MATH 212 Exam 1, Spring 2016

Quiz 2: MATH 212 Instructors: Mohammad El Smaily & Wael Mahboub November 6, 2015 Duration: 70 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 /10 2 /20 3 /15 4 /40 5 /15 Total /100 1 [10 points] Problem 1. Consider the function F(x) defined over R by F(x) = ∞ � n=0 an cos �nπx 7 � + bn sin �nπx 7 � , where, for each n ≥ 0, an and bn are given by an = 1 7 � 7 −7 e−4x cos �nπx 7 � dx and bn = 1 7 � 7 −7 e−4x sin �nπx 7 � dx. (a) (5 points) Find F(0). Then, deduce the value of the sum ∞ � n=0 an. (Justify your answers). (b) (5 points) Find F(7) and then deduce the value of the sum ∞ � n=0 (−1)nan. Page 2 of 10 [20 points=12+8] Problem 2. Consider the heat problem with homogeneous Dirichlet boundary conditions ut = uxx for t > 0, 0 < x < 2; u(t, 0) = u(t, 2) = 0; u(0, x) = x for 0 ≤ x ≤ 2. (a) (12 points) Write down the solution u(t, x) in a series format. Page 3 of 10 (b) (8 points) Find a lower bound for the time needed so that u(t, x) is approximately − 4 πe−π2t/4sin(πx 2 ) with an error less than 0.01. Page 4 of 10 [15 points] Problem 3. Consider the sequence of continuous functions fn(x) =            1, x ≥ 0 nx + 1, −1 n ≤ x < 0 0, x < −1 n. (a) (8 points) Find the pointwise limit f(x) over R and sketch its graph. (b) (7 points) Does {fn(x)}n converge uniformly to f over R? Justify your answer. Page 5 of 10 [40 points] Problem 4. Let u(t, x) be the solution to the heat problem ut(t, x) = uxx + bu + c, t > 0, 0 < x < 1 u(t, 0) = u(t, 1) = 0, t > 0 u(0, x) = 400, where b and c are two constants. Denote the equilibrium solution of the above problem by uE(x) and let v(t, x) = e−bt(u(t, x) − uE(x)). (a) (10 points) Show that v(t, x) satisfies the simpler PDE vt = vxx, for all t > 0, x ∈ (0, 1) with boundary conditions v(t, 0) = v(t, 1) = 0. Page 6 of 10 (b) (10 points) Set b = π2 4 . Show that equilibrium solution uE(x) is given by uE(x) = 4c π2 � cos �πx 2 � + sin �πx 2 � − 1 � Help: you will need to solve an inhomogeneous ordinary differential equation (after setting b = π2 4 ). Page 7 of 10 (c) (15 points) You can assume the result of part (a) here. Write down v(t, x) in a series format. Help: make sure that you compute the initial condition v(0, x) in order to find the coefficients of the series. You may also use the formulæ cos �πx 2 � sin nπx = 1 2 [sin(n + 1/2)πx + sin(n − 1/2)πx] and sin �πx 2 � sin nπx = 1 2 [cos(n − 1/2)πx − cos(n + 1/2)πx] . Page 8 of 10 (d) (5 points) Conclude a series formula for the solution u(t, x) of the original problem, where b = π2 4 . Page 9 of 10 [15 points] Problem 5. Compute the Fourier series of the function sign(x) defined by 1 for x ≥ 0, and −1 for x < 0. Page 10 of 10