MATH 212 Final, Spring 2009

Final Exam: MATH 212 (Introductory PDEs) Instructors: Mohammad El Smaily & Wael Mahboub December 11, 2015 Duration: 120 minutes Name (Last, First): Student number: Circle your instructor’s name and your section’s number: M. El Smaily Section: 1 (from 11:00 am to 12:00 pm) W. Mahboub Section: 2 (from 12:00 pm to 1:00pm ), Section 3 (from 1:00pm to 2:00pm) For marker’s use only Problem Score 1 /15 2 /15 3 /10 4 /12 5 /13 6 / 12 7 /13 8 /10 Total /100 1 [15 points=5+7+3] Problem 1. The goal of this problem is to solve the PDE ux − 2uy − 5u = 2x2 − 3xy − 2y2 with the initial condition u(x, 0) = ex. We will proceed as follows: (a) Let t = x − 2y and s = 2x + y and v(t, s) = u(x, y). Use the PDE satisfied by u(x, y) and the chain rule to show that v satisfies the PDE 5vt − 5v = st. (b) Use e−t as an integrating factor to solve the PDE obtained in part (a) above. Page 2 of 16 (c) Deduce u(x, y) interms of x and y (use the initial conditions in the process). Page 3 of 16 [15 points=6+6+3] Problem 2. Let f(x) = � 1 for − 1 ≤ x ≤ 1 0 otherwise and g(x) = � 1 for − 2 ≤ x ≤ 2 0 otherwise. (a) Find f ∗ g(x). Page 4 of 16 (b) Find the Fourier transform �f(k) and �g(k). Page 5 of 16 (c) Find the Fourier transform � f ∗ g(k) using a method of your choice. [10 points=5+5] Problem 3. Consider the PDE ut = uxx + t sin x, ux(t, 0) = −1, ux(t, π) = 0 u(0; x) = cos x. Let E(t) = � π 0 u(t, x)dx denote the total thermal energy at time t. (a) Compute dE(t) dt in terms of t Page 6 of 16 (b) Conclude E(t) in terms of t. Page 7 of 16 [12 points] Problem 4. Solve using a method of your choice the initial value problem ut + ux − 3u = t for t > 0, u(0, x) = x2. Page 8 of 16 [13 points=5+5+3] Problem 5. Consider the following boundary value problem �������� uxx(x, y) + uyy(x, y) − 16u(x, y) + 8(ux + 4u) = 0, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, u(x, 0) = 2 sin 3x, u(x, 1) = 0, u(1, y) = u(0, y) = 0. (a) Let v(x, y) = e4xu(x, y). Show that v(x, y) verify a planar Laplace equation (L) with suitable boundary conditions. Page 9 of 16 (b) Write v(x, y) in a series format–compute the coefficients of the trigonometric terms. (c) Deduce a series format for u(x, y). Page 10 of 16 [12 points=5+7] Problem 6. Consider the heat problem with inhomogeneous Dirichlet boundary conditions ����� ut(t, x) = uxx(t, x) for 0 ≤ x ≤ 2, t ≥ 0, u(t, 0) = 0, u(t, 2) = 2, and the initial datum u(0, x) = � x if 0 ≤ x ≤ 1 2 − x if 1 < x ≤ 2 (a) Find the equilibrium state u∗(x). Page 11 of 16 (b) Using the function v(t, x) = u(t, x) − u∗(x), find u(x, y). Page 12 of 16 [13 points=4+6+3] Problem 7. Consider the sequences of functions {fn(x)} and {gn(x)} defined by fn(x) = n nx + 1, x ∈ (0, 1) and gn(x) = � 1 + x n �n , x ∈ R. (a) Find the pointwise limits f and g of {fn(x)} and {gn(x)} respectively. Page 13 of 16 (b) Show that {fn(x)} converges to f uniformly on [a, 1) for every 0 < a < 1 but not over (0, 1). Page 14 of 16 (c) Does {gn(x)} converge uniformly to g over R? Justify your answer. Page 15 of 16 [10 points] Problem 8. Show that, if v(t, x) > 0 is any positive solution to the linear heat equation vt = γvxx, then u(t, x) = ∂ ∂x (−2γ log v(t, x)) solves Burgers’ equation ut + uux = uxx. Page 16 of 16