Consider the one-dimensional wave equation au at au = 16- t0 ax2' with boundary conditions au(0,t)...

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Consider the one-dimensional wave equation a²u at² a²u = 16- t≥0 ax2' with boundary conditions au(0,t) = 0, xu(x,t) = 0, t > 0. and initial conditions u(x, 0) = x²(x-x)², au(x, 0) = 8(cos2x - sin² x) - 12 cos 3x. (a) Solve this IBVP. (13 marks) (6 marks) (b) Show that the solution is genuine for x = [0,л] and t≥ 0. (c) Prove that the solution to the IBVP is unique by employing the "energy method" for an appropriate function v, where the energy functional is defined by 1 F[u](t) = = = ½ √* [(a,v(x,t))² + 16(@xv(x, t))²] dx. 2 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. (6 marks) Consider the one-dimensional wave equation a²u at² a²u = 16- t≥0 ax2' with boundary conditions au(0,t) = 0, xu(x,t) = 0, t > 0. and initial conditions u(x, 0) = x²(x-x)², au(x, 0) = 8(cos2x - sin² x) - 12 cos 3x. (a) Solve this IBVP. (13 marks) (6 marks) (b) Show that the solution is genuine for x = [0,л] and t≥ 0. (c) Prove that the solution to the IBVP is unique by employing the "energy method" for an appropriate function v, where the energy functional is defined by 1 F[u](t) = = = ½ √* [(a,v(x,t))² + 16(@xv(x, t))²] dx. 2 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. (6 marks)