Discrete Fourier Inversion of Linear Inhomogeneity

Solutions to tensor, vector, and scalar linear inhomogeneous partial differential equations can be obtained by discrete Fourier inversion of the linear system. The inverse (Green’s function) problem can be cast into sets of single, double, and triple summation/integration expressions by using transcendental eigenfunction expansions in certain three-dimensional triply-orthogonal geometries whose solutions admit simple and R-separation of variables. By reversing and collapsing traditional ordering schemes for the inverse problem, one can derive new special function addition theorems for asymmetric, axisymmetric and cylindrical, orthogonal curvilinear coordinate geometries. In this paper, we introduce such important applications as inhomogeneous Laplace, Helmholtz, wave, Schrödinger, heat, Klein–Gordon, Laplace–Beltrami, biharmonic, triharmonic as well as mixed higher-order harmonic and linear inhomogeneous partial differential equations. Compact expressions are seen to exist in spherical geometries through the utilization of separation angle spherical linear systemfunction expansions.