The heat equation ∂u/∂t = ∂2u/∂x2 starts from a temperature distribution u at t = 0 and follows it for t > 0 as it quickly becomes smooth. The heat equation ∂u/∂t = ∂2u/∂x2 starts from a temperature distribution u at t = 0 and follows it for t > 0 as it quickly becomes smooth.