Vanshing vertical viscosity limit of anisotropic magnetohydrodynamics equation with non-slip boundary condition

In this paper, we study the zero-viscosity limit of the anisotropic incompressible magnetohydrodynamics (MHD) equations with no-slip boundary conditions in $ \mathbb{R}_+^2 $. Assuming that the initial velocity and magnetic fields are sufficiently regular, we justify the validity of the Prandtl boundary layer expansion. Additionally, we provide an $ L^\infty $ estimate for the error equation by using multi-scale analysis as the vertical viscosity vanishes. Furthermore, we derive the optimal convergence rate for the solution.