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

Status of the differential transformation method

Further to a recent controversy on whether the differential transformation method (DTM) for solving a differential equation is purely and solely the traditional Taylor series method, it is emphasized that the DTM is currently used, often only, as a technique for (analytically) calculating the power series of the solution (in terms of the initial value parameters). Sometimes, a piecewise analytic continuation process is implemented either in a numerical routine (e.g., within a shooting method) or in a semi-analytical procedure (e.g., to solve a boundary value problem). Emphasized also is the fact that, at the time of its invention, the currently-used basic ingredients of the DTM (that transform a differential equation into a difference equation of same order that is iteratively solvable) were already known for a long time by the "traditional"-Taylor-method users (notably in the elaboration of software packages --numerical routines-- for automatically solving ordinary differential equations). At now, the defenders of the DTM still ignore the, though much better developed, studies of the "traditional"-Taylor-method users who, in turn, seem to ignore similarly the existence of the DTM. The DTM has been given an apparent strong formalization (set on the same footing as the Fourier, Laplace or Mellin transformations). Though often used trivially, it is easily attainable and easily adaptable to different kinds of differentiation procedures. That has made it very attractive. Hence applications to various problems of the Taylor method, and more generally of the power series method (including noninteger powers) has been sketched. It seems that its potential has not been exploited as it could be. After a discussion on the reasons of the "misunderstandings" which have caused the controversy, the preceding topics are concretely illustrated.Comment: To appear in Applied Mathematics and Computation, 29 pages, references and further considerations adde

A survey on fuzzy fractional differential and optimal control nonlocal evolution equations

We survey some representative results on fuzzy fractional differential equations, controllability, approximate controllability, optimal control, and optimal feedback control for several different kinds of fractional evolution equations. Optimality and relaxation of multiple control problems, described by nonlinear fractional differential equations with nonlocal control conditions in Banach spaces, are considered.Comment: This is a preprint of a paper whose final and definite form is with 'Journal of Computational and Applied Mathematics', ISSN: 0377-0427. Submitted 17-July-2017; Revised 18-Sept-2017; Accepted for publication 20-Sept-2017. arXiv admin note: text overlap with arXiv:1504.0515