Use the Forward-Difference method to approximate the solution to the following parabolic partial differential equations.
a. ∂u / ∂t - 4 / π2 ∂2u / ∂x2 = 0, 0 < x < 4, 0 < t;
u(0, t) = u(4, t) = 0, 0 < t,
u(x, 0) = sin π/4 x. 1 + 2 cos π / 4 x, 0≤ x ≤ 4.
Use h = 0.2 and k = 0.04, and compare your results at t = 0.4 to the actual solution u(x, t) = e−t sin π2 x + e−t/4 sin π / 4 x.
b. ∂u / ∂t - 1 π2 ∂2u / ∂x2 = 0, 0 < x < 1, 0 < t;
u(0, t) = u(1, t) = 0, 0 < t,
u(x, 0) = cos π (x - 1/2), 0≤ x ≤ 1.
Use h = 0.1 and k = 0.04, and compare your results at t = 0.4 to the actual solution u(x, t) = e−t cos π(x - 1/2).