On the Application of Fractional Derivatives to the Study of Memristor Dynamics

In this chapter, the authors report their work on the application of fractional derivative to the study of the memristor dynamic where the effects of the parasitic fractional elements of the memristor have been studied. The fractional differential equations of the memristor and the memristor-based circuits under the effects of the parasitic fractional elements have been formulated and solved both analytically and numerically. Such effects of the parasitic fractional elements have been studied via the simulations based on the obtained solutions where many interesting results have been proposed in the work. For example, it has been found that the parasitic fractional elements cause both charge and flux decay of the memristor and the impasse point breaking of the phase portraits between flux and charge of the memristor-based circuits similarly to the conventional parasitic elements. The effects of the order and the nonlinearity of the parasitic fractional elements have also been reported.

The Invariant Subspace Method for Solving Fractional Partial Differential Equations

In this chapter, the authors discuss the effectiveness of the invariant subspace method (ISM) for solving fractional partial differential equations. For this purpose, they have chosen a nonlinear time fractional partial differential equation (PDE) with variable coefficients to be investigated through this method. One-, two-, and three-dimensional invariant subspace classifications have been performed for this equation. Some new exact solutions have been obtained using the ISM. Also, the authors give a comparison between this method and the homogeneous balance principle (HBP).

Properties of Spatial-Rotation Transformations of Fractional Derivatives

The transformation properties of the fractional derivatives under spatial rotation in two-dimensional space and for both the Riemann-Liouville and Caputo definitions are investigated and derived in their general form. In particular, the transformation properties of the fractional derivatives acting on scalar fields are studied and discussed. The study of the transformation properties of fractional derivatives is an essential step for the formulation of fractional calculus in multi-dimensional space. The inclusion of fractional calculus in the Lagrangian and Hamiltonian dynamical formulation relies on such transformation. Specific examples on the transformation of the fractional derivatives of scalar fields are discussed.

Revisiting Anomalous g-Factors for Charged Leptons in a Fractional Coarse-Grained Approach With Axiomatic Local Metric Derivatives

This contribution sets out to extend the concept of helicity so as to include it in a fractional scenario with a low-level of fractionality. To accomplish this goal, the authors write down the left- and the right-handed Weyl equations from first principles in this extended framework. Next, by coupling the two different fractional Weyl sectors by means of a mass parameter, they arrive at the fractional version of Dirac's equation, which, whenever coupled to an external electromagnetic field and reduced to the non-relativistic regime, yields a fractional Pauli-type equation. From the latter, they are able to present an explicit expression for the gyromagnetic ratio of charged fermions in terms of the fractionality parameter. They then focus their efforts to relate the coarse-grained property of space-time to fractionality and to the (g-2) anomalies of the different leptonic species. To do this, they build up an axiomatic local metric derivative that exhibits the Mittag-Leffler function as eigenfunction and is valid for low-level fractionality, whenever the order parameter is close to 1.

Fractional-Order-Based Inferential Control

This chapter presents the application of fractional differential operator in modelling and control of a three-tank interacting level process. In cases where the usage of sensors for the measurement of primary variable, which is the level of third tank in present case, is physically or economically not feasible, the measurement of secondary variable (i.e., second tank level) is used to determine the level of third tank for control purpose, known as inferential control scheme. The process is modeled and linearized around the operating points, resulting in third order plant, which is approximated to lower order integer and non-integer model. Both conventional integer order PI (IO-PI) & PID (IO-PID) and fractional order PI (FO-PI) & (FO-PID) controllers are implemented for this inferential control. Extensive simulation studies performed using MATLAB validate the supremacy of non-integer order model and controller over integer order model and controller. Genetic algorithm (GA) is being applied for both, firstly for reduced order model approximation and secondly for controller tuning.

Generalized Brinkman Type Dusty Fluid Model for Blood Flow

This chapter is dedicated to studying the magnetic blood flow with uniformly distributed magnetite dusty particles (MDP) in a cylindrical tube. For this purpose, the two-phase fractional Brinkman type fluid model is considered. The fractional governing equations are modeled in the cylindrical coordinate system taking into consideration the magnetization of the fluid due to the applied magnetic field. The fractional governing equations are subjected to physical initial and boundary conditions. The joint Laplace and Hankel transform is employed to develop exact analytical solutions. The obtained solutions are computed numerically and plotted in different graphs. It is noticed that for a long time the blood and MDP velocities increase with increasing values of the fractional parameter. In contrast, this effect reverses for a shorter time. In the case of the magnetic parameter, both velocities are decreased with increasing values of the magnetic parameter.

Apparent Measure, Integral of Real Order, and Relative Dimension

In this chapter, the author introduces a concept of apparent measure in R^n and associates the concept of relative dimension (of real order) that depends on the geometry of the object to measure and on the distance that separates it from an observer. At the end, the author discusses the relative dimension of a Cantor set. This measure enables us to provide a geometric interpretation of the Riemann-Liouville's integral of order αϵ├]0,1], and based on this interpretation, the author introduces a modification on the Riemann-Liouville's integral to make it symmetrical and then introduces a new fractional derivative that exploits at the same time the right and the left fractional derivatives.

On the Approximate Solution of Caputo-Riesz-Feller Fractional Diffusion Equation

In this work, a space-time fractional diffusion equation with spatial Riesz-Feller fractional derivative and Caputo fractional time derivative is introduced. The continuation of the solution of this fractional equation to the solution of the corresponding integer order equation is proved. Also, a very useful Riesz-Feller fractional derivative is proved; the property is essential in applying iterative methods specially for complex exponential and/or real trigonometric functions. The analytic series solution of the problem is obtained via the optimal homotopy analysis method (OHAM). Numerical simulations are presented to validate the method and to highlight the effect of changing the fractional derivative parameters on the behavior of the obtained solutions. The results in this work are originally extracted from the author's work.

Synchronization of Fractional-Order Hyperchaotic Finance Systems Using Sliding Mode Control Techniques

In this chapter, the basic concepts of fractional-order dynamical systems are presented, and the synchronization methodologies of fractional order chaotic dynamical systems are established using slide mode control techniques. Through observation of the different phase portraits and time-series graphs of fractional order finance systems through utilization of the fractional calculus and computer simulation, the authors have obtained that the lowest dimension of fractional order hyper chaotic finance system is 3.90, which is less than 4. Bifurcation diagrams and Lyapunov exponents of fractional order hyper chaotic finance system are calculated to justify the chaos in the systems. Synchronization of two identical fractional-order hyper chaotic finance systems are achieved using sliding mode control techniques.

Dynamics of Some Discretized Fractional-Order Differential Equations

This chapter deals with fractional-order differential equations and their discretization. First of all, a discretization process for discretizing ordinary differential equations with piecewise constant arguments is presented. Secondly, a discretization method is proposed for discretizing fractional-order differential equations. Stability of fixed points of the discretized equations are investigated. Numerical simulations are carried out to show the dynamic behavior of the resulting difference equations such as bifurcation and chaos.