A Numerical Method for Poisson-Boltzmann Equation Using the Lambert W Function

The Poisson-Boltzmann equation is a well-known nonlinear differential equation that is fundamental in plasma theory. However, since it has nonlinear characteristics, the conditions and situations for obtaining analytical solutions are limited. Therefore, research and development on ways to obtain solution through numerical analysis methods have been continued. In this paper, a numerical method using the Lambert W function to solve the nonlinear Poisson-Boltzmann equation is explained. To investigate the applicability of the current method, three cases were assumed: a uniform ion density, the collisionless sheath of cold ion plasma and a nonuniform ion density. Solutions can also obtained even if the density of the ions is given as a function of the potential. In this case, the iterative steps can be integrated and simplified further.