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Applied/Numerical Analysis Qualifying Exam
January 11, 2010
Policy on misprints: The qualifying exam committee tries to proofread exams as carefully as possible. Nevertheless, the exam may contain a few misprints. If you are convinced a problem has been stated incorrectly, in- dicate your interpretation in writing your answer. In such cases, do not interpret the problem so that it becomes trivial.
Part 1: Applied Analysis
Instructions: Do any 3 of the 4 problems in this part of the exam. Show all of your work clearly. Please indicate which of the 4 problems you are skipping.
- Let Lu = (^) dxd
(1+x)dudx
. Find the Green’s function for Lu = f , u(0) = 0 and u′(1) = 0.
- This problem concerns Mallat’s multiresolution analysis (MRA).
(a) Define the term multiresolution analysis. For the Haar MRA, state the scaling function φ, the wavelet ψ, the approximation spaces Vj , the dilation (or scaling) relation, and the wavelet spaces Wj. (b) Use the scaling and wavelet coefficients given below to derive the decomposition and reconstruction formulas for the Haar MRA.
sjk = 2j
R
f (x)φ(2j^ x − k)dx and djk = 2j
R
f (x)ψ(2j^ x − k)dx.
(c) Let f be compactly supported and continuous on R. Show that sjk is the average of f (x) over the interval [k · 2 −j^ , (k + 1) · 2 −j^ ], where sjk is given in part 2b. What role does this formula play in the initialization step of a wavelet analysis? (One or two sentences will suffice.)
- A chain having uniform linear density ρ = 1 hangs between the points (-1,0) and (1,0). (The positive y direction is downward; the acceleration due to gravity is g = 1.) The total mass m, which is fixed, and the total energy E of the chain are
m =
− 1
1 + y′^2 dx > 2 and E[y] =
− 1
y
1 + y′^2 dx
Assuming that the chain hangs in a shape that minimizes the energy, find the shape of the hanging chain. (Hint: the integrand of the func- tional to be minimized doesn’t depend on x.)
- Let H be a complex (separable) Hilbert space, with 〈·, ·〉 and ‖ · ‖ being the inner product and norm.
(a) Let λ ∈ C be fixed. If K : H → H is a compact linear operator, show that the range of the operator L = I − λK is closed. (b) Briefly explain why the operator Ku(x) :=
0 (3 + 4xy
(^2) )u(y)dy is compact on H = L^2 [0, 1]. Determine the values of λ ∈ C for which u = f + λKu has a solution for all f ∈ L^2 [0, 1]. State the theorem that you are using to answer the question.
(a) Which of the two elements is unisolvent? Prove it! (b) Show that the unisolvent element leads to a finite element space, which is not H^1 -conforming.
- Consider the following initial boundary value problem: find u(x, t) such that ut − uxx + u = 0 , 0 < x < 1 , t > 0 ux(0, t) = ux(1, t) = 0 , t > 0 u(x, 0) = g(x), 0 < x < 1.
(a) Derive the semi-discrete approximation of this problem using lin- ear finite elements over a uniform partition of (0, 1). Write it as a system of linear ordinary differential equations for the coefficient vector. (b) Further, derive discretizations in time using backward Euler and Crank-Nicolson methods, respectively. (c) Show that both fully discrete schemes are unconditionally stable with respect to the initial data in the spatial L^2 (0, 1)-norm.
