MATH 224 Final, Fall 2005

Math 224 – Final Exam (Fall 05) B. Shayya Problem 1 For 0 < ϵ ≤ π, let fϵ be the function in H(T) given on [−π, π) by fϵ(x) =      1 − |x| ϵ if |x| ≤ ϵ, 0 if ϵ < |x| ≤ π. (i) (5 pts) Show that �f(n) =      ϵ 2π if n = 0, 2 sin2(nϵ/2) πϵn2 if n = ±1, ±2, . . .. (ii) (3 pts) Find the Fourier series of fπ (so here, ϵ = π). (iii) (4 pts) Write the series of part (ii) in real form. (iv) (4 pts) Show that ∞ � k=0 1 (2k + 1)2 = π2 8 . Problem 2 Let f ∈ C(T) ∩ PS(T) and consider the BVP (∗)      ut(x, t) = uxx(x, t), (x, t) ∈ R × (0, ∞) u(x, 0) = f(x), x ∈ R. (i) (12 pts) Use Fourier series to find a solution of (∗). (ii) Put f = fπ, where fπ(x) = 1 − |x|/π is as defined in Problem 1. Notice that since fπ is real-valued and even, a solution u(x, t) of (∗) (restricted to the region [0, π]×[0, ∞) in the xt-plane) now represents the temperature at point x and time t of a one-dimensional body that occupies the interval [0, π], has initial temperature u(x, 0) = fπ(x), and is insulated along its length and at both ends. (ii)-(a) (6 pts) Use part (i) and part (ii) of Problem 1 to find the temperature u(x, t) for t > 0. (It is something of the form 1/2 + �∞ k=0 ak(t) cos(2k + 1)x.) (ii)-(b) (4 pts) Find a number T > 0 such that 0.4 < u(x, t) < 0.6 for all x when t > T. (ii)-(c) (2 pts) What point x0 of the body shows no change in temperature as t increases? (iii) (4 pts) Write the solution obtained in part (i) as a convolution with the heat kernel ht(x) = ∞ � n=−∞ e−n2teinx, x ∈ R. (iv) (4 pts) Use the functions fϵ from Problem 1 to give a heuristic argument, based on the physical properties of the diffusion of heat, to explain why the heat kernel ht(x) is expected to be nonnega- tive everywhere on R. Problem 3 Let f ∈ S(R2). Use the Fourier transform to find a solution for each of the fol- lowing PDEs. (i) (8 pts) uxx(x, y) + 2uyy(x, y) + 3ux(x, y) − 4u(x, y) = f(x, y), (x, y) ∈ R2. (ii) (8 pts) uxxxx(x, y) − uyy(x, y) + 2u(x, y) = f(x, y), (x, y) ∈ R2. Problem 4 True or False: (i) (4 pts) If f ∈ C2(T) and A ∈ C are such that f ′′(x) = Af(x) for all x ∈ R, then Im A = 0? (ii) (4 pts) If f is a bounded function in C(Rd) with ∥f∥L1(Rd) < ∞ and �f ∈ S(Rd), then f ∈ S(Rd)? (iii) (4 pts) There is a function f ∈ H(T) whose Fourier series is ∞ � n=1 einx n ? (iv) (4 pts) There is a bounded function f on Rd with ∥f∥L1(Rd) < ∞ whose Fourier transform is �f(ξ) = 1 (1 + |ξ|)d ? Problem 5 (i) (8 pts) State and prove Plancherel’s theorem. (ii) (6 pts) Let f ∈ S(R8). Prove that � R5 ���� � R8 e−2πi(ξ,0)·uf(u) du ���� 2 dξ = � R5 ���� � R3 f(x, y) dy ���� 2 dx. Problem 6 (6 pts) Suppose φ ∈ C(T), φ(x) ̸= 0 for all x ∈ R, and lim n→+∞ � 2π 0 e−inx2φ(x2) dx = 0. Prove that lim n→−∞ � 2π 0 e−inx2φ(x2) dx = 0.