Homework/1

$\newcommand\p{\partial}$ Due October 9.

1. Change in moments captured by a diffusion-advection equation. This problem illustrates how the diffusion coefficient $D$ controls the change in second moment of the solution and how $v$ controls the change in the first moment of the solution.
   1. Show that the non-dimensional equation \[ \frac{\p w}{\p \tau} = \frac{\p^2 w}{\p z^2}\] (on the whole real line) is satisfied by the function \[w(z,\tau)=\frac{1}{\sqrt{4\pi \tau}} \exp\left(-\frac{z^2}{4\tau}\right)\] which has a zero mean for all time and linearly increasing variance.
   2. Use changes of variables to turn the solution in part (a) into a solution to the dimensional equations
      1. $\displaystyle \frac{\p u}{\p t} = D \frac{\p^2 u}{\p x^2}$
      2. $\displaystyle \frac{\p u}{\p t} = D \frac{\p^2 u}{\p x^2}-v\frac{\p u}{\p x}$
      where $u(x,t)$ is the concentration (molecules per unit length) of substance whose total amount (i.e. integral over all of space) is $u_T$.
2. Conserved quantity. Describe a physical system (a real one if possible but a made-up one is ok) that is well-modeled by the following system of equations for the functions $u(x,t), v(x,t), p(t)$ and $q(t)$: \[ \frac{\p u}{\p t} = D_1 \frac{\p^2 u}{\p x^2} - c_1 \frac{\p u}{\p x} - a u + b v, \] \[ \frac{\p v}{\p t} = D_2 \frac{\p^2 v}{\p x^2} - c_2 \frac{\p v}{\p x} + a u - b v, \] \[ -D_1 \frac{\p u}{\p x}(0,t) + c_1 u(0,t) = -\alpha \frac{V_1}{A} (u(0,t)-p), \quad -D_1 \frac{\p u}{\p x}(L,t) + c_1 u(L,t) = 0, \] \[ -D_2 \frac{\p v}{\p x}(0,t) + c_2 v(0,t) = 0, \quad -D_2 \frac{\p v}{\p x}(L,t) + c_2 v(L,t) = \beta \frac{V_2}{A} (v(L,t)-q), \] \[ \frac{d p}{d t} = \alpha (u(0,t)-p) , \quad p(0)=p_0, \] \[ \frac{d q}{d t} = \beta (v(L,t)-q) , \quad q(0)=0, \] where $D_i, a, b, c_i, \alpha$, and $\beta$ are all constants. Give a physically reasonable constraint for any parameter that needs one to prevent unreasonable (e.g. negative) values of $u,v, p$ or $q$. Show that some quantity is conserved throughout the system. That is, find a function $Q(t)$ with the property that $dQ/dt = 0$.
3. Telegrapher's Equation. Consider a collection of particles that can move to the left or to the right at a constant velocity \(c\). Switching from left-moving to right-moving occurs with a rate constant \(\alpha_1\) and switching from right-moving to left-moving occurs with rate constants \(\alpha_2\).
   1. Write down equations for the density of left- and right- movers ($u(x,t)$ and $v(x,t)$ respectively). Derive a single equation for $Y(x,t)=u(x,t)+v(x,t)$. $Z(x,t)=u(x,t)-v(x,t)$ will be useful along the way as will some mixed partial derivatives.
   2. Nondimensionalize using arbitrary time and space scales \(\tau\) and \(L\). What are the characteristic time and space scales for the equation?
   3. Under what temporal scaling does the single-time-derivative term dominate the double-time-derivative term? Under what spatial scaling do the diffusion and transport terms balance? Under what temporal and spatial scaling is the equation (approximately) an advection-diffusion equation? What condition on the \(\alpha_i\) must hold for such a scaling to exist?
4. First passage time calculation. Consider a particle diffusing within the interval $[0,L]$, and under the influence of a linear potential (with diffusion coefficient $D$ and constant velocity $v$). The left edge is reflecting and the right edge is absorbing. Derive a boundary value problem for the mean first passage time (MFPT) for a particle starting at position $x$ from the equation for the full distribution discussed in class. Assuming that $\epsilon=vL/D$ is small, approximate the MFPT using a Taylor expansion to high enough order so that the $v$ dependence is not lost. List all space and time scales in the problem and describe the small-$\epsilon$ regime in terms of these scales.