Consider a thin rod of length L with an initial temperature fix) throughout. Its ends are held at zero temperature for all time. The temperature u(x, t) in the rod is governed by the following equation, boundary and initial conditions

∂^2w = c2 ∂^2u (t ≥ 0, 0 ≤ x ≤ L)

∂x^2 ∂x^2

u(t,0) = 0, u(L,t) = 0, t ≥ 0

u(x,0) = f(x), 0 ≤ x ≤ L

(a) Solve for u(x,t)

(b) Find u(x, t) for the specific case of f(x) = 100, L = pi and c = 1.