Consider the partial differential equation for heat in a one-dimensional rod with temperature u(x, t):
∂u/∂t = D * ∂²u/∂x²
Assume initial condition: u(x, 0) = x - 6
and boundary conditions:
∂u/∂x(0, t) = 0, ∂u/∂x(8, t) = 0
(a) Determine the steady state temperature distribution:
u(x) =
(b) Find the full time dependent solution using Fourier series with trigonometric functions chosen appropriately for the boundary conditions.
u(x, t) = A₀ + b₀x + Σ(n=1 to ∞) aₙe⁻ᴰωₙ²ᵗ cos(ωₙx) + Σ(n=1 to ∞) bₙe⁻ᴰωₙ²ᵗ sin(ωₙx)
where ωₙ = n pi/8
A₀ =
b₀ = 0
aₙ =
bₙ =