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MATH 333: Partial Differential Equations
Problem Set 2, Final version
Due Date: Tues., Sept. 21, 2010
Read Sections 2.1–2.2 from the Olver text. Along with that, you might read Lesson 19
from Farlow’s book.
1.13 Suppose u(t, x), v(t, x) are C2 functions defined on R2 and satisfy the first order system
of partial differential equations ut = vx , vt = ux .
(a) Show that both u and v are classical solutions to the wave equation wtt = wxx .
Which result from vector calculus do you need to justify the conclusion?
{b} [Optional; a very good exercise using vector calculus] Conversely, given a
classical solution u(t, x) to the wave equation, can you construct a function
v(t, x) such that u(t, x), v(t, x) form a solution to the first order system?
2.1.3 Find the general solution u(t, x) to the following PDEs. Consider the domain of
definition to be x ∈ R, t > 0.
(c) ut = x − t,
(e) ux + tu = 0,
(f) utt + 4u = 1.
∂2 u
2.1.6 Solve the partial differential equation
= 0, (x, y) ∈ R2 , for u(x, y).
∂x ∂y
?8 Suppose we wish to solve the 1st-order linear nonhomogeneous Cauchy problem
ut + a(x, t)ux + b(x, t)u = f (x, t), x ∈ R, t > 0
u(x, 0) = φ(x),
x ∈ R.
(1)
Consider the related problems
and
vt + a(x, t)vx + b(x, t)v = 0, x ∈ R, t > 0
v(x, 0) = φ(x),
x ∈ R,
(2)
wt + a(x, t)wx + b(x, t)w = f (x, t), x ∈ R, t > 0
w(x, 0) = 0,
x ∈ R.
(3)
MATH 333
Problem Set 2
2
Without attempting to solve problems (2), (3) (i.e., without expressions for their
solutions v and w), show that when v and w solve (2), (3) respectively, then u = v + w
solves (1). (Don’t forget to verify that u satisfies the appropriate initial condition.)
2.2.2 Solve the initial value problems and graph the solutions at times t = 1, 2 and 3:
2
(a) ut − 3ux = 0, x ∈ R, t > 0, u(0, x) = e−x ;
1
(c) ut + ux + u = 0, x ∈ R, t > 0, u(0, x) = arctan x.
2
2.2.5 Solve the initial value problem ut + 2ux = sin x,
2.2.17 (a) Solve the initial value problem ut − xux = 0,
x ∈ R, t > 0,
x ∈ R, t > 0,
u(0, x) = sin x.
u(0, x) = (x2 + 1)−1 .
(b) Graph the solution at time t = 0, 1, 2, 3.
(c) What is limt→∞ u(t, x)?
2.2.12 A sensor at position x = 1 monitors the concentration of a pollutant u(t, 1) as a function of t for t ≥ 0. Assuming the pollutant is transported with wave speed c = 3, at
what positions x can you determine the initial concentration u(0, x)?
PS2—Final version