Rod with ends at 0°C, u(x,0)=3sin(πx/l)

Problem. u_t = c² u_xx on 0 < x < l with both ends at 0°C: u(0, t) = u(l, t) = 0, and u(x, 0) = 3 sin(πx/l).

Step 1: general solution (separation of variables, ends at 0°C):

u(x, t) = Σ_{n=1}^{∞} Bₙ sin(nπx/l) e^{−c²n²π²t/l²}

Step 2: initial condition.

3 sin(πx/l) = Σ Bₙ sin(nπx/l)

Comparing both sides directly (no integration is needed because the initial profile is already a single sine mode): B₁ = 3 and Bₙ = 0 for n ≥ 2.

Answer:

u(x, t) = 3 sin(πx/l) e^{−c²π²t/l²}

Check: u(0, t) = u(l, t) = 0 ✓, u(x, 0) = 3 sin(πx/l) ✓, and u_t = −(c²π²/l²) u = c² u_xx ✓. The temperature decays exponentially to 0°C.