This tutorial review of fractal-Cantorian spacetime and fractional calculus begins with Leibniz's notation for derivative without limits which can be generalized to discontin- uous media like fractal derivative and q-derivative of quantum calculus. Fractal spacetime is used to elucidate some basic properties of fractal which is the foundation of fractional calculus, and El Naschie's mass-energy equation for the dark energy. The variational itera- tion method is used to introduce the definition of fractional derivatives. Fractal derivative is explained geometrically and q-derivative is motivated by quantum mechanics. Some effec- tive analytical approaches to fractional differential equations, e.g., the variational iteration method, the homotopy perturbation method, the exp-function method, the fractional com- plex transform, and Yang-Laplace transform, are outlined and the main solution processes are given.

A Tutorial Review on Fractal Spacetime and Fractional Calculus

Projective synchronization for fractional neural networks

In this paper, we consider the existence and uniqueness of weak solutions for a class of fractional superdiffusion equations with initial-boundary conditions. For a multidimensional fractional drift superdiffusion equation, we just consider the simplest case with divergence-free drift velocity $u \in L^{2}(\Omega)$
 only depending on the spatial variable x. Finally, exploiting the Schauder fixed point theorem combined with the Arzela-Ascoli compactness theorem, we obtain the existence and uniqueness of weak solutions in the standard Banach space $C([0,T]; H_{0}^{1}(\Omega))$
 for a class of fractional superdiffusion equations.

/pdf/existence-and-uniqueness-of-weak-solutions-for-a-class-of-l0gkkyq2ke.pdf

Existence and uniqueness of weak solutions for a class of fractional superdiffusion equations

In this paper, we propose a nonlinear fractional order model in order to explain and understand the outbreaks of influenza A(H1N1). In the fractional model, the next state depends not only upon its current state but also upon all of its historical states. Thus, the fractional model is more general than the classical epidemic models. In order to deal with the fractional derivatives of the model, we rely on the Caputo operator and on the Grunwald–Letnikov method to numerically approximate the fractional derivatives. We conclude that the nonlinear fractional order epidemic model is well suited to provide numerical results that agree very well with real data of influenza A(H1N1) at the level population. In addition, the proposed model can provide useful information for the understanding, prediction, and control of the transmission of different epidemics worldwide. Copyright © 2013 John Wiley & Sons, Ltd.

A fractional order epidemic model for the simulation of outbreaks of influenza A(H1N1)

This paper is devoted to study the problem of modified projective synchronization of fractional-order chaotic system. Base on the stability theorems of fractional-order linear system, active sliding mode controller is proposed to synchronize two different fractional-order systems. Moreover, the controller is robust to the bounded noise. Numerical simulations are provided to show the effectiveness of the analytical results.

Modified projective synchronization of fractional-order chaotic systems via active sliding mode control

Projective synchronization of chaotic fractional-order energy resources demand–supply systems via linear control

A homotopy perturbation transformation method (HPTM) which is based on homotopy perturbation method and Laplace transform is first applied to solve the approximate solution of the fractional nonlinear equations. The nonlinear terms can be easily handled by the use of He's polynomials. Illustrative examples are included to demonstrate the high accuracy and fast
convergence of this new algorithm.

https://downloads.hindawi.com/journals/aaa/2012/752869.pdf

Approximate Solutions of Fractional Nonlinear Equations Using Homotopy Perturbation Transformation Method

This paper is devoted to investigating a generalized nonlinear Fokker–Planck diffusion equation with external force and absorption. We first investigate the integer nonlinear anomalous diffusion, and we obtain the corresponding exact solution expressed by q -exponential function. The solutions of nonlinear diffusion equation with one-fractional derivative and multi-fractional derivative are also studied in detail, and the solutions can have a compact behavior or a long tailed behavior.

Exact solutions for a generalized nonlinear fractional Fokker–Planck equation

We implement relatively new analytical technique, the Homotopy perturbation method, for solving nonlinear fractional partial differential equations arising in predator-prey biological population dynamics system. Numerical solutions are given, and some properties exhibit biologically reasonable dependence on the parameter values. And the fractional derivatives are described in the Caputo sense.

/pdf/numerical-solutions-of-a-fractional-predator-prey-system-40wec3giv2.pdf

Numerical Solutions of a Fractional Predator-Prey System

We propose a new chaotic financial system by considering ethics involvement in a four-dimensional financial system with market confidence. We present a five-dimensional conformable derivative financial system by introducing conformable fractional calculus to the integer-order system. We propose a discretization scheme to calculate numerical solutions of conformable derivative systems. We illustrate the scheme by testing hyperchaos for the system.

/pdf/modeling-discretization-and-hyperchaos-detection-of-wqmzgqxzhl.pdf

Yanqin Liu

Papers

Projective synchronization of chaotic fractional-order energy resources demand–supply systems via linear control

Approximate Solutions of Fractional Nonlinear Equations Using Homotopy Perturbation Transformation Method

Exact solutions for a generalized nonlinear fractional Fokker–Planck equation

Numerical Solutions of a Fractional Predator-Prey System

Modeling, discretization, and hyperchaos detection of conformable derivative approach to a financial system with market confidence and ethics risk