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Math 425 Exam 1 Jerry L. Kazdan
March 3, 2011 12:00 – 1:
Directions This exam has three parts, Part A, short answer, has 1 problem (12 points). Part B has 5 shorter problems (7 points each, so 35 points). Part C has 3 traditional problems (15 points each so 45 points). Total is 92 points. Closed book, no calculators or computers– but you may use one 3′′^ × 5 ′′^ card with notes on both sides.
Part A: Short Answer (1 problems, 12 points).
- Let S and T be linear spaces and A : S → T be a linear map. Say V and W are particular solutions of the equations AV = Y 1 and AW = Y 2 , respectively, while Z 6 = 0 is a solution of the homogeneous equation AZ = 0.
Answer the following in terms of V , W , and Z. a) Find some solution of AX = 3Y 1. b) Find some solution of AX = − 5 Y 2. c) Find some solution of AX = 3Y 1 − 5 Y 2. d) Find another solution (other than Z and 0) of the homogeneous equation AX = 0. e) Find two solutions of AX = Y 1. f) Find another solution of AX = 3Y 1 − 5 Y 2.
Part B: Short Problems (5 problems, 7 points each so 35 points)
B–1. U = (1, 1 , 0 , 1) and V = (− 1 , 2 , 1 , −1) are orthogonal vectors in R^4.
Write the vector X = (1, 1 , 1 , 2) in the form X = aU + bV + W , where a, b are scalars and W is a vector perpendicular to U and V.
B–2. Find u(x, t) that satisfies ux − 2 ut = 1 with u(x, 0) = 0.
B–3. Let u(x, t) be a solution of the wave equation
utt = 4uxx, for − ∞ < x < ∞, t ≥ 0 ,
with the (continuous) initial conditions
u(x, 0) = f (x), ut(x, 0) = g(x).
Find the largest interval J = {a ≤ x ≤ b} where changing f (x) or g(x) at any point of J can change (“influence”) the value of u(0, 3). In other words, in the (x, t) plane, find all the points on the x-axis that are in the domain of dependence of (0, 3).
B–4. Find the general solution u(x, y) of uxy = 4y.
B–5. Let u(x, y) and v(x, y) be a solutions of the Laplace equation ∆u = 0, ∆v = 0 in a bounded region Ω in the plane. If u > v on the boundary of Ω, what, if anything, can you conclude about the relationship between u and v inside Ω? Justify your assertion.
Part C: Traditional Problems (3 problems, 15 points each so 45 points)
C–1. Find the motion u(x, t) of a clamped string { 0 ≤ x ≤ π}
utt = uxx,
with initial and boundary conditions:
u(x, 0) = 0, ut(x, 0) = 15 sin 5x, and u(0, t) = u(π, t) = 0.
C–2. Let u(x, y) satisfy ∆u − u = 0 in a bounded region Ω ⊂ R^2 with u(x, y) = 0 on the boundary of Ω. Use Green’s identity to show that u(x, y) = 0 throughout Ω.
C–3. Let u(x, t) be the temperature of a rod of length L that satisfies
ut = uxx − ru for 0 < x < L, t > 0 ,
where r > 0 is a constant [this is related to the heat equation but assumes that heat radiates out into the air along the rod]. Assume u satisfies the initial condition u(x, 0) = f (x).
Define the total heat energy by E(t) = (^12)
∫ L
0
u^2 (x, t) dx.
a) If u also satisfies the Dirichlet boundary conditions
u(0, t) = 0, u(L, t) = 0
(the ends of the rod are held at temperature 0), show that E(t) is a decreasing function of t. b) Show that even if u satisfies Neumann boundary conditions
ux(0, t) = 0, ux(L, t) = 0
(the ends of the rod are insulated), E(t) is still a decreasing function of t. c) [Extra credit!] Show that in either of the above cases lim t→∞
E(t) = 0.
