The core principle is that linear partial differential equations governing unidirectional transport with volumetric sources (e.g., body forces) can be decomposed into a steady-state solution satisfying the inhomogeneous source term and a transient homogeneous response. This superposition allows reduction to an ordinary differential equation via separation of variables, provided the initial condition for the transient component accounts for the negative value of the steady state at $t=0$. In oscillatory regimes, complex variable substitution exploits linearity to decouple temporal frequency from spatial decay profiles, characterized by dimensionless groups such as the Reynolds number scaled by angular frequency ($\text{Re}\Omega$), which dictate penetration depth and phase lag.
The core principle is that linear partial differential equations governing unidirectional transport with volumetric sources (e.g., body forces) can be decomposed into a steady-state solution satisfyi…