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MATH 401 FINAL EXAM – April 13, 2012
No notes or calculators allowed. Time: 2.5 hours. Total: 50 pts.
- Consider the ODE problem for u(x): { u′′^ + u′^ − 2 u = f (x), 0 < x < 1 u′(0) + u(0) = 0, u(1) = 2
(a) (4 pts.) Write down the problem satisfied by the Green’s function Gx(y) = G(x; y) for problem (1) (but do not try to solve it). (b) (2 pts.) Express the solution u(x) of (1) in terms of Gx(y). (c) (4 pts.) Now suppose the BC at x = 1 is changed to u(1) + au′(1) = 2. For what value of a is a solvability condition on f (x) required for (1).
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- (a) (3 pts.) Write down the problem satisfied by the Green’s function G(y, τ ; x, t) for the following initial - boundary value problem for the heat equation on an interval: (^)
∂u ∂t =^
∂^2 u ∂x^2 0 < x < π, t >^0 u(x, 0) = u 0 (x) 0 ≤ x ≤ π, u(0, t) = 0 u(π, t) = 0
and express the solution u(x, t) in terms of the Green’s function. (b) (4 pts.) Find the Green’s function as an eigenfunction expansion. (c) (3 pts.) Suppose u 0 (x) ≥ 0, u 0 6 ≡ 0. Use the maximum principle for the heat equation to show that u(x, t) > 0 for t > 0.
- Let D = (0, 1) × (0, 1) = {(x 1 , x 2 ) | 0 < x 1 < 1 , 0 < x 2 < 1 } be the unit square in R^2 , let f (x) be a smooth function on D, and consider the variational problem
min u∈C^2 (D)
D
(1 + x 1 )|∇u(x)|^2 − f (x)u(x)
dx.
(a) (3 pts.) Determine the problem (Euler-Lagrange equation plus BCs) that a minimizing function would solve. (b) (3 pts.) Is there a solvability condition on f (x) required for the Euler- Lagrange problem to have a solution? If so, what is the effect of this condition in the original variational problem? (c) (4 pts.) For f (x) = x 1 x 2 − 1 /4, find an approximate minimizer, using a Rayleigh-Ritz approach, with three trial functions v 1 (x) = 1, v 2 (x) = x 1 , v 3 (x) = x 2.
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