Nonlocal continuum electrostatic models have been used numerically in protein simulations, but analytic solutions have been absent. In this paper, two modified nonlocal continuum electrostatic models, the Lorentzian Model and a Linear Poisson-Boltzmann Model, are presented for a monatomic ion treated as a dielectric continuum ball. These models are then solved analytically using a system of differential equations for the charge distributed within the ion ball. This is done in more detail for a point charge and a charge distributed within a smaller ball. As the solutions are a series, their convergence is verified and criteria for improved convergence is given
