Recently, in the direction of developing realistic mathematical models, there are a
number of works that extended the ordinary differential equation to the fractionalorder
equation.
Fractional-order
models
are
thought
to
provide
better
agreement
with
the
real
data
compared
with
the
integer-order
models.
The
fractional
logistic
equation
is
one of the equations that has been getting the attention of researchers due to its
nature in predicting population growth and studying growth trends, which assists in
decision making and future planning. This research aims to propose the numerical
solution for the fractional logistic equation. Two different solving methods, which are
the Adam’s-type predictor-corrector method and the Q-modified Eulerian numbers,
were successfully applied to two versions of the fractional-order logistic equation,
which are the fractional modified logistic equation and the fractional logistic equation,
respectively. The fractional modified logistic equation, which involved the extended
Monod model, was solved by the Adam’s-type predictor-corrector method and was
applied in estimating microalgae growth. The results show that the fractional modified
logistic equation agreed with the real data of microalgae growth. Meanwhile, a closedform
solution by the Q-modified Eulerian numbers was proposed for the fractional
logistic equation. These modified Eulerian numbers were obtained by modifying the
Eulerian polynomials in two variables. Interestingly, these modified polynomials
corresponded to the polylogarithm
p
Li z( ) of the negative order and with a negative
real argument, z . The proposed method via the modified Eulerian numbers can
provide the generalised solution for an arbitrary value. The proposed method was
shown to achieve numerical convergence. The numerical experiment shows that this
method is highly efficient and accurate since the absolute error obtained from the
subtraction of the exact and proposed solution is considerably small
