If the left end of a laterally insulated bar extending from x = 0 to x = 1 is insulated, the boundary condition at x = 0 is u_n(0, t) = u_x(0, t) = 0. Show that, in the application of the explicit method given by (5), we can compute u_oj+1 by the formula

u_oj+1 = (1 - 2r)u_0j + 2_ru1j.

Apply this with h = 0.2 and r = 0.25 to determine the temperature u(x, t) in a laterally insulated bar extending from x = 0 to 1 if u(x, 0) = 0, the left end is insulated and the right end is kept at temperature g(t) = sin 50/3πt. Use 0 = ∂_u0j/∂x = (u_1j - u_-1j)/2h.