In this paper, the start-up from rest of the electroosmotic flow of Maxwell fluids in a rectangular microchannel with asymmetric high zeta potentials at the walls is studied. By appropriately combining the momentum equation with the rheological Maxwell model, a hyperbolic partial differential equation to determine the flow field is obtained. The dimensionless mathematical model is solved using Green's functions and the separation-of-variables method. Because the high zeta potentials at the walls of the microchannel are taken into account, the electrical potential distribution is numerically solved. We show that the flow is characterized by two relevant parameters that control the electroosmotic flow: the dimensionless relaxation time of the viscoelastic fluid, (lambda) over bar (1), and the dimensionless zeta potentials of the walls, (zeta) over bar (i). The most important results of this paperare that the required time to reach the steady state depends on the assumed values of the relaxation time and that the velocity profiles exhibit a singular transient oscillatory behavior because of the competition between viscous, elastic and electroosmotic forces. For asserting the correctness of the semi-analytical solution of the electroosmotic flow, an analytical solution was determined by considering low zeta potential, and a very good agreement between them was found. (c) 2015 Elsevier B.V. All rights reserved.