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APPLIED ANALYSIS/NUMERICAL ANALYSIS QUALIFYING
EXAMINATION
JANUARY 2009
Policy on misprints. The qualifying examination committee tries to proofread the ex- aminations as carefully as possible. Nevertheless, there may be a few misprints. If you are convinced that a problem has been stated incorrectly, indicate your interpretation in writing your answer. In such cases, do not interpret the problem so that it becomes trivial.
- Part I: Applied Analysis
Work 3 out of 4 problems of this part.
Problem 1. Consider the equation Lu = f, λ 1 (u) = 0, λ 2 (u) = 0, where L is a sec- ond order linear differential operator. A Green’s function g(x, y) for L must satisfy λ 1 (g(x, y)) = 0, λ 2 (g(x, y)) = 0, where y is fixed, λ 1 and λ 2 are liner functionals, and g is considered as a function of x.
(a) List the other properties g(x, y) must satisfy.
(b) Consider the equation u′′(x) = f (x), u(0) = 0,
0 u(t)dt^ = 0.^ Find the Green’s function for this equation. (Hint: the Green’s function has the form u 1 (·)u 2 (·) where u 1 is a solution to u′′(x) = 0, u(0) = 0 while u 2 is a solution to u′′(x) = 0.
(c) Write down a solution to u′′(x) = f (x), u(0) = 0,
0 u(t)dt^ = 0.
Problem 2. (a) State the Courant Minimax Principle.
(b) Prove an inequality relating the eigenvalues of a symmetric matrix before and after one of its diagonal elements is increased.
(c) Use this inequality and the minimax principle to show that the smallest eigenvalue of the following matrix is negative:
1
2
Problem 3. Let K be a compact, self-adjoint operator on a Hilbert space H and suppose (I − λK) is bounded below, i.e., inf||u||=1 ||(I − λK)u|| > 0.
(a) Explain why (I − λK)u = f can always be solved whenever f ∈ H.
(b) Explain how to solve (I − λK)u = f explicitly in terms of the eigenfunctions of K.
Problem 4. (a) Prove the following theorem: If {Pn} is a sequence of projections with the property that ||Pnu − u|| → 0 as n → ∞ for every u ∈ H, and if (I − λK)−^1 exists, then un, the solution of (I−λK)un = Pnf , converges to the solution of (I−λK)u = f as n → ∞.
(b) Apply this theorem to sketch a way to find an approximate solution of the integral equation
u(x) +
0
k(x, y)u(y)dy = f (x)
using piecewise linear finite elements. For simplicity assume k(x, y) and f (x) are contin- uous functions of their arguments and define φk(x) to be the piecewise linear continuous functions with φk(xj ) = δk,j and linear on all the intervals [xj , xj+1] where xj = j/n. Also assume that {Pn} are interpolating projections.
- Part II: Numerical Analysis
Work 2 out of 3 problems of this part.
Problem 1. Let Ω = (0, 1) and u be the solution of the boundary value problem
u(4)^ − (k(x)u′)′^ + q(x)u = f (x), for x ∈ Ω, u(0) = u′′(0) = 0, u(1) = 0, u′′(1) + βu′(1) = γ,
where k(x) ≥ 0, q(x) ≥ 0, f (x), γ, and β > 0 are given data.
(a) Derived the weak formulation of this problem. Specify the appropriate Sobolev spaces and show that the corresponding bilinear form is coercive.
(b) Suggest a finite element approximation to this problem using piece-wise polynomial functions over a uniform partition of Ω into subintervals with length h.
(c) Derive an error estimate for the FE solution.
Problem 2. Let Ω = (0, 1)^2 and u be the solution of the second order elliptic problem:
−∆u := −ux 1 x 1 − ux 2 x 2 = f (x), for x ∈ Ω, ∂u ∂n
