Approximate the solution to the following partial differential equation using the Backward-Difference method.
∂u / ∂t - 1 / 16 ∂2u / ∂x2 = 0, 0 < x < 1, 0 < t;
u(0, t) = u(1, t) = 0, 0 < t, u(x, 0) = 2 sin 2πx, 0≤ x ≤ 1.
Use m = 3, T = 0.1, and N = 2, and compare your results to the actual solution u(x, t) = 2e−(π2/4)t sin 2πx.