18.311 — MIT (Spring 2015)
Rodolfo R. Rosales (MIT, Math. Dept., E17-410, Cambridge, MA 02139).
March 5, 2015.
Problem Set # 03. Due: Friday March 13.
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 ExID24 statement: Check a pde implicit solution . . . . . . . . . . . . . . . . . . .
1.2 Statement: Linear 1st order PDE #16 (ut + 21 x3 ux = −u; data on t = 0 and |x| ≤ 1) .
1.3 Statement: Haberman problem 57.01 . . . . . . . . . . . . . . . . . . . . . . . . . .
Cars in a road . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Statement: Haberman problem 57.02 . . . . . . . . . . . . . . . . . . . . . . . . . .
Explore a velocity field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Problem 5803a. Calculating the car flux q from data . . . . . . . . . . . . . . . . .
1.6 Problem 6901a. Rate of change of the flux for a moving observer . . . . . . . . . .
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2 Special Problems.
2.1 Statement: Linear 1st order PDE #15 (ut + 2 t x ux = 4 t u; data on t = 0) . . . . . . . . . . .
2.2 Statement: Linear 1st order PDE #19 (ut − 21 t2 x ux = 0; data on x = 1) . . . . . . . . . . . .
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Regular Problems.
1.1
ExID24 statement: Check a pde implicit solution
By direct substitution, show that
yields a solution to the equation:
u = f y − u2 ex ,
(1.1)
ux + 2 u2 uy = u.
(1.2)
Here f is an arbitrary function. Note that:
1. Equation (1.1) defines u implicitly. You have to find ux and uy by implicit differentiation.
2. Equation (1.2) is nonlinear.
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Pages, quite often, become separated from the rest of the package.
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1.2
Statement: Linear 1st order PDE #16 (ut + 12 x3 ux = −u; data on t = 0 and |x| ≤ 1)
Consider the following problem in space-time
ut + 12 x3 ux = −u,
with u(x, 0) = f (x) given, for − 1 ≤ x ≤ 1.
(1.3)
1. Show that u = u(x, t), for t ≥ 1, is completely determined by (1.3). That is, for any −∞ < x < ∞
and t ≥ 1, u = u(x, t) can be written in terms of f using (1.3).
2. When 0 < t < 1, for which values of x is the solution determined by (1.3)? How about t < 0?
3. Let f (x) = x2 . In this case, write an explicit formula for the solution.
Hint: solve the equations for the characteristics, and write them as functions of time x = x(t). Then plot
the ones that start (time t = 0) from −1 ≤ x ≤ 1.
1.3
Statement: Haberman problem 57.01
Assume that there are an infinite number of cars (of infinitesimal length) on a roadway, each labeled
with a number β, ranging from β = 0 to β = 1 (let β = 0 correspond to the first car on the left, and
β = 1 to the last car on the right.) Suppose that the car labeled β moves at a velocity u = 30 + 15 β (i.e.
dx/dt = 30 + 15 β) and starts at a position x(0) = β L.
(a) Show that the car’s velocities steadily increase from 30 to 45 as β ranges from 0 to 1.
(b) Show that the car’s initial positions range from 0 to L as β ranges from 0 to 1.
(c) Consider the car labeled β. Show that: x = (30 + 15 β) t + β L and u = 30 + 15 β.
(d) Eliminate β from these two equations to derive the velocity field
u=
1.4
15 x + 30 L
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15 t + L
(1.4)
Statement: Haberman problem 57.02
Suppose that the velocity field is given by
30 x + 30 L
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15 t + L
(a) Determine the motion of a car that starts at x = L/2 at t = 0.
Hint: Why does dx/dt = (30 x + 30 L)/(15 t + L)? Solve this O.D.E. It is separable.
u=
(1.5)
(b) Show that u is constant along straight lines in the space–time plane, but the cars do not move at
constant velocity.
Is this a realistic velocity field?
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1.5
Problem 5803a. Calculating the car flux q from data
Consider the process of calculating the traffic flow q, from car “crossing times”, as follows:
At any road location x0 and time t0 , take the interval |t − t0 | ≤ 0.5 ∆t (∆t ”small”) and
count the number of cars, N , that cross through x = x0 . Then: q(x0 , t0 ) = N/∆t.
(1.6)
This definition depends on the choice of ∆t, but the idea is to take ∆t small, but large enough that N is
“large”. In this case, provided that the number of cars per unit time (flux) does not change too fast, a
“good” measurement of q can be obtained.
NOTE 1: Since cars are not points, the prescription in (1.6) has an un-resolved issue: What to do with
cars which do not fully cross through x = x0 during the counting interval. We could spend a lot of energy
on this issue, but, for the purpose of this problem, replacing the cars by points (e.g.: the position of their
front bumper) is good enough. In addition, when a “point” is at x = x0 right at the edge of the time
interval, do not count it. Fancier definitions will not change the basic point this exercise aims to illustrate,
but will succeed in making the problem harder, and the point behind it less obvious.
First, notice that the defined flux has a discontinuity every time an additional2 car is included — such
discontinuities appear as we either change ∆t, or move in space and/or time. QUESTIONS:
(a) What is the jump ∆q in the flux (i.e.: the magnitude of the discontinuity)?
(b) If cars are equally spaced (ρ = ρc = constant), and moving at constant speed (u = uc = constant),
how long must the measuring interval ∆t be so that the discontinuities in the flux are less than 5
percent of the flux? Namely: so that ∆q < 0.05 q. In this case, how many cars cross through x0 in
the time interval ∆t?
(c) What do the formulas you derived in b tell you when ρc = 50 cars per km and uc = 72 kph?
From item (b) a formula of the form ∆t > ∆c will follow — where ∆c depends on what size of error (i.e.:
the jumps in ∆q/q) you are willing to tolerate (in this problem 1/20). Let then T be a typical length scale
over which the traffic road conditions change. Then the continuum approximation, in which we replace
the cars by a density, will make sense as long as ∆c T .
NOTE 2: It is, in principle, possible that two cars may be added (or dropped) from the counting interval
in (1.6) as either of ∆t, t0 , or x0 changes. These situations are rare. For simplicity, ignore them in your answer.
1.6
Problem 6901a. Rate of change of the flux for a moving observer
Derive the formula that gives the rate of change of the flux for an observer moving with the traffic. The
formula has the form dq
dt = F (ρ, ρx ), where F is a function that you need to find, and the time derivative
dq
is from the observer’s point of view. Recall that q = Q(ρ), ρt + qx = 0, and c = c(ρ) = dρ
= wave speed.
The flux q here is the car flux as measured from the ground. What car flux q̃ does the observer see?
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See note 2.
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Special Problems.
2.1
Statement: Linear 1st order PDE #15 (ut + 2 t x ux = 4 t u; data on t = 0)
Consider the following initial value problem, for −∞ < x < ∞,
ut + 2 t x ux = 4 t u,
x2
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1 + x4
with u(x, 0) =
(2.7)
1. Use the method of characteristics to solve the problem. Write an explicit formula for the solution.
2. For any fixed x, what is the value of limt→∞ u?
3. Sketch the solution, as a function of x, for a few selected times.
4. What is the solution for arbitrary initial data, u(x, 0) = f (x)?
2.2
Statement: Linear 1st order PDE #19 (ut − 21 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
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1 + t6
(2.8)
1. Use the method of characteristics to solve the problem. Write an explicit formula for the solution.
2. Where is the solution defined by (2.8)?
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.
THE END.
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