1 Given Name: Family Name: I.D.# MAT3320 Assignment 4 Total: 10 marks. Due date: Tuesday, July 18, on or before 4:00pm. In MATH Department (585 King Edward), there is a Drop-Box. You need to put your assignment into the box on or before 4:00pm on the due date. Late assignments will not be accepted. 1. (2 marks) The solution of Laplace's equation uxx + uyy = 0, 0 < x < L, 0 < y < M with the boundary conditions u(x, M ) = f (x), u(x, 0) = u(0, y) = u(L, y) = 0 is given by u(x, y) = X bn sinh \u0010 ny \u0011 n=1 L sin \u0010 nx \u0011 L . Find the solution of Laplace's equation uxx + uyy = 0 within R = {(x, y) : 0 < x < 3, 0 < y < 2} with BC : u(x, 0) = 0, u(x, 2) = x 3, u(0, y) = 0, u(3, y) = 0. 2. (3 marks) The solution of utt = c2 (uxx + uyy ), (x, y) R = [0, a] [0, b], t > 0, subject to BC : u(x, y, t) = 0 for t > 0 and (x, y) R (boundary of R), is u(x, y, t) = X X (Amn cos[mn t] + Bmn sin[mn t]) sin n=1 m=1 where mn = c q m2 a2 + mx ny sin , a b n2 . b2 Solve utt = 25(uxx + uyy ), (x, y) R = [0, 3] [0, 2], t > 0, subject to BC : u(x, y, t) = 0 for t > 0 and (x, y) R , ICs : u(x, y, 0) = 0, ut (x, y, 0) = sin(3x) sin(4y), (x, y) R. 3. (2 marks) Find the solution u(r, ) of 2 1 cot urr + ur + 2 u + 2 u = 0, r r r such that u(0, ) is bounded, and u(4, ) = cos(2). r < 4, 2 4. (3 marks) Evaluate Z 3 x5 J2 (2x)dx. 0 Remark. Your solution should be a linear combination of J3 (6) and J4 (6)