  The presented paper is a comprehensive analysis of two dark energy (DE)
cosmological models wherein exact solutions of the Einstein field equations
(EFEs) are obtained in a model-independent way (or by cosmological
parametrization). A simple parametrization of the Hubble parameter ($H$) is
considered for the purpose in the flat Friedmann-Lemaitre-Robertson-Walker
(FLRW) background. The parametrization of $H$ covers some known models for some
specific values of the model parameters involved. Two models are of special
interest which show the behavior of cosmological phase transition from the
deceleration in the past to acceleration at late-times. The model parameters
are constrained with $57$ points of Hubble datasets together with the $580$
points of Union $2.1$ compilation supernovae datasets and baryonic acoustic
oscillation (BAO) datasets. With the constrained values of the model
parameters, both the models are analyzed and compared with the standard
$\Lambda $CDM model and showing nice fit to the datasets. Two different
candidates of DE is considered, cosmological constant $% \Lambda $ and a
general scalar field $\phi $ and their dynamics are discussed on the
geometrical base built. The geometrical and physical interpretations of the two
models in consideration are discussed in details and the evolution of various
cosmological parameters are shown graphically. The age of the Universe in both
models are also calculated. Various cosmological parametrization schemes used
in the past few decades to find exact solutions of the EFEs are also summarized
at the end which can serve as a unified reference for the readers.

General Physics