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 α-prescription for the viscosity, and all 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 θ. 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. We solve the algebraic equations via the Newton– Raphson method, and investigate the ADAF properties over a wide range of 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 that may arise for solving ADAF differential equations near the polar axis. In order to check the reliability of our approach, we recover some widely studied solutions. Further, we introduce a new varying α viscosity model. New outflow and inflow solutions for ADAFs are also presented, using Fourier expansion series.

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