Terms Of Fractional Derivatives Research Articles

Duhem histeresisin Riemann-Liouville kesirli türevi ile modellenmesi

In the view of memory effect of hysteresis, this work aims to interpret hysteresis nonlinearities in terms of Riemann-Liouville fractional derivative which is a singular operator with memory and hereditary properties. For this purpose, Duhem hysteresis, a model defined by a first order differential equation, is considered and adapted to a fractional order differential equation. Since the fractional order Duhem hysteresis cannot be solved by an analytical scheme, Grünwald-Letnikov approximation is used to obtain numerical solutions. Thus, the effect of fractional order derivative to Duhem hysteresis is demonstrated with graphics obtained by this approximation and plotting using MATLAB. As a result, it is observed that the fractional order model exhibits hysteresis behavior for the orders that are smaller than 1.

A new model of dengue fever in terms of fractional derivative.

It is eminent that the epidemiological patterns of dengue are threatening for both the global economy and human health. The experts in the field are always in search to have better mathematician models in order to understand the transmission dynamics of epidemics models and to suggest possible control or the minimization of the infection from the community. In this research, we construct a new fractional-order system for dengue infection with carrier and partially immune classes to visualize the intricate dynamics of dengue. By using the basics of fractional theory, we determine the fundamental results of the proposed fractional-order dengue model. We obtain the basic reproduction number $R_0$ by next generation method and present the results based on it. The stability results are established for the infection-free state of the system. Moreover, sensitivity of $R_0$ is analyzed through partial rank correlation coefficient(PRCC) method to show the importance of different parameters in $R_0$. The influence of different input factors is shown on the output of basic reproduction number $R_0$ numerically. Our result showed that the threshold parameter $R_0$ can be decreased by increasing vaccination and treatment in the system. Finally, we illustrate the solution of the suggested dengue system through a numerical scheme to notice the influence of the fractional-order $\vartheta$ on the system. We observed that the fractional-order dynamics can explain the complex system of dengue infection more precisely and accurately rather than the integer-order dynamics. In addition, we noticed that the index of memory and biting rate of vector can play a significant part in the prevention of the infection.

<i>S</i>-function associated with fractional derivative and double Dirichlet average

The object of this article is to investigate the double Dirichlet averages of S-functions. Representations of such relations are obtained in terms of fractional derivative. Some interesting special cases are also stated.

Finite difference method for a nonlinear fractional Schrödinger equation with Neumann condition

Abstract In this paper, a special case of nonlinear fractional Schrödinger equation with Neumann boundary condition is considered. Finite difference method is implemented to solve the nonlinear fractional Schrödinger problem with Neumann boundary condition. Previous theoretical results for the abstract form of the nonlinear fractional Schrödinger equation are revisited to derive new applications of these theorems on the nonlinear fractional Schrödinger problems with Neumann boundary condition. Consequently, first and second order of accuracy difference schemes are constructed for the nonlinear fractional Schrödinger problem with Neumann boundary condition. Stability analysis show that the constructed difference schemes are stable. Stability theorems for the stability of the nonlinear fractional Schrödinger problem with Neumann boundary condition are presented. Additionally, applications of the new theoretical results are presented on a one dimensional nonlinear fractional Schrödinger problem and a multidimensional nonlinear fractional Schrödinger problem with Neumann boundary conditions. Numerical results are presented on one and multidimensional nonlinear fractional Schrödinger problems with Neumann boundary conditions and different orders of derivatives in fractional derivative term. Numerical results support the validity and applicability of the theoretical results. Numerical results present the convergence rates are appropriate with the theoretical findings and construction of the difference schemes for the nonlinear fractional Schrödinger problem with Neumann boundary condition.

Dynamics and control of periodic and non-periodic behavior of Duffing vibrating system with fractional damping and excited by a non-ideal motor

This paper deals with a non-ideal system with memory by possessing a fractional damping term. The system is characterized by additional cubic nonlinearity. Based on the time series form of the numerical simulations of this non-ideal model, the nonlinear dynamical response of the system is investigated. A DC electric motor with limited power supply driving by an unbalanced rotating mass provides the non-ideal excitation. To distinguish between periodic and non-periodic behaviors, they are used three different mathematical tools, which are the 0–1 test, scale index and wavelet technique. The response of the considered system is investigated with respect to the fractional damping derivative term and the voltage applied in the DC motor, which is considered as a control parameter. Numerical results showed that, the scale index, 0–1 test and the wavelet transform technique defined very accurately the periodic and non-periodic motions.

Stochastic P-bifurcation in a bistable Van der Pol oscillator with fractional time-delay feedback under Gaussian white noise excitation

The stochastic P-bifurcation behavior of a bistable Van der Pol system with fractional time-delay feedback under Gaussian white noise excitation is studied. Firstly, based on the minimal mean square error principle, the fractional derivative term is found to be equivalent to the linear combination of damping force and restoring force, and the original system is further simplified to an equivalent integer order system. Secondly, the stationary Probability Density Function (PDF) of system amplitude is obtained by stochastic averaging, and the critical parametric conditions for stochastic P-bifurcation of system amplitude are determined according to the singularity theory. Finally, the types of stationary PDF curves of system amplitude are qualitatively analyzed by choosing the corresponding parameters in each area divided by the transition set curves. The consistency between the analytical solutions and Monte Carlo simulation results verifies the theoretical analysis in this paper.

Finite difference scheme for a fractional telegraph equation with generalized fractional derivative terms

In this paper, a finite difference scheme is presented for the Generalized Time-Fractional Telegraph Equation (GTFTE) defined using Generalized Fractional Derivative (GFD) terms introduced recently. The generalization of fractional derivatives is done by introducing scale and weight functions, and for their particular choices, GFD reduces to Caputo and Riemann–Liouville derivatives. We present the solution behaviour of the GTFTE by changing the weight and scale functions in GFD. The convergence and the stability of the finite difference scheme (FDS) are also presented, and for the numerical simulation of the FDS, we consider examples which validate our numerical method.

Series solutions of nonlinear conformable fractional KdV-Burgers equation with some applications

In this paper, the non-linear fractional KdV-Burgers equation (KdVBE) in terms of conformable fractional derivative (CFD) is reconstituted instead of the Caputo fractional derivative and the series solution of this case is also presented by using the residual power series (RPS) method. Moreover, five important and interesting applications related to the fractional KdVBE are given and discussed in order to show the behavior of the surface graphs of the solutions. More clarifications: Firstly, we compare the solutions of the conformable fractional KdVBE and the Caputo fractional KdVBE. Secondly, in order to demonstrate the generality, potential and superiority of the RPS method, we discuss the simplicity of this method compared with other methods. Thirdly, we present the approximate solutions with graphical results of a time-CFD, space-CFD and time-space-CFD non-linear fractional KdVBEs. Finally, the results indicate that the CFD is very suitable for modeling the KdVBE and computations show that our proposed method for solving the conformable fractional KdVBE does not have mathematical requirements which implies that it is very effective as well as for providing the numerical solutions and more flexible in choosing the initial guesses approximations.

Spectral Approximations for Nonlinear Fractional Delay Diffusion Equations with Smooth and Nonsmooth Solutions

A fully discrete scheme is proposed for the nonlinear fractional delay diffusion equations with smooth solutions, where the fractional derivative is described in Caputo sense with the order $\alpha$ ($0 \lt \alpha \lt 1$). The scheme is constructed by combining finite difference method in time and Legendre spectral approximation in space. Stability and convergence are proved rigorously. Moreover, a modified scheme is proposed for the equation with nonsmooth solutions by adding correction terms to the approximations of fractional derivative operator and nonlinear term. Numerical examples are carried out to support the theoretical analysis.

Application of fractional differential equation to interpret the dynamics of dissolved heavy-metal uptake in streams at a wide range of scales

Fractional differential equations (FDEs) provide promising models to simulate non-Fickian transport in heterogeneous systems such as geological media, but the FDEs have seldom been used to model reactive transport across a wide range of spatial scales. To fill the knowledge gap, this study proposed a fractional-order advection-dispersion-reaction (fADR) model to quantify the dynamics of dissolved heavy metals, such as manganese (Mn), moving in streams with oxidative precipitation and sorption in storage zones. The fADR model was applied to fit the Mn concentration profiles documented in the literature at the hyporheic flow path scale (0.30m in length), the reach scale (100-3000m), and the basin scale ( ∼ 20000 m) at Pinal Creek, Arizona. Numerical results showed that compared with standard transport models such as the OTIS model and the single-rate mass transfer model, the fADR model can better capture the observed plumes at all scales. This is because the space fractional-diffusive term in the fADR model can capture super-diffusive jumps of dissolved metals driven by turbulence at the bedform scale and flooding in a decade-long time scale. Meanwhile, the time fractional derivative term in the fADR model describes complex solute retention due to multiple-rate mass exchange between the mobile zone (stream or the hyporheic flux) and various storage regimes with different properties (such as streambed sediments and stagnant portions in the hyporheic zone). This does not rely on the equilibrium chemistry condition for solutes in storage zones assumed by standard hyporheic-exchange models. In addition, the first-order reaction in the fADR model can efficiently characterize the mass decline of Mn downstream resulting from enhanced Mn oxidation (such as oxidation of MN(II) to +3 or +4 oxidation states) due to the input of streamflow with increased pH and dissolved oxygen and/or groundwater recharge with high dissolved metals into the hyporheic zone. The decoupled super-diffusion and retention for Mn can exhibit scale-dependent behaviors due to the evolution of driving mechanisms, which can be characterized by the parsimonious, phenomenological FDE by adjusting the indexes. Therefore, the application of FDEs helps us to interpret the physical and geochemical processes in streams across scales.

Prediction of foam impulse response through combination of hereditary and fractional derivative approaches

PurposeThe purpose of this paper is to cover an experimental investigation of the impulse response of the foam-mass system (FMS) to unveil some of the foam dynamic behavior features needed to optimize the impact comfort of seat-occupant system. The equation of motion of the studied system is modeled as a sum of a linear elastic, pneumatic damping and viscoelastic residual forces. An identification methodology based on two separated calibration processes of the viscoelastic parameters was developed.Design/methodology/approachThe viscoelastic damping force representing the foam short memory effects was modeled through the hereditary formulation. Its parameters were predicted from the free vibrational response of the FMS using iterative Prony method for autoregressive–moving–average model. However, the viscoelastic residual force resulting in the long memory effects of the material was modeled with fractional derivative term and its derivative order was predicted from previous cyclic compression standards.FindingsThe coefficients of the motion law were determined using closed form solution approach. The predictions obtained from the simulations of the impulse and cyclic tests are reasonably accurate. The physical interpretations as well as the mathematical correlations between the system parameters were discussed in details.Originality/valueThe prediction model combines hereditary and fractional derivative formulations resulting in short and long physical memory effects, respectively. Simulation of impulse and cyclic behavior yields good correlation with experimental findings.

Stochastic P-bifurcation in a tri-stable Van der Pol system with fractional derivative under Gaussian white noise

In this paper, we study the tri-stable stochastic P-bifurcation problem of a generalized Van der Pol system with fractional derivative under Gaussian white noise excitation. Firstly, using the principle for minimal mean square error, we show that the fractional derivative term is equivalent to a linear combination of the damping force and restoring force, so that the original system can be transformed into an equivalent integer order system. Secondly, we obtain the stationary Probability Density Function (PDF) of the system’s amplitude by the stochastic averaging, and using the singularity theory, we find the critical parametric conditions for stochastic P-bifurcation of amplitude of the system, which can make the system switch among the three steady states. Finally, we analyze different types of the stationary PDF curves of the system amplitude qualitatively by choosing parameters corresponding to each region divided by the transition set curves, and the system response can be maintained at the small amplitude near the equilibrium by selecting the appropriate unfolding parameters. We verify the theoretical analysis and calculation of the transition set by showing the consistency of the numerical results obtained by Monte Carlo simulation with the analytical results. The method used in this paper directly guides the design of the fractional order controller to adjust the response of the system.

Numerical and analytical investigations for neutral delay fractional damped diffusion-wave equation based on the stabilized interpolating element free Galerkin (IEFG) method

A delay PDE is different from a PDE in which it depends not only on the solution at a present stage but also on the solution at some past stage(s). In the current paper, we develop interpolating stabilized EFG method for a neutral delay PDE with fractional derivative in terms of Caputo fractional derivatives. The considered model in this paper is a generalized form of the other equations such that it contains a delay term. Thus, the existence and uniqueness of its analytic solution have been proven by the separation variable technique. At first, the temporal direction is discretized with a finite difference scheme. Then the EFG method has been employed to discrete the spatial direction. The stability of time-discrete scheme is studied by using the energy method. Also, we show the convergence order of the developed technique is O(τ3−α) in the temporal direction and the convergence order in spatial direction is O(rp+1). Finally, two test problems have been considered to demonstrate the capability and efficiency of the proposed numerical technique.

A numerical method for two-dimensional nonlinear modified time-fractional fourth-order diffusion equation

In this paper, we consider the finite element method for two-dimensional nonlinear modified time-fractional fourth-order diffusion equation. In order to avoid using higher order elements, we introduce an intermediate variable [Formula: see text] and translate the fourth-order derivative of the original problem into a second-order coupled system. We discretize the fractional time derivative terms by using the [Formula: see text]-approximation and discretize the first-order time derivative term by using the second-order backward differentiation formula. In the fully discrete scheme, we implement the finite element method for the spatial approximation. Unconditional stability of the fully discrete scheme is proven and its optimal convergence order is obtained. Numerical experiments are carried out to demonstrate our theoretical analysis.

The Lerch zeta function as a fractional derivative

We derive and prove a new formulation of the Lerch zeta function as a fractional derivative of an elementary function. We demonstrate how this formulation interacts very naturally with basic known properties of Lerch zeta, and use the functional equation to obtain a second formulation in terms of fractional derivatives.

Analytical solutions of fractal-hydro-thermal model for two-phase flow in thermal stimulation enhanced coalbed methane recovery

Thermal stimulation is a useful supplementary mining technique for the enhancement of coalbed methane recovery. This technique couples the temperature change with gas-water two-phase flow in the mining process. Many integer dimension hydro-thermal models have been proposed but cannot well describe this coupling because two-phase flow and heat conduction are usually non-linear, tortuous and fractal. In this study, a fractal-hydro-thermal coupling model is proposed to describe the coupling between heat conduction and two-phase flow behaviors in terms of fractional time and space derivatives. This model is analytically solved through the fractal travelling-wave method for pore pressure and production rate of gas and water. The analytical solutions are compared with the in-situ coalbed methane production rate. Results show that our proposed fractal-hydro-thermal model can describe both heat and mass transfers in thermal stimulation enhanced coalbed methane recovery.

A Survey on Fractional Order Control Techniques for Unmanned Aerial and Ground Vehicles

In recent years, numerous applications of science and engineering for modeling and control of unmanned aerial vehicles (UAVs) and unmanned ground vehicles (UGVs) systems based on fractional calculus have been realized. The extra fractional order derivative terms allow to optimizing the performance of the systems. The review presented in this paper focuses on the control problems of the UAVs and UGVs that have been addressed by the fractional order techniques over the last decade.

Dynamic cobweb models with conformable fractional derivatives

In this paper, basic and more realistic dynamic cobweb models are developed in terms of conformable fractional derivatives. The general solutions and stability criteria for the proposed models are given. Moreover, the developed models are illustrated with examples on several time scales.

Stochastic P-Bifurcation of a Bistable Viscoelastic Beam with Fractional Constitutive Relation under Gaussian White Noise

In this paper, we study the stochastic P-bifurcation problem for axially moving of a bistable viscoelastic beam with fractional derivatives of high order nonlinear terms under Gaussian white noise excitation. First, using the principle for minimum mean square error, we show that the fractional derivative term is equivalent to a linear combination of the damping force and restoring force, so that the original system can be simplified to an equivalent system. Second, we obtain the stationary Probability Density Function (PDF) of the system’s amplitude by stochastic averaging, and using singularity theory, we find the critical parametric condition for stochastic P-bifurcation of amplitude of the system. Finally, we analyze the types of the stationary PDF curves of the system qualitatively by choosing parameters corresponding to each region within the transition set curve. We verify the theoretical analysis and calculation of the transition set by showing the consistency of the numerical results obtained by Monte Carlo simulation with the analytical results. The method used in this paper directly guides the design of the fractional order viscoelastic material model to adjust the response of the system.

Fractional optimal control of an HIV/AIDS epidemic model with random testing and contact tracing

In this paper, we present a fractional model for transmission dynamics of HIV/AIDS with random testing and contact tracing and incorporate into the model the control parameters of condom use and antiretroviral therapy (ART) aimed at controlling the spread of HIV/AIDS epidemic. The stability of equilibria of the model is discussed using the stability theorem and using the fractional La-Salle invariance principle for fractional differential equations. The effect of the fractional derivative order $$\alpha $$ ( $$ 0.6\le \alpha <1 $$ ) on the HIV/AIDS epidemic model is investigated. The numerical results show that the derivative order $$\alpha $$ can play the role of precautionary measures against infection transmission, treatment of infection and delay in accepting HIV test. We give a general formulation for a fractional optimal control problem, in which the state and co-state equations are given in terms of the left fractional derivatives. We develop the Forward–Backward sweep method using the Adams-type predictor–corrector method to solve the fractional optimal control of the model. The results show that implementing the ART treatment control of diagnosed HIV-infected people together with contact tracing results in a significant decrease in the number of undiagnosed HIV-infected population and AIDS people. Also, implementing all the control efforts decreases significantly the number of diagnosed and undiagnosed HIV-infected population and AIDS people. In addition, the values of the controls are reduced when the fractional derivative order $$\alpha $$ limits to 1 ( $$ 0< \alpha < 1$$ ).