Electroosmotic flow in nanochannels with non-uniform wall potential is investigated. While several researchers have presented results for the case of periodic potential and sudden change in potential, most of the previous work in this area is based on the Debye-Hückel approximation and the validity of Boltzmann distribution for ionic species. In this paper, the nonlinear distributions of potential, velocity and mole fractions are calculated numerically based on the Poisson-Nernst-Planck model including convective effects. The distribution of potential and species concentration are shown to be different from the Boltzmann distribution at large applied electric field strength. The convective effect is also investigated and found to be negligible.