  We implement a new semi-analytical approach to investigate radially
self-similar solutions for the steady-state advection-dominated accretion flows
(ADAFs). We employ the usual $\alpha$-prescription for the viscosity and all
the components of the energy-momentum tensor are considered. In this case, in
the spherical coordinate, the problem reduces to a set of eighth-order,
nonlinear differential equations with respect to the latitudinal angle
$\theta$. Using the Fourier expansions for all the flow quantities, we convert
the governing differential equations to a large set of nonlinear algebraic
equations for the Fourier coefficients. Using the Newton-Raphson method we
solve the algebraic equations and ADAF properties are investigated over a wide
range of the model parameters. We also show that the implemented series are
truly convergent. The main advantage of our numerical method is that it does
not suffer from the usual technical restrictions which may arise for solving
ADAF differential equations near to the polar axis. In order to check the
reliability of the approach, we recover some widely studied solutions. We also
introduce a new varying $\alpha$ viscosity model and new outflow and inflow
solutions for ADAFs are presented using Fourier expansion series.
