Use the Forward-Difference method to approximate the solution to the following parabolic partial differential equations.
a. ∂u / ∂t − ∂2u / ∂x2 = 0, 0 < x < 2, 0 < t;
u(0, t) = u(2, t) = 0, 0 < t,
u(x, 0) = sin 2πx, 0≤ x ≤ 2.
Use h = 0.4 and k = 0.1, and compare your results at t = 0.5 to the actual solution u(x, t) = e−4π2t sin 2πx. Then use h = 0.4 and k = 0.05, and compare the answers.
b. ∂u / ∂t − ∂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.
Use h = π/10 and k = 0.05, and compare your results at t = 0.5 to the actual solution u(x, t) = e−t sin x.