Summary: This study investigates viscoelastic fluid, which possesses both viscous and elastics properties, by employing a fractional derivative model to reveal the stress relaxation phenomenon with distance. The spatial-fractional derivative used in the momentum conservation equation is the Riemann-Liouville type derivative. Further, we use non-uniform boundary conditions subject to the boundary layer equations of the fluid flowing through a semi-infinite permeable flat surface. Owing to the fractional derivative model and non-uniform boundary conditions, this problem is complex. Thus, a finite difference scheme is applied after the coupled continuity equation and momentum equation are decoupled and linearized. The accuracy, convergence, and stability of the numerical method are presented. It is shown that non-uniform mass transfer through a permeable surface considerably affects the velocity boundary layer. By illustrating the physical interactions between the velocity fields in the boundary layer and the permeation mode in the surface, this paper predicts the velocity distributions with varying permeable surface, and also provides the possibility of changing the velocity fields by altering the permeable sheet. The results and numerical technique used in this study will help in the understanding of fractional calculus investigation in engineering.