u(\infty, t) = 0$.
(a) What are the dimensions of $D$ and $A$.
(b) Use dimensional reduction to derive a form of $u(x, t)$. Is there a similarity variable,
call it $\eta$. Be sure your similarity variable vanishes at $x = 0$ so $\eta = 0$.
(c) Use the class notes to transform the diffusion problem into an ODE-BVP. Verify that
the boundary conditions are consistent with the IC and BC of the diffusion problem.
