A deep theoretical discussion proves that in Joglekar's and Biolek's models the memductance-flux relation of a memristor driven by a sign-varying voltage source may only exhibit single-valuedness and multi-valuedness respectively. This manuscript derives a novel boundary condition-based Model for memristor nanostructures. Unlike previous models, the proposed one allows for closed-form solutions. More importantly, subject to the nonlinear behavior under exam, this model enables a suitable tuning of boundary conditions, which may result in the detection of both single-valued and multi-valued memductance-flux relations under certain sign-varying inputs of interest. The large class of modeled dynamics include all behaviors reported in the legendary paper revealing the existence of memory-resistance at the nano scale.