Description

Recall that, the solution u for 1D heat equation
( 2
∂tu = ∂xu, in (t,x) ∈ [0,∞) × R,
u|t=0 = φ(x), for x ∈ R,
is given by
Z
u(t,x) = S(t,x − y)φ(y)dy, where.
R
1. What are the types of the following equations.
(a) (3 points)
(b) (3 points) 9∂x2u + 6∂xyu + ∂y2u + ∂xu = 0.
(c) (4 points) 4∂x2u − 12∂xyu + 9∂y2u + ∂yu = 0.
2. Solve the following PDE.
(a) (5 points) ∂xu + 2∂yu − 4u = ex+y with u(x,0) = sin(x2).
(b) (5 points) = 0 with u(0,x) = sinx.
(c) (5 points) x∂tu − t∂xu = u, t,x > 0 with u(0,x) = x2.
3. (a) (5 points) State the definition of a well-posed PDE problem.
(b) (5 points) Is the following problem well-posed? Why?
, for x ∈ B1(0), for x ∈ ∂B1(0).
(c) (10 points) Verifying that solves the following problem
for (t,x) ∈ (−∞,+∞) × (0,π),
, for t ∈ (−∞,+∞),
for x ∈ [0,π],
1
for all positive integer n. How does the energy change when t → ±∞.
(d) (10 points) Is the following problem well-posed? Why?
∂tv(t,x) = ∂x2v(t,x),
 v(t,0) = v(t,π) = 0,
 v(0,x) = 0, for (t,x) ∈ (−∞,0) × (0,π),
for t ∈ (−∞,0), for x ∈ [0,π].
4. Derive the solution formula for the following problems.
(a) (10 points) Solve
(∂tv(t,x) = ∂x2v(t,x) + v(t,x), for (t,x) ∈ R+ × R,
v(0,x) = φ(x),
(b) (10 points) Solve for t = 0,
∂tv(t,x) = ∂x2v(t,x),
 v(0,x) = φ(x),
 v(t,0) = 0,
(c) (10 points) Solve for (t,x) ∈ R+ × R+,
for t = 0, for x = 0.
∂tv(t,x) = ∂x2v(t,x),
 v(0,x) = φ(x),
∂xv(t,0) = 0, for (t,x) ∈ R+ × R+,
for t = 0, for x = 0.
5. (7 points) Suppose u is harmonic in B1(0) {0} ⊂ R2 and satisfies
u(x) = o(log(|x|)), as |x| → 0.
Show that u can be defined at 0 so that it is C2 and harmonic in B1(0).
6. (8 points) Consider the following exterior Dirichlet problem
(∆u(x) = 0, for x ∈ R3 B1(0),
(1)
u(x) = 0, for x ∈ ∂B1(0). Show that there exists unique solution u to (1) such that
.
*************** END OF THE QUESTIONS ***************
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