A numerical model that simulates the steady-state current-voltage curve and the time-dependent response of a dye-sensitized solar cell with a single continuity equation is derived. It is shown that the inclusion of the multiple-trapping model, the quasi-static approximation and non-linear recombination kinetics leads to a continuity equation for the total electron density in the photoanode with an electron density-dependent diffusion coefficient and a density-dependent pseudo-first order recombination constant. All parameters in the model can be related to quantities accessible experimentally. The required power exponents are taken from impedance spectroscopy measurements at different voltages. The model provides new insights into the physical interpretation of the power exponents. Modeling examples involving a high-efficiency TiO2-based dye solar cell and a ZnO-based dye solar cell are presented. It is demonstrated that the model reproduces the transient behavior of the cell under small perturbations. The spatial dependence of the recombination rate and the influence of film thickness and of voltage dependent injection efficiency on cell performance are studied. The implications of the model are discussed in terms of efficiencies potentially attainable in dye-sensitized solar cells and other kinds of solar cells with a diffusional mechanism of charge transport.