Numerical solution for fractionalorder logistic equation

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