The proper treatment of linear inhomogeneity in 3-space and 1-time
Solutions to tensor, vector, and scalar linear partial differential equations are obtained through the inversion of the linear problem by a convolution integral whose kernel is given by the infinite-extent Green's function for the appropriate linear partial differential operator. The inverse (Green's function) problem can be cast into sets of single, double, and triple summation/integration expressions using transcendental eigenfunction expansions in certain quadric and cyclidic geometries which admit a separation of variables. By reversing and collapsing traditional ordering schemes for the inverse problem, one can derive new special function representations for quadric and cyclidic, corporeal, axisymmetric and cylindrical, orthogonal, curvilinear, coordinate geometries. In this talk, we treat such important applications as the inhomogeneous Laplace, Helmholtz, wave, diffusion, biharmonic, and triharmonic equations, as well as for higher order inhomogeneous problems.