Battery discharging model on fractal time sets

This article is devoted to propose and investigate the fractal battery discharging model, which is one of the well-known models with a memory effect. It is presented as to how non-locality affects the behavior of solutions and how the current state of the system is affected by its past. Firstly, we present a local fractal solution. Then we solve the non-local fractal differential equation and examine the memory effect that includes the Mittag-Leffler function with one parameter. For that aim, the local fractal and non-local fractal Laplace transforms are used to achieve fractional solutions. In addition, the simulation analysis is performed by comparing the underlying fractal derivatives to the classical ones in order to understand the significance of the results. The effects of the fractal parameter and the fractional parameter are discussed in the conclusion section.

New variational theory for coupled nonlinear fractal Schrödinger system

Purpose The purpose of this paper is the coupled nonlinear fractal Schrödinger system is defined by using fractal derivative, and its variational principle is constructed by the fractal semi-inverse method. The approximate analytical solution of the coupled nonlinear fractal Schrödinger system is obtained by the fractal variational iteration transform method based on the proposed variational theory and fractal two-scales transform method. Finally, an example illustrates the proposed method is efficient to deal with complex nonlinear fractal systems. Design/methodology/approach The coupled nonlinear fractal Schrödinger system is described by using the fractal derivative, and its fractal variational principle is obtained by the fractal semi-inverse method. A novel approach is proposed to solve the fractal model based on the variational theory. Findings The fractal variational iteration transform method is an excellent method to solve the fractal differential equation system. Originality/value The author first presents the fractal variational iteration transform method to find the approximate analytical solution for fractal differential equation system. The example illustrates the accuracy and efficiency of the proposed approach.

Variational principle and its fractal approximate solution for fractal Lane-Emden equation

Purpose The purpose of this paper is to describe the Lane–Emden equation by the fractal derivative and establish its variational principle by using the semi-inverse method. The variational principle is helpful to research the structure of the solution. The approximate analytical solution of the fractal Lane–Emden equation is obtained by the variational iteration method. The example illustrates that the suggested scheme is efficient and accurate for fractal models. Design/methodology/approach The author establishes the variational principle for fractal Lane–Emden equation, and its approximate analytical solution is obtained by the variational iteration method. Findings The variational iteration method is very fascinating in solving fractal differential equation. Originality/value The author first proposes the variational iteration method for solving fractal differential equation. The example shows the efficiency and accuracy of the proposed method. The variational iteration method is valid for other nonlinear fractal models as well.

Local fractional derivative: A powerful tool to model the fractal differential equation

In this paper, the modified Fornberg-Whitham equation is described by the local fractional derivative for the first time. The fractal complex transform and the modified reduced differential transform method are successfully adopted to solve the modified local Fornberg-Whitham equation defined on fractal sets.

Operational Procedures in the Theory of the Drug Release from Chitosan Hydrogels

We build a theoretical model based on a generalization of harmonic applications of Misner-type. It results a sine-Gordon type fractal differential equation whose elliptical solutions can describe, through a convenient choice of fractal dynamic constants, various modes of drug release. Thus, the entire class of empirical models (Higuchi, Korsmeyer-Peppas, Peppas-Sahlin) describing the drug release processes can be dispensed with.