MATH 212 Final, Spring 2016

Final Exam : MATH 212 (Introductory PDEs) Instructor: Wael Mahboub May 13, 2016 Duration: 120 minutes Name (Last, First): Student number: For marker’s use only Problem Score 1 /10 2 /20 3 /10 4 /15 5 /15 6 / 20 7 / 10 Total /100 1 [10 points=5+5] Problem 1. Consider the following PDE ut = uxx + 2ux − 3u. (a) Let v(t, x) = e3tu(t, x). Use the PDE satisfied by u(x, y) to show that v satisfies the PDE. vt + 2vx = vxx. (1) (b) Let ξ = x − 2t, and W(t, ξ) = v(t, x). Use the PDE satisfied by v(t, x) and chain rule to show that W satisfies a heat equation. Page 2 of 12 [25 points=3+2+5+5+5] Problem 2. Consider the following boundary value problem uxx + uyy − u = 0, u(0, y) = u(π, y) = u(x, 0) = 0, u(x, π) = f(x). (a) Show that separating as u(x, y) = v(x).w(y), we get a seperation constant λ and two ordinary differential equations v′′ − v − λv = 0. (2) w′′ + λw = 0 (3) (b) Write the boundary conditions that v and w must satisfy. Page 3 of 12 (c) Show that the ODE (2) admits a non trivial solution, if and only if λ = −1 − n2 where n is a integer. Page 4 of 12 (d) Show that for each positive integer n, for λ = −1 − n2, the ODE (3) has a solution involving the hyperbolic sine function (Hint: Use w(0) = 0). (e) Conclude a series solution of the form u(x, y) = ∞ � n=1 cnun(x, y) where the constants cn must be computed in terms of the function f. Page 5 of 12 [10 points] Problem 3. Solve the following boundary value problem. ����� ∆u = 0, x2 + y2 < 1, u(x, y) = x, x2 + y2 = 1. . Where ∆ is the Laplace Operator (You’re not required to give a proof of your formulas). Page 6 of 12 [15 points=10+5] Problem 4. Let D be the square {(x, y) ∈ R2/ 0 ≤ x < π, 0 ≤ y < π}. 1. Solve the heat problem �������� ut = uxx + uyy, (x, y) ∈ D, u(t, 0, y) = u(t, π, y) = u(t, x, 0) = u(t, x, π) = 0, u(0, x, y) = f(x, y), t ≥ 0. Where f is a continuous function in D. (You’re not required to give a proof of your formulas). Page 7 of 12 2. Give the solution in the case f(x, y) = 2sin(x)sin(2y) − 3sin(2x)sin(5y). [15 points=6+6+3] Problem 5. Let f(x) = � 2 for − 3 ≤ x ≤ 3 0 otherwise (a) Find the Fourier transform �f(k). Page 8 of 12 (b) Find f ∗ f(x). (c) Find the Fourier transform � f ∗ f(k). Page 9 of 12 [20 points=6+7+7] Problem 6. Consider the following heat problem. �������� ut = uxx − 2tu, (x ∈ R, t ≥ 0), u(0, x) = e−x2, u(t, x) → 0 as |x| → ∞, uniformly in t. 1. Show that the Fourier transform of e−x2 is given by e−x2 = e −k2 4 √ 2 . (Hint: Use � ∞ −∞ e−x2dx = √π). 2. Let ˆu denote the Fourier transform of u with respect to x. Show that ˆu(t, k) = e−k2(t+ 1 4 )−t2 √ 2 . Page 10 of 12 3. Show that the solution is given by u(t, x) = e−t2 √π � ∞ 0 e−k2(t+ 1 4 )cos(kx)dx. [10 points=6+4] Problem 7. Let f(x) = sin x for 0 ≤ x < π. (a) Expand the function f(x) = sin x in a Fourier cosine series in the range 0 ≤ x < π. Page 11 of 12 (b) Use the series to evaluate the sum ∞ � k=1 1 4k2 − 1. Page 12 of 12