<p>A novel exponential integrator has been developed to address time-dependent partial differential equations encountered in complex fluid flow problems. This method is formulated as an explicit two-stage scheme that attains second-order accuracy in time. For spatial discretization, a compact finite difference approach is utilized, which improves both accuracy and computational efficiency. Comprehensive stability and convergence analyses validate the robustness of this integrator. The method is applied to simulate the electro-osmotic flow of an Eyring-Powell non-Newtonian fluid over a stationary plate, a situation relevant to microfluidic and bioengineering applications, effectively capturing the intricate interaction between electric double layer forces and the fluid’s rheological properties with high precision. Electro-osmotic fluids find applications in geotechnical engineering, environmental remediation, and microfluidic devices. The model can be reduced to dimensionless time-dependent partial differential equations, which are then solved using the proposed method. This new scheme is compared with existing first- and second-order methods from the literature. Across most time step sizes, the proposed approach shows superior accuracy, as demonstrated by the comparison table. The constructed figures show that velocity profile raises by growing Helmholtz Smoluchowski velocity and the electroosmotic parameter.</p>