MATH 451 Projects
Homework #5
A river with parallel straight banks b units apart has stream velocity
given by v(x,y) = v(x)j. The functional describing
possible paths y = y(x) of minimal transit time for the boat to take
across the river was derived in class. Find extremals for this model.
Homework #8
A river with parallel straight banks b units apart has stream velocity
given by v(x,y) = v(x)j. The functional describing
possible paths y = y(x) of minimal transit time for the boat to take
across the river was derived in class and the possible extremals were
found in Homework #5. Assume that the velocity profile v(x) is
parabolic with maximum velocity of b2 in the center of
the stream and zero velocity at both banks. Find the path y = y(x) of
minimal transit time, find the minimal transit time, and find the
steering angle across the river for all x.
Homework #9
Consider the differential equation (D) -u''(x) + q(x) u(x) = f(x), a < x <
b, with u(a) = u(b) = 0. Determine an equivalent minimization problem (M)
and variational problem (V). Show that under certain assumptions these
three problems are equivalent, carefully stating these assumptions.
Develop a discrete finite element formulation for solving this problem
using a fixed grid spacing h and the standard linear hat functions.
Identify the matrix formulation completely.
Homework #10
Existence, uniqueness, maximum principle and continuous dependence on
the data of the advection-diffusion equation.
Project
The ongoing development and analysis of a traffic flow problem.
- April 3 Analyze the traffic flow on a heavily travelled
street with a stop light.
- April 5 The traffic flow is on a one-way street with only one
lane. Thus the problem is one-dimensional (x) and time-dependent (t).
The street remains one lane its entire length. View the traffic from
afar so that individual vehicles (and their lengths) are
indistinguishable and thus consider the traffic density.
- April 7 An appropriate model would be the standard
conservation law for the traffic density involving a flux function and a
nonhomogeneous term. Thus it would take the form pt +
qx = f.
- April 10 The time domain is readily thought of as t>0 while
the spatial domain might be finite or infinite. We will need boundary
and/or initial conditions.
- April 12 Let the units of t be seconds and the units of x be
meters. Also the traffic density p has units of cars per meter. The
equation then tells us that the units of q are cars per second. There
are no smooth sources or sinks (we are ignoring side streets) so the
nonhomogeneous term f is zero. Hence pt + qx = 0.
Only look at one cycle of the light. Additional cycles might produce
periodic motion over a longer time period. So start with a red light
that then changes to a green light.
- April 14 To develop the model for the flux
function q, ignore the stop light initially. Because of the bunching of
traffic, the flux is not proportional to the negative gradient of the
density or we can't write q = -D px (it is not Fickian
diffusion). As the speed of the cars varies the flux in a fixed length
of road changes (the greater the speed the greater the flux).
Similarly, as the traffic density varies the flux varies (the greater
the density the greater the flux). So we use q = pu. This can be
thought of as q is proportional to the traffic density with
proportionality a varying factor u or q is directly proportional to both
p and u. Hence pt + (up)x = 0.
- April 17 Modeling u as a constant velocity c gives the linear
wave equation pt + c px = 0. However this means
that every car travels at the same constant speed. Not very realistic
for this model. Modeling u as a variable velocity c(x,t) that depends
on position and space might work. This leads to pt + c(x,t)
px = 0. However, we don't know that dependence and we might
not be able to write an equation (algebraic or differential) that
describes the dependence. We do know that the dependence of u is
reflected in the fact that as the traffic density changes, so does the
traffic velocity. So model u as u = u(p). Now determine the dependence
on p. Using u = u(p), note that as p increases, u should
decrease. One possibility is that u is inversely proportional to p.
However, remember that we will have a speed limit on the road (hence a
maximum u denoted umax) and we might have cars
bumper-to-bumper that brings traffic to a standstill. Denote the
traffic density of bumper-to-bumper traffic by pmax. We will
assume a decreasing function u = u(p) between the points
(0,umax) and (pmax,0).
- April 19 Using u = u(p), we will
assume a linearly decreasing function u = u(p) between the points
(0,umax) and (pmax,0). Thus we arrive at u(p) =
umax (1-p/pmax). Hence our model is the nonlinear
wave equation pt + qx = 0, where q = pu =
umax p (1-p/pmax).
- April 24 Using the linear one-way wave equation,
pt + c px = 0, as our guide, we can specify
initial data on the whole real line or we can specify initial data on a
finite interval [a,b] or semiinfinite interval x>a and specify boundary
data (for all time t) along x = a.
- April 28 We choose to specify the problem on the whole real
line and thus will specify the initial data there as well. Starting
with a red stoplight at t=0 we turn the light to green and watch the
traffic flow. Hence we have p(x,0) = p0(x), where
p0(x) = pmax for x<0 and p0(x) = 0 for
x>0. Thus our complete model is given by pt + qx
= 0, x in R, t>0, where q = pu = umax p
(1-p/pmax) and p(x,0) = p0(x), x in R.
- Due May 4 Discuss the solution of this model including the
use of characteristics and the initial data. Discuss weak solutions,
jump conditions, shocks, rarefactions, and the entropy condition.