  An integral equation method is presented for the 1D steady-state
Poisson-Nernst-Planck equations modeling ion transport through membrane
channels. The differential equations are recast as integral equations using
Green's 3rd identity yielding a fixed-point problem for the electric potential
gradient and ion concentrations. The integrals are discretized by a combination
of midpoint and trapezoid rules and the resulting algebraic equations are
solved by Gummel iteration. Numerical tests for electroneutral and
non-electroneutral systems demonstrate the method's 2nd order accuracy and
ability to resolve sharp boundary layers. The method is applied to a 1D model
of the K$^+$ ion channel with a fixed charge density that ensures cation
selectivity. In these tests, the proposed integral equation method yields
potential and concentration profiles in good agreement with published results.
