Let u be the solution to the initial boundary value problem for the Heat Equation,
∂tu(t, x) = 5 ∂"xu(t, x), t ∈ (0, ∞), x ∈ (0, 1);
with initial condition u(0, x) = f(x), where f(0) = 0 and f'(1) = 0, and with boundary conditions
u(t, 0) = 0, ∂xu(t, 1) = 0.
Using separation of variables, the solution of this problem is
u(t, x) = ∑ c_n v_n(t) w_n(x),
with the normalization conditions
v_n(0) = 1, w_n(1/(2n-1)) = 1.
a. (5/10) Find the functions w_n, with index n ≥ 1.
w_n(x) =
b. (5/10) Find the functions v_n, with index n ≥ 1.
v_n(t) =