Alfvén wave current drive (AWCD) in low-aspect ratio tokamaks (LARTs) is studied numerically. For this, the full-wave equation with a Vlasov-based dielectric tensor is solved by relaxation techniques, subject to appropriate boundary conditions at the plasma centre and at the plasma-vacuum interface, as well as the concentric antenna current sheet and at the external metallic wall. A systematic investigation of the physical characteristics of the AWCD generated in LARTs when kinetic effects are considered is carried out and illustrative results are presented and discussed.