Question 3. (24 marks) Consider the one-dimensional wave equation Fu Fu at2 Oxs1, 120 with boundary conditions u (0, t) = 0, u(1, t) = 0, 1>0. and initial conditions u(x, 0) = x* - 2x'+x, du(x,0) = 12x sin 3xx cos 3xx (a) Solve this IBVP. (1 1 marks) (b) Show that the solution is genuine for x e [0, 1] and t 2 0. (6 marks) (c) Prove that the solution to the IBVP is unique by employing the "energy method" for an appropriate function , where the energy functional is defined by E[. ] ( 1) = = [ [(2,0(x,1)2 + (a,0 (x, t))?|dx. You will need to demonstrate that this functional is 0 for all t > 0 and then show why this implies that any solution to is unique. (7 marks)