The separation of variables method is applied to solve unsteady-state diffusion equations in spherical coordinates by decomposing a scalar field into spatial eigenfunctions and temporal decay functions. This theoretical framework relies on the orthogonality properties of spherical Bessel functions, which serve as the complete basis set for radial transport problems where geometric symmetry dictates that second-order differential operators yield solutions distinct from cylindrical or Cartesian counterparts. The principle establishes that steady-state solutions vanish in transient analyses defined relative to infinite ambient boundaries, requiring homogeneous boundary conditions derived at the origin and surface to determine discrete eigenvalues via integer constraints on Bessel function zeros.
The separation of variables method is applied to solve unsteady-state diffusion equations in spherical coordinates by decomposing a scalar field into spatial eigenfunctions and temporal decay functio…