The Separation of Variables method decomposes linear partial differential equations with homogeneous spatial boundary conditions into a product solution consisting of independent steady and transient components. By enforcing the PDE to separate into ordinary differential equations where each side depends on only one variable, the theory establishes that these sides must equal discrete eigenvalues derived from orthogonal basis functions (eigenfunctions). Within transport phenomena, this mathematical technique transforms unsteady state problems with inhomogeneous initial conditions into solvable systems defined by an infinite series expansion over a complete set of orthogonal spatial modes.
The Separation of Variables method decomposes linear partial differential equations with homogeneous spatial boundary conditions into a product solution consisting of independent steady and transient…