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This paper presents a theory for analyzing porous functionally graded (PFG), rectangular in plan, size-dependent shells at large deflections. Three types of porosity are considered in the study. The theory is based on the Kirchhoff-Love kinematic model, incorporating the modified coupled stress theory (MCST) to account for size-dependent factors. The PDEs that are valid for the shells are obtained by means of the Hamiltonian principle. To obtain reliable results, the governing partial differential equations (PDEs) are transformed into ordinary differential equations (ODEs) by means of the second-order finite difference method (FDM), the Bubnov-Galerkin method (BGM) in higher approximations, and either the variational iteration method (VIM) or the extended Kantorovich method (EKM). The numerical methods and algorithms are validated through convergence studies. Among the numerical methods employed, preference is given to the variational iterations method, specifically the extended Kantorovich method, for its effectiveness in obtaining accurate results. The study also investigates the influence of the porosity type, the size-dependent parameter, and the porosity index on the stress-strain state of nano-shells. Through analysis, the optimum type of pore distribution is identified, which can enhance the mechanical performance of the shells. The findings contribute to the understanding of the stress-strain state of nano-shells and the influence of porosity characteristics, ultimately aiding in the development of improved materials and structures. [ABSTRACT FROM AUTHOR]