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An integral equation method is presented for the 1D steady-statePoisson-Nernst-Planck equations modeling ion transport through membranechannels. The differential equations are recast as integral equations usingGreen's 3rd identity yielding a fixed-point problem for the electric potentialgradient and ion concentrations. The integrals are discretized by a combinationof midpoint and trapezoid rules and the resulting algebraic equations aresolved by Gummel iteration. Numerical tests for electroneutral andnon-electroneutral systems demonstrate the method's 2nd order accuracy andability to resolve sharp boundary layers. The method is applied to a 1D modelof the K$^+$ ion channel with a fixed charge density that ensures cationselectivity. In these tests, the proposed integral equation method yieldspotential and concentration profiles in good agreement with published results.

Integral equation method for the 1D steady-state Poisson-Nernst-Planck equations