  In order to solve the Bethe-Salpeter equation (BSE) in the Minkowski space,
we first introduce the Nakanishi integral representations of the Bethe-Salpeter
amplitude (BSA) and the Bethe-Salpeter wave function (BSWF). We then derive the
explicit integral equations for the corresponding spectral functions from the
BSE for states of $2$ scalar particles bound by a scalar-particle exchange
interaction, where the propagators of constituents are allowed to be fully
dressed. These integral equations are subsequently solved numerically in the
variation of the Wick-Cutkosky model with massive exchange particles, where an
algorithm of adaptive mash grid is proposed. The equations and algorithm we
develop here serve as the foundation of Minkowski-space formulation of BSE for
bound states of fermions.
