18.311 — MIT (Spring 2015)
Rodolfo R. Rosales (MIT, Math. Dept., E17-410, Cambridge, MA 02139).
February 26, 2015.
Problem Set # 02. Due: Friday March 6.
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 ExID21 statement: Check a pde implicit solution . . . . . .
1.2 ExID51 statement: Directional derivatives . . . . . . . . . .
1.3 ExID61 statement: Implicit Taylor Expansions . . . . . . . .
1.4 ExPD03 statement: Solve the one directional wave equation
1.5 Statement: TFPa02. Near constant initial data (linearize) .
1.6 Statement: TFPa06. Solve a first order linear pde . . . . . .
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2 Special Problems.
2.1 ExID73 statement: Change of variables for a pde . . . . . . . . . . . . . . . . . . .
2.2 Statement: TFPa07. Solve a first order linear pde . . . . . . . . . . . . . . . . . . .
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Regular Problems.
1.1
ExID21 statement: Check a pde implicit solution
By direct substitution, show that
is a solution of the equation:
u = f (y − x2 − V (u) x),
(1.1)
ux + (2 x + V (u)) uy = 0.
(1.2)
Here f is an arbitrary function and V = V (u) is a given 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, except for the trivial case when V is a constant function.
1
Pages, quite often, become separated from the rest of the package.
1
1.2
ExID51 statement: Directional derivatives
Calculate the derivatives indicated below
du
1. Express
— for u = u(x(s), y(s)) — in terms of x, y, and u, given that:
ds
x(s) = es , y(s) = s es , and u satisfies x ux + (y + x) uy = 2 x + u.
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du
— for u = u t2 , ln(t),
2. Express
dt
t
and t.
— in terms of the partial derivatives of u = u(x, y, z)
∂u
3. Let u = ex sin(y), and let (r, θ) be the polar coordinate radius and angle. Calculate
and
∂r
∂u
in terms of x and y.
∂θ
1.3
ExID61 statement: Implicit Taylor Expansions
For the examples below, calculate the Taylor expansion for f (x) up to the order indicated — for
example: f (x) = a0 + a1 x + a2 x2 + O(x3 ), for some coefficients
an . Do not use a calculator to
√
evaluate constants that appear in the expansions √
— e.g.,
√ 2/π or cos(3). On the other hand, do
simplify when possible — e.g., tan(π/4) = 1 or 2/ 2 = 2.
1. Expand up to O(x4 ), where f (x) is defined by x = (1 + f ) sin(f ), with f (0) = 0.
2. Expand up to O(x4 ), where f (x) is defined by f = 1 + x + x2 f 3 , with f (0) = 1.
Hint. Write f (x) = f (0) + a1 x + a2 x2 + . . ., substitute into the equation for f , expand, and find
PN
P
n
N +1
n
) you can conclude An = Bn .
the coefficients. Recall: from N
n=0 Bn x + O(x
n=0 An x =
Small challenge: can you prove this?
1.4
ExPD03 statement: Solve the one directional wave equation
Consider the one directional wave equation for u = u(x, t), where c is a constant:
ut + cux = 0.
(1.3)
Introduce the new independent variables η = x − c t and ξ = x + c t, and change variables to write
the equation for u as a function of these new variables: u = u(η, ξ). Using the transformed form of
the equation, integrate it to show that it must be
u = f (η),
(1.4)
for some arbitrary function f . Hence, the solutions to (1.3) must have the form u = f (x − c t).
1.5
Statement: TFPa02. Near constant initial data (linearize)
Let q =
4 qm
ρ2j
ρ (ρj − ρ), where qm is the maximum flow rate and ρj is the jamming concentration.
At time t = 0, the concentration is specified by
ρ(x, 0) = ρ0 + ρj f (x),
2
(1.5)



0

1−

for |x| ≥ D, 

and D > 0 is some constant distance.
(1.6)
x2

for |x| ≤ D, 
2
D
If 0 < 1 is some small number (e.g.: = 0.1), use linear theory to determine the concentration
ρ(x, t) at a later time. In particular, set ρ0 = 0.75 ρj , find the wave speed and draw in space–
time (x–t plane) the characteristics marking the front and back of the disturbance. On the same
graph, plot the trajectories of the vehicles (which move with the space average velocity u) whose
initial positions are x = 0 and x = 2 D.
where f (x) = 
Repeat the exercise for ρ0 = 0.25 ρj and ρ0 = 0.5 ρj . What is un-usual about the last case?
For the purposes of the plotting, assume space and time units where D = 1 and qm = ρj —
questions: is this possible? What are the units?
1.6
Statement: TFPa06. Solve a first order linear pde
If a 6= 0, b, and c are constants, show that any function u = u(x, y) that satisfies the partial differential equation
a ux + b uy = c u ,
(1.7)
must be of the form
u = ec x/a F (b x − a y) ,
(1.8)
where F is an arbitrary function.
Hint: Use ξ = b x − a y and η = x as the new independent variables.
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2.1
Special Problems.
ExID73 statement: Change of variables for a pde
Let u = u(x, t) be a solution of the heat
ut = uxx .
(2.9)
What equation does φ = − u1 ux satisfy?
Hint. Calculate φt and use the equation for u. Calculate φx and write it in terms of u, uxx , and φ2 .
Then compute φxx . You should now be able to write φt in terms of φ, φx , and φxx .
2.2
Statement: TFPa07. Solve a first order linear pde
If c and β are a constants, find the solution of
ut + c ux = −β 2 u ,
with initial condition u(x, 0) = sin x.
THE END.
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(2.10)