18.311 — MIT (Spring 2015)
Rodolfo R. Rosales (MIT, Math. Dept., E17-410, Cambridge, MA 02139).
April 6, 2015.
Problem Set # 05. Due: Friday April 17.
Turn it in before 3:00 PM, in the boxes provided in Room E18-366.
IMPORTANT:
— Turn in the regular and the special problems stapled in two SEPARATE packages.
— Print your name in EACH PAGE of your answers.1
— In page one of each package print the names of the other members of your group.
Contents
1 Regular Problems
1.1 Statement: Linear 1st order PDE #17 (ut + 21 t2 x ux = 0; data on x = 1) .
1.2 Statement: Linear 1st order PDE #20 (ut − 21 t2 x ux = 0; data on x = −1)
1.3 Statement: Information loss and traffic flow shocks . . . . . . . . . . . .
1.4 Statement: Haberman problem 71.07 (observers) . . . . . . . . . . . . . .
1.5 Statement: Haberman problem 74.01 (solve initial value problem) . . . . .
1.6 Statement: Haberman problem 74.03 (show not a traffic flow model & solve
1.7 Statement: Haberman problem 77.06 (Burger’s traveling waves) . . . . . .
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2 Special Problems
2.1 DiAn28 statement: Non-dimensional form (nonlinear damped spring) . . . . . . . . . . . . . . .
2.2 Statement: Initial value problem with Q quadratic # 01a . . . . . . . . . . . . . . . . . . .
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Regular Problems
1.1
Statement: Linear 1st order PDE #17 (ut + 12 t2 x ux = 0; data on x = 1)
Consider the following signaling problem, for time −∞ < t < ∞,
ut + 12 t2 x ux = 0,
with u(1, t) =
1
.
1 + t6
(1.1)
1. Use the method of characteristics to solve the problem. Write an explicit formula for the solution.
2. Where is the solution defined by (1.1)?
Be careful here, as the formula you will obtain in item 1 gives values for a larger region.
3. Do a plot/sketch of a few selected characteristic curves.
1.2
Statement: Linear 1st order PDE #20 (ut − 21 t2 x ux = 0; data on x = −1)
Consider the following signaling problem, for time −∞ < t < ∞,
ut − 21 t2 x ux = 0,
with u(−1, t) =
1
.
1 + t6
(1.2)
1. Use the method of characteristics to solve the problem. Write an explicit formula for the solution.
1
Pages, quite often, become separated from the rest of the package.
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2. Where is the solution defined by (1.2)?
Be careful here, as the formula you will obtain in item 1 gives values for a larger region.
3. Do a plot/sketch of a few selected characteristic curves.
1.3
Statement: Information loss and traffic flow shocks
In this problem we will show that shocks in traffic flow are associated with information loss, and will derive
a formula that quantifies this loss. For simplicity, consider periodic solutions to the traffic flow equation
ρt + qx = 0,
periodic of period T > 0,
(1.3)
where the traffic flux is given by q = Q(ρ). Assume that there is a single shock per period, at x = σ(t).
The shock velocity, s, is given by the Rankine-Hugoniot jump condition
dσ
qR − qL
=s=
,
dt
ρR − ρL
(1.4)
where ρR (resp.: ρL ) is the value of ρ immediately to the right (resp.: the left) of the shock discontinuity,
qR = Q(ρR ), and qL = Q(ρL ). In addition, the shock satisfies the entropy condition
cL > s > cR ,
dQ
is the characteristic speed, cR = c(ρR ), and cL = c(ρL ). Now, let
dρ
Z
1 2
I = I(t) =
ρ (x, t) dx.
period 2
Z ρR
dI
1
1. Show that:
= (qL − qR ) (ρL + ρR ) +
ρ c(ρ) dρ.
dt
2
ρL
(1.5)
where c =
(1.6)
(1.7)
Hint. For any short enough time period, a constant “a” exists such that a < σ(t) < a + T during the
Z σ
Z a+T
1 2
1 2
time interval. Given this, write I in the form I =
ρ dx +
ρ dx, before taking the time
2
2
a
σ
derivative. Then use the fact that, in each of the two intervals
Z ρρ satisfies
1
ρt = −c(ρ) ρx = − fx where f =
s c(s) ds.
(1.8)
ρ
0
In addition, use (1.4) to eliminate σ̇ from the resulting formulas.
dI
2. Show that:
<0
(1.9)
dt
by interpreting the right hand side in (1.7) geometrically, as (the negative of) the area between two
d
curves — such area known to be positive. Hint. Use ρ c =
(ρ q) − q and (1.11).
dρ
Remark 1.1 Recall that, in traffic flow, q = Q(ρ) is a concave function,2 vanishing at both ρ = 0 and the
jamming density ρj > 0, positive for 0 < ρ < ρj , with a maximum (the road capacity qm ) at some value
0 < ρm < ρj . Because of this, the conditions (1.4–1.5) are equivalent to:3
The shock velocity is given by the slope of the secant
line joining (ρL , qL ) to (ρR , qR ). Further ρL < ρR .
d2 Q
< 0. Note also that only solutions satisfying 0 ≤ ρ ≤ ρj are acceptable.
dρ2
3
Thus shocks move slower than cars, and the density a driver sees increases as the car goes through a shock.
2
Concave means that
2
(1.10)
Note that
For ρL < ρ < ρR , the curve q = Q(ρ) is above the secant line,
because Q is concave.
(1.11)
Remark 1.2 How does I in (1.6) related to “information”? Basically, we argue that a function contains
the least amount of information when it is a constant, and that the more “oscillations” it has, the more
information it carries. A (rough) measure of this is given by
Z
1
J =
(ρ − ρ)2 dx,
(1.12)
2
period
where ρ is the average value of ρ (note that conservation guarantees that ρ is a constant in time). AlterP
natively, write ρ the Fourier series ρ = n ρn (t) ei n 2 π x/T for ρ. Then the amount of “oscillation” in ρ
P
can be characterized by n6=0 12 |ρn |2 , which is the same as (1.12). It is easy to see that
J = I − T ρ2 .
(1.13)
Hence J˙ = İ, so that (1.9) indicates that the shock is driving ρ towards a constant value.
The notion of “information” that we use here is not the same as that in “information theory” (which
requires a stochastic process of some kind). Neither is it the same as the related notion of “entropy” of
statistical mechanics (for this we would need a statistical theory 4 for traffic flow). Nevertheless, S = −J
plays the same role for (1.3) that entropy plays for gas dynamics.
Finally, we point out that it can be shown that: for any convex function h = h(ρ) (i.e.: h0 0 > 0)
Z
d
h(ρ) dx < 0 when shocks are present,
dt period
though showing this is a bit more complicated than for the special case h =
1.4
1
2
ρ2 .
Statement: Haberman problem 71.07 (observers)
Consider two moving observers (possibly far apart), both moving at the same velocity V , such that the
number of cars the first observer passes is the same as the number passed by the second observer.
∆q
(a) Show that V =
.
∆ρ
(b) Show that the average density between the two observers stays a constant.
1.5
Statement: Haberman problem 74.01 (solve initial value problem)
Assume that u(ρ) = um (1 − ρ/ρj ), where um is the speed limit and ρj is the jamming density. For the
initial conditions:

for x < 0,
 ρ0
ρ(x, 0) =
(1.14)
ρ (L − x)/L for 0 ≤ x ≤ L,
 0
0
for L < x,
where 0 < ρ0 < ρj and 0 < L, determine and sketch ρ(x, t).
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Such theories exist.
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1.6
Statement: Haberman problem 74.03 (show not a traffic flow model & solve IVP)
Consider the (non-dimensionalized) partial differential equation
ρt − ρ2 ρx = 0 ,
−∞ < x < ∞ and t > 0.
(1.15)
(a) Why can’t this equation model a traffic flow problem?
(b) Solve this P.D.E. by the method of characteristics, subject to the initial conditions:

 1
ρ(x, 0) =
1−x

0
1.7
for x < 0,
for 0 ≤ x ≤ 1,
for 1 < x.
(1.16)
Statement: Haberman problem 77.06 (Burger’s traveling waves)
Consider Burger’s equation, as derived in exercise 77.5:
2ρ
ρt + umax 1 −
ρx = ν ρxx .
ρmax
(1.17)
Suppose that a solution exists as a density wave, moving without change of shape at velocity V
ρ = f (x − V t).
(1.18)
(a) What O.D.E. is satisfied by f ?
(b) Integrate this differential equation once. By graphical techniques show that a solution exists, such
that ρ → ρ2 as x → ∞ and ρ → ρ1 as x → −∞, only if ρ2 > ρ1 . Roughly sketch this solution, and
give a physical interpretation of this result.
(c) Show that the velocity of wave propagation, V , is the same as the shock velocity separating ρ = ρ1
from ρ = ρ2 (occurring if ν = 0).
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Special Problems
2.1
DiAn28 statement: Non-dimensional form (nonlinear damped spring)
dx̃
k x̃
d2 x̃
Consider a nonlinear damped mass-spring system
m 2 +ν
=
,
(2.19)
1 + α x̃2
dt̃
dt̃
where x̃ = x̃ t̃ is the position of the center of mass, m > 0 is the mass, ν > 0 is the (constant) damping
coefficient, k > 0 is the spring constant for small deviations, α > 0 is a coefficient that characterizes the
spring nonlinearity, and tildes (i.e.: x̃ and t̃) denote dimensional variables. The spring coefficient is
Since the spring to gets weaker as it stretches, the nonlinearity is called “soft”.
k
.
1 + α x̃2
1. What are the dimensions of ν, k, and α?
2. Introduce a-dimensional variables 5 x and
t, so that the equation takes the form
d2 x
6
+
dx
= f (x),
(2.20)
dt2
dt
where γ is a constant without dimensions, and f is a function that involves no free constants.6
5
γ
That is: x = x̃/L and t = t̃/T , for appropriate choices of a length L and a time T .
That is, no letter constants in it.
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2.2
Statement: Initial value problem with Q quadratic # 01a
Consider the traffic flow equation
ρt + qx = 0,
(2.21)
for a flow q = Q(ρ) that is a quadratic function of ρ. In this case c = dQ/dρ is a conserved quantity, and
the problem (including shocks, if any) can be entirely formulated in terms of c, which satisfies
1 2
ct +
c
= 0.
(2.22)
2
x
1. Consider the initial value problem determined by (2.22) and 7
c(x, 0) = 0 for x ≤ 0
and c(x, 0) = 2
√
x for x ≥ 0.
(2.23)
Without actually solving the problem, argue that the solution to this problem must have the form
c = t f (x/t2 ) for t > 0,
for some function f .
(2.24)
Hint. Let c = c(x, t) be the solution to the problem. For any constant a > 0, define C = C(x, t) by C =
1
c(a2 x, a t). What problem does C satisfy? Use now the fact that the solution to (2.22–2.23) is unique
a
to show that (2.24) must apply, by selecting a appropriately at any fixed time t > 0.
2. Use the method of characteristics to solve the problem in (2.22–2.23). Write the solution explicitly
for all t > 0, and verify that it satisfies (2.24). Warning: the solution involves a square root. Be careful
to select the correct sign, and to justify your choice.
3. For the solution obtained in item 2, evaluate cx at x = 0 for t > 0. Note that this derivative is
discontinuous there, so it has two values.
THE END.
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In a traffic problem, c must satisfy c(ρj ) ≤ c ≤ c(0). Ignore the fact that this does not apply for (2.22).
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