18.311 — MIT (Spring 2015)
Rodolfo R. Rosales (MIT, Math. Dept., E17-410, Cambridge, MA 02139).
March 20, 2015.
Problem Set # 04. Due: Friday April 3.
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
1.1
DiAn27 statement: Non-dimensional form (beam equation boundary value problem) . . . . . . .
1
1.2
ExPD01 statement: General solution of the 1D wave equation . . . . . . . . . . . . . . . . .
2
1.3
ExPD02 statement: Initial value problem for the 1D wave equation . . . . . . . . . . . . . .
2
1.4
Statement: Haberman problem 63.07 (flow density curve fitting Lincoln Tunnel data) . . . . . .
2
1.5
Statement: Haberman problem 72.01 (waiting time at light – braking distance theory) . . . . .
3
1.6
Statement: Haberman problem 77.04 (nonlinearity matters) . . . . . . . . . . . . . . . . . . .
3
1.7
Statement: Linear 1st order PDE #06 (general sln. for ux + 2 x uy = y + x u) . . . . . . . . . .
3
2 Special Problems.
4
2.1
Statement: Haberman problem 73.01 (solve initial value problem) . . . . . . . . . . . . . . . .
4
2.2
Statement: Haberman problem 74.02 (solve initial value problem) . . . . . . . . . . . . . . . .
4
1
Regular Problems.
1.1
DiAn27 statement: Non-dimensional form (beam equation boundary value problem)
Consider the problem (Beam equation Boundary Value Problem)
∂ 4 ũ
∂ 2 ũ
+
β
= 0,
∂ x̃4
∂ t̃2
0 < x̃ < d and t̃ > 0,
(1.1)
with some initial data and the boundary conditions: ũ and ũx̃ vanish at both x̃ = 0 and x̃ = d. Here (a)
β > 0, and d > 0 are dimensional constants; (b) ũ measures the deviation of the beam from straight; and
(c) tildes (i.e.: ũ, x̃, and t̃) denote dimensional variables.
1. What are the dimensions of ũ, β, and d?
1
Pages, quite often, become separated from the rest of the package.
1
2. Introduce a-dimensional variables 2 u, x,
and t, so that the equation takes the form
utt + uxxxx = 0,
(1.2)
where 0 < x < 1.
1.2
ExPD01 statement: General solution of the 1D wave equation
Consider the wave equation for u = u(x, t), where c > 0 is a constant,
utt − c2 uxx = 0.
(1.3)
Introduce the new independent variables η = x − c t, and ξ = x + c t. Change variables to write the
equation for u as a function of these new variables: u = u(η, ξ). Using this transformed form of the
equation, integrate it twice to show that it must be
u = f (η) + g(ξ),
(1.4)
for some arbitrary functions f and g. This shows that any solution of the wave equation (1.3) must have the
form u = f (x − c t) + g(x + c t).
1.3
ExPD02 statement: Initial value problem for the 1D wave equation
In problem ExPD01 we showed that the general solution of the 1-D wave equation ut t − c2 ux x = 0,
with c > 0 a constant, has the form u(x, t) = f (x − c t) + g(x + c t) — where f and g are arbitrary
functions. Using this result, show that if initial values
u(x, 0) = U (x)
and ut (x, 0) = V (x),
(1.5)
are given for the wave equation, then the solution is
u=
1
1
(U (x − c t) + U (x + c t)) +
2
2c
Z
x+c t
V (s) ds.
(1.6)
x−c t
In particular, show that if U and V are periodic functions of x (of period P ) then:
u is periodic in x (of period P ) and periodic in t (of period T = P/c).
Use the GBNS GScheme script in the 18311 MatLab Toolkit and check how closely the numerical calculation
reproduces the periodicity in time (for initial conditions where, say, U is periodic of period 2 and V ≡ 0.)
Note that in the GBNS GScheme, c = 1.
1.4
Statement: Haberman problem 63.07 (flow density curve fitting Lincoln Tunnel data)
Consider exercise 61.3. Suppose that the drivers accelerate in such a fashion that
α = ut + u ux = −
2
a2
ρx ,
ρ
where a > 0 is a constant.
That is, x = x̃/L, t = t̃/T , and u = ũ/U , for appropriate choices of a length L, a time T , and units of ũ, U .
2
(1.7)
(a) Physically interpret this situation.
(b) If u only depends on ρ, and the equation for conservation of cars is valid, show that
du
a
=− .
dρ
ρ
(1.8)
(c) Solve the differential equation in part (b), subject to the condition that u(ρmax ) = 0. The resulting
flow–density curve fits quite well to the Lincoln Tunnel data.
(d) Show that a is the velocity that corresponds to the road’s capacity.
(e) Discuss objections to the theory for small densities.
1.5
Statement: Haberman problem 72.01 (waiting time at light – braking distance theory)
Show that if u = u(ρ) is determined by “braking distance theory” (see exercise 61.2), then the waiting time
per car after a traffic light turns green is the same as the human reaction time for braking.
Recall that (below the speed limit, which certainly applies in the circumstances of this problem) “braking
distance theory” gives the following density-flow velocity relationship
1 u
=
+ 1 L,
ρ
V
(1.9)
where V is the trigger velocity in the law,3 and L is the car length. The trigger velocity V is related to
the reaction time τ by the equation V τ = L.
1.6
Statement: Haberman problem 77.04 (nonlinearity matters)
Show [ ρ2 ] 6= [ ρ ]2 .
1.7
Statement: Linear 1st order PDE #06 (general sln. for ux + 2 x uy = y + x u)
Consider the p.d.e.
ux + 2 x uy = y + x u.
(1.10)
Part 1. Write the characteristic form for this equation, and use it to write the general solution u = u(x, y)
to the p.d.e. — the general solution should involve an arbitrary function f = f (·).
Part 2 (challenge). Find u = u(x, y) if u = u(0, y) = 1 + y 2 for 1 < y < 2. In which region is the solution
u uniquely defined by this? That is: What is the region Ω of the x-y plane with the following property: if
u1 and u2 are both solutions of (1.10) such that uj (0, y) = 1 + y 2 for 1 < y < 2 and j = 1, 2, then u1 = u2
in Ω. Give examples of solutions such that u1 6= u2 outside Ω.
3
V is defined by: For each amount V in car velocity, the separation between cars must augment by one car length.
3
2
2.1
Special Problems.
Statement: Haberman problem 73.01 (solve initial value problem)
Assume that the traffic density is
initially given by
ρ(x, 0) =



 ρmax
for x < 0,
1
2
ρmax


 0
for 0 < x < a,
(2.11)
for a < x,
where a > 0 is some fixed length, and that the car flow velocity is related to the car density by
ρ
ρ
ρ
u = umax 1 −
=⇒ q = umax 1 −
ρ =⇒ c = umax 1 − 2
.
ρmax
ρmax
ρmax
(2.12)
Sketch the initial density. Determine and sketch the density at all later times.
2.2
Statement: Haberman problem 74.02 (solve initial value problem)
Assume that u(ρ) = um (1 − ρ2 /ρ2j ), where um is the speed limit and ρj is the jamming density. For the
initial conditions:
ρ(x, 0) =



 ρ0
for x < 0,
ρ0 (L − x)/L


 0
for 0 < x < L,
for L < x,
where 0 < ρ0 < ρj and 0 < L, determine and sketch ρ(x, t).
THE END.
4
(2.13)