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Question 2. (20 marks) Consider the one-dimensional wave equation $=c2%, 0<x<1 and :90, (la) subject to the mixed boundary conditions
a(0, t) = 6xu(l, t) = 0. (1b)
(a) Use separation of variables to determine the general (formal) solution of (la) and (1b). (3 marks) (b) Use eigenfunction orthogonality to find a (formal) solution that also satisfies the initial conditions u(x, 0) = x(3 - x2), 6,u(x, 0) = x(x - 2). (1c) [To make life easier, you may use a computer to calculate at most two integrals. You must indicate which with a “(by computer)” remark in your solution.] (5 marks)
MAST20030, Assignment 3 2
(c) Show that this is a genuine solution for x E [0, 1] and t 2 0. (6 marks) (d) Prove that the solution to the IBVP (la—c) is unique by employing the “energy method” for an appropriate
function e, where the energy functional is defined by E[v](t) = g f I [(6,0(x, n)2 + c2(6xv(x, of] dx.
0 You will need to demonstrate that this functional is 0 for all t 2 0 and then show why this implies that any
solution to (la—c) is unique. (6 marks)

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Part a Let us consider t 2 2 u = c 2 x 2 2 u ------------------------------(**) we seek a ... View the full answer

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