Approximate the solution to the wave equation
∂2u / ∂t2 − ∂2u / ∂x2 = 0, 0 < x < π, 0 < t;
u(0, t) = u(π, t) = 0, 0 < t,
u(x, 0) = sin x, 0≤ x ≤ π,
∂u / ∂t (x, 0) = 0, 0 ≤ x ≤ π,
using the Finite-Difference Algorithm with h = π/10 and k = 0.05, with h = π/20 and k = 0.1, and then with h = π/20 and k = 0.05. Compare your results at t = 0.5 to the actual solution u(x, t) = cos t sin x.