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MATH 401 FINAL EXAM – April 18, 2011
No notes or calculators allowed. Time: 2.5 hours. Total: 50 pts.
- Consider the ODE problem for u(x): { Lu := (p(x)u′)′^ + q(x)u = f (x), 1 < x < 2 u(1) = 2, u(2) = 5
(p(x), q(x), and f (x) are given functions).
(a) (2 pts.) Write down the problem satisfied by the Green’s function Gx(y) = G(x; y) for problem (1). (b) (2 pts.) Express the solution u(x) of (1) in terms of the Green’s function Gx(y). (c) (3 pts.) If p(x) = 1/x^2 and q(x) = 2/x^4 , find the Green’s function G(x; y) for (1). (d) (3 pts.) Again with p(x) = 1/x^2 and q(x) = 2/x^4 , find the solvability condition on f (x) for (1) if the boundary conditions are changed to u′(1) = 2, u(2) = 5.
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- Let D be a bounded domain in Rn, and let {φj (x)}∞ j=1 be a complete, orthonormal set of eigenfunctions for −∆ on D with zero (Dirichlet) BCs: −∆φj (x) = λj φj (x), λ 1 ≤ λ 2 ≤ λ 3 ≤ · · ·. (a) (4 pts.) Write down the problem satisfied by the Green’s function G(y, τ ; x, t) for the following problem for a heat-type equation
ut = ∆u + u x ∈ D, t > 0 u(x, 0) = u 0 (x) x ∈ D u(x, t) ≡ 0 x ∈ ∂D
and express the solution u(x, t) in terms of the Greens function. (b) (4 pts.) Find the Green’s function as an eigenfunction expansion. (c) (2 pts.) Under what condition will typical solutions grow with time?
- Consider the variational problem
min u∈C^4 ([0,1])
0
(u′′(x))^2 +
(u(x))^2 − exu(x)
dx.
(a) (5 pts.) Determine the problem (Euler-Lagrange equation plus BCs) that a minimizer u(x) solves. (b) (5 pts.) Find an approximate minimizer, using a Rayleigh-Ritz approach, with two trial functions v 1 (x) = ex, v 2 (x) = e−x.
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