School of Physics, University of Exeter,

Stocker Road, Exeter, EX4 4QL

United Kingdom

E-mail: hcohl@falcon.spartanet.sk

**The proper treatment of linear inhomogeneity in 3-space
and 1-time**

**Abstract:**

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.