Consider the implicit central finite difference scheme U n 1 j U n j = (U n 1 j1 2U n 1 j U n 1 j 1 ), j = 1, . . . , J 1, n = 1, 2, . . . for the model problem (EQ) ut uxx = 0, (x, t) = (0, 1) (0, ) (BC) u(0, t) = u(1, t) = 0, t 0 (1) (IC) u(x, 0) = u0(x), 0 x 1. 1 (a) Prove the implicit difference system can be written in the vector form as (I B)Un 1 = Un, n = 0, 1, . . . . where Un = [U n 1 , U n 2 , . . . , U n J1]T , B = 2 1 1 2 . . . . . . . . . 1 1 2 , J = 1 x, = t (x)2 . (b) Use the spectral decomposition B = Y Y 1 = Y P 1Y T , = 1 . . . J1 , where Y = sin x1 sin 2x1 . . . sin(J