MATH 212 Final, Fall 2007

Math 212 , Fall 2007 Instructor : Friedemann Brock Final Exam, 28 January 2008 1. (10 points) Consider the Dirichlet problem in a square, (P) uxx + uyy = −F(x, y), (0 ≤ x ≤ π, 0 ≤ y ≤ π), u(0, y) = u(π, y) = u(x, 0) = u(x, π) = 0, where F is a given continuous function. Find a formula for the solution u of (P) in form of a double Fourier series which involves terms of the form sin mx sin ny, (m, n ≥ 1). Hint : The function v(x, y) ≡ vmn(x, y) := (m2 + n2)−1 sin mx sin ny satisfies the equation vxx + vyy = − sin mx sin ny. 2. Find the Fourier transforms of the following functions: (a) (10 points) f1(x) = � x if − 1 < x < 1 0 otherwise. (b) (10 points) f2(x) = 2e−(x−1)2/2. 3. Consider the following heat problem with dissipation: (Q) ∂u ∂t = ∂2u ∂x2 − tu, (x ∈ IR, t > 0), u(x, 0) = e−|x|, (x ∈ IR), u(x, t) → 0 as |x| → ∞, uniformly in t. (a) (10 points) Let ˆu denote the Fourier transform of u w.r.t. the variable x. Obtain an ODE for ˆu. Show that ˆu(ξ, t) = e−ξ2t−(t2/2) 2 1 + ξ2. (1) (b) (5 points) Use formula (1) to show that the solution of (Q) is given by u(x, t) = 2 πe−t2/2 � +∞ 0 e−ξ2t cos ξx 1 + ξ2 dξ. (2) (c) (5 points) Use the well-known property F−1 � ˆf(ξ)ˆg(ξ) � = (f ∗ g)(x), (3) (|f|2, |g|2 integrable), to obtain an alternative representation formula for the solution u of (Q).