Recently, we proposed a weak-form discretization scheme to derive second-order difference equations from the governing equation of the scattering problem. In this paper, under the scope of the proposed scheme, numerical expressions for the waveguide boundary conditions are derived as perfectly absorbing conditions for input and output ports. The waveguide boundary conditions play an important role in extracting the quasi-localized wave as an eigenstate with a complex eigenvalue. The wave-number dependence of the resonance curve in Fano resonance is reproduced by using a semi-analytic model that is developed on the basis of the phase change relevant to the S-matrix. The reproduction confirms that the eigenstate with a complex eigenvalue does cause the observed Fano resonance.