The Poisson-Boltzmann equation (PBE) is a second-order partial differential equation that can be used to describe the ion distribution of colloidal systems. In its classical form, only the electrostatic interactions between ions and interface are considered. The aim of this work is to study the solution of PBE by second order spline finite elements method applied to a one-dimensional system. This method is used to solve the classical and the modified PB equation, which considers van der Waals dispersion (non-electrostatic) interactions between ions and macro-particles. The method consists in dividing of the domain into a number of subdomains, in which the dependent variable is approximated by a second order polynomial. The algorithm developed is validated by comparing the numerical results of the classical PBE with results from the literature.