  Fractional differential calculus is a mathematical tool that has found
applications in the study of social and physical behaviors considered
``anomalous''. It is often used when traditional integer derivatives models
fail to represent cases where the power law is observed accurately. Fractional
calculus must reflect non-local, frequency- and history-dependent properties of
power-law phenomena. This tool has various important applications, such as
fractional mass conservation, electrochemical analysis, groundwater flow
problems, and fractional spatiotemporal diffusion equations. It can also be
used in cosmology to explain late-time cosmic acceleration without the need for
dark energy. We review some models using fractional differential equations. We
look at the Einstein--Hilbert action, which is based on a fractional derivative
action, and add a scalar field, $\phi$, to create a non-minimal interaction
theory with the coupling, $\xi R \phi^2 $, between gravity and the scalar
field, where $\xi$ is the interaction constant. By employing various
mathematical approaches, we can offer precise schemes to find analytical and
numerical approximations of the solutions. Moreover, we comprehensively study
the modified cosmological equations and analyze the solution space using the
theory of dynamical systems and asymptotic expansion methods. This enables us
to provide a qualitative description of cosmologies with a scalar field based
on fractional calculus formalism.
