Study for System of Nonlinear Differential Equations with Riemann-Liouville Fractional Derivative

In this work, we study existence theorem of the initial value problem for the system of fractional differential equations where Dα denotes standard Riemann-Liouville fractional derivative, 0 and A ?is a square matrix. At the same time, power-type estimate for them has been given.

THE NONEMPTINESS AND COMPACTNESS OF MILD SOLUTION SETS FOR RIEMANN-LIOUVILLE FRACTIONAL DELAY DIFFERENTIAL VARIATIONAL INEQUALITIES

This paper investigates the nonemptiness and compactness of the mild solution set for a class of Riemann-Liouville fractional delay differential variational inequalities,which are formulated by a Riemann-Liouville fractional delay evolution equation and a variational inequality.Our approach is based on the resolvent technique and a generalization of strongly continuous semigroups combined with Schauder's fixed point theorem.

High-speed train cooperativecontrol based on fractional-ordersliding mode adaptive algorithm

Purpose–This study aims to propose an adaptive fractional-order sliding mode controller to solve the problem of train speed tracking control and position interval control under disturbance environment in moving block system,so as to improve the tracking efficiency and collision avoidance performance.Design/methodology/approach–The mathematical model of information interaction between trains is established based on algebraic graph theory,so that the train can obtain the state information of adjacent trains,and then realize the distributed cooperative control of each train.In the controller design,the sliding mode control and fractional calculus are combined to avoid the discontinuous switching phenomenon,so as to suppress the chattering of sliding mode control,and a parameter adaptive law is constructed to approximate the time-varying operating resistance coefficient.Findings–The simulation results show that compared with proportional integral derivative(PID)control and ordinary sliding mode control,the control accuracy of the proposed algorithm in terms of speed is,respectively,improved by 25%and 75%.The error frequency and fluctuation range of the proposed algorithm are reduced in the position error control,the error value tends to 0,and the operation trend tends to be consistent.Therefore,the control method can improve the control accuracy of the system and prove that it has strong immunity.Originality/value–The algorithm can reduce the influence of external interference in the actual operating environment,realize efficient and stable tracking of trains,and ensure the safety of train control.

Nonlinear feedback synchronisation control between fractional-order and integer-order chaotic systems

This paper focuses on the synchronisation between fractional-order and integer-order chaotic systems. Based on Lyapunov stability theory and numerical differentiation, a nonlinear feedback controller is obtained to achieve the synchronisation between fractional-order and integer-order chaotic systems. Numerical simulation results are presented to illustrate the effectiveness of this method.

Improved quantum bacterial foraging algorithm for tuning parameters of fractional-order PID controller

The quantum bacterial foraging optimization(QBFO)algorithm has the characteristics of strong robustness and global searching ability. In the classical QBFO algorithm, the rotation angle updated by the rotation gate is discrete and constant,which cannot affect the situation of the solution space and limit the diversity of bacterial population. In this paper, an improved QBFO(IQBFO) algorithm is proposed, which can adaptively make the quantum rotation angle continuously updated and enhance the global search ability. In the initialization process, the modified probability of the optimal rotation angle is introduced to avoid the existence of invariant solutions. The modified operator of probability amplitude is adopted to further increase the population diversity.The tests based on benchmark functions verify the effectiveness of the proposed algorithm. Moreover, compared with the integerorder PID controller, the fractional-order proportion integration differentiation(PID) controller increases the complexity of the system with better flexibility and robustness. Thus the fractional-order PID controller is applied to the servo system. The tuning results of PID parameters of the fractional-order servo system show that the proposed algorithm has a good performance in tuning the PID parameters of the fractional-order servo system.

Existence of positive solutions for integral boundary value problem of fractional differential equations

In this paper,we concern ourselves with the existence of positive solutions for a type of integral boundary value problem of fractional differential equations with the fractional order linear derivative operator. By using the fixed point theorem in cone,the existence of positive solutions for the boundary value problem is obtained. Some examples are also presented to illustrate the application of our main results.

Numerical Solutions of Fractional Differential Equations by Using Fractional Taylor Basis

In this paper, a new numerical method for solving fractional differential equations(FDEs) is presented. The method is based upon the fractional Taylor basis approximations. The operational matrix of the fractional integration for the fractional Taylor basis is introduced. This matrix is then utilized to reduce the solution of the fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of this technique.

Numerical Methods for Solving Space Fractional Partial Differential Equations Using Hadamard Finite-Part Integral Approach

We introduce a novel numerical method for solving two-sided space fractional partial differential equations in two-dimensional case.The approximation of the space fractional Riemann-Liouville derivative is based on the approximation of the Hadamard finite-part integral which has the convergence order O(h^3-a),where h is the space step size and α∈(1,2)is the order of Riemann-Liouville fractional derivative.Based on this scheme,we introduce a shifted finite difference method for solving space fractional partial differential equations.We obtained the error estimates with the convergence orders O(τ+h^3-a+h^β),where τ is the time step size and β>0 is a parameter which measures the smoothness of the fractional derivatives of the solution of the equation.Unlike the numerical methods for solving space fractional partial differential equations constructed using the standard shifted Griinwald-Letnikov formula or higher order Lubich's methods which require the solution of the equation to satisfy the homogeneous Dirichlet boundary condition to get the firstorder convergence,the numerical method for solving the space fractional partial differential equation constructed using the Hadamard finite-part integral approach does not require the solution of the equation to satisfy the Dirichlet homogeneous boundary condition.Numerical results show that the experimentally determined convergence order obtained using the Hadamard finite-part integral approach for solving the space fractional partial differential equation with non-homogeneous Dirichlet boundary conditions is indeed higher than the convergence order obtained using the numerical methods constructed with the standard shifted Griinwald-Letnikov formula or Lubich's higher order approximation schemes.

Nonnegative Solutions for a Riemann-Liouville Fractional Boundary Value Problem

We investigate the existence of nonnegative solutions for a Riemann-Liouville fractional differential equation with integral terms, subject to boundary conditions which contain fractional derivatives and Riemann-Stieltjes integrals. In the proof of the main results, we use the Banach contraction mapping principle and the Krasnosel’skii fixed point theorem for the sum of two operators.

ON A COUPLED INTEGRO-DIFFERENTIAL SYSTEM INVOLVING MIXED FRACTIONAL DERIVATIVES AND INTEGRALS OF DIFFERENT ORDERS

By applying the standard fixed point theorems,we prove the existence and uniqueness results for a system of coupled differential equations involving both left Caputo and right Riemann-Liouville fractional derivatives and mixed fractional integrals,supplemented with nonlocal coupled fractional integral boundary conditions.An example is also constructed for the illustration of the obtained results.

Some New Delay Integral Inequalities Based on Modified Riemann-Liouville Fractional Derivative and Their Applications

By using the properties of modified Riemann-Liouville fractional derivative, some new delay integral inequalities have been studied. First, we offered explicit bounds for the unknown functions, then we applied the results to the research concerning the boundness, uniqueness and continuous dependence on the initial for solutions to certain fractional differential equations.

Resolution method for overlapping peaks based on the fractional-order differential

Equations between the differential order and the maximum of the fractional-order differential for the specified peak signals are developed based on the variation of the maximum of the specified peak signals at different orders. Also, equations between the differential order and the zero-crossing of the fractional-order differential of the specified peak signals are proposed according to the variation of the zero-crossing of the specified peak signals at different orders. Characteristic paramters of the Gaus- sian peak, Lorentzian peak, and Tsallis peak can be estimated using estimator I and estimator II which are obtained by the equations above. As a result, a new method is presented to resolve the overlapped peaks signal. Firstly, a fractional-order differential of the specified peak signals is obtained with the fractional-order differentiation filter. Then, characteristic paramters of the specified peak signals can be extracted using estimator I and estimator II. Finally, the Tsallis peak is used as a model to assign the overlapping peak signals correctly. Experimental results show that the proposed method is efficient and effective for the simulated overlapping peaks and detected overlapping voltammetric peak signals.

Closed-form solutions to fractional-order linear differential equations

The definitions and properties of widely used fractional-order derivatives are summarized in this paper.The characteristic polynomials of the fractional-order systems are pseudo-polynomials whose powers of the complex variable are non-integers.This kind of systems can be approximated by high-order integer-order systems,and can be analyzed and designed by the sophisticated integer-order systems methodology.A new closed-form algorithm for fractional-order linear differential equations is proposed based on the definitions of fractional-order derivatives,and the effectiveness of the algorithm is illustrated through examples.

Image Enhancement Using Adaptive Fractional Order Filter

Image enhancement is an important preprocessing task as the contrast is low in most of the medical images,Therefore,enhancement becomes the mandatory process before actual image processing should start.This research article proposes an enhancement of the model-based differential operator for the images in general and Echocardiographic images,the proposed operators are based on Grunwald-Letnikov(G-L),Riemann-Liouville(R-L)and Caputo(Li&Xie),which are the definitions of fractional order calculus.In this fractional-order,differentiation is well focused on the enhancement of echocardiographic images.This provoked for developing a non-linear filter mask for image enhancement.The designed filter is simple and effective in terms of improving the contrast of the input low contrast images and preserving the textural features,particularly in smooth areas.The novelty of the proposed method involves a procedure of partitioning the image into homogenous regions,details,and edges.Thereafter,a fractional differential mask is appropriately chosen adaptively for enhancing the partitioned pixels present in the image.It is also incorporated into the Hessian matrix with is a second-order derivative for every pixel and the parameters such as average gradient and entropy are used for qualitative analysis.The wide range of existing state-of-the-art techniques such as fixed order fractional differential filter for enhancement,histogram equalization,integer-order differential methods have been used.The proposed algorithm resulted in the enhancement of the input images with an increased value of average gradient as well as entropy in comparison to the previous methods.The values obtained are very close(almost equal to 99.9%)to the original values of the average gradient and entropy of the images.The results of the simulation validate the effectiveness of the proposed algorithm.

A Novel Analytical Technique of the Fractional Bagley-Torvik Equations for Motion of a Rigid Plate in Newtonian Fluids

The current paper is concerned with a modified Homotopy perturbation technique.This modification allows achieving an exact solution of an initial value problem of the fractional differential equation.The approach is powerful,effective,and promising in analyzing some classes of fractional differential equations for heat conduction problems and other dynamical systems.To crystallize the new approach,some illustrated examples are introduced.

Mathematical Study of A Memory Induced Biochemical System

In this work, to study the effect of memory on a bi-substrate enzyme kinetic reaction, we have introduced an approach to fractionalize the system, considering it as a threecompartmental model. Solutions of the fractionalized system are compared with the corresponding integer-order model. The equilibrium points of the fractionalized system are derived analytically. Their stability properties are discussed from numerical aspect. We determine the changes of the substances due to the changes of "memory effect". The effect is discussed critically from the perspective of product formation. We have also analyzed the memory induced system with a control measure in view of optimizing the product. Our numerical result reveals that the solutions of the fractionalized system, when it is free from memory, are in good agreement with the integer-order system.It is noticed that the effect of memory influences the reaction in the forward direction and assists in yielding the product more quickly. However, an extensive use of memory makes the system slower, but introduction of a control input makes the reaction faster. It is possible to overcome the slowness of the reaction due to the undue effect of memory by appropriate use of a control measure.

A Fractional Drift Diffusion Model for Organic Semiconductor Devices

Because charge carriers of many organic semiconductors(OSCs)exhibit fractional drift diffusion(Fr-DD)transport properties,the need to develop a Fr-DD model solver becomes more apparent.However,the current research on solving the governing equations of the Fr-DD model is practically nonexistent.In this paper,an iterative solver with high precision is developed to solve both the transient and steady-state Fr-DD model for organic semiconductor devices.The Fr-DD model is composed of two fractionalorder carriers(i.e.,electrons and holes)continuity equations coupled with Poisson’s equation.By treating the current density as constants within each pair of consecutive grid nodes,a linear Caputo’s fractional-order ordinary differential equation(FrODE)can be produced,and its analytic solution gives an approximation to the carrier concentration.The convergence of the solver is guaranteed by implementing a successive over-relaxation(SOR)mechanism on each loop of Gummel’s iteration.Based on our derivations,it can be shown that the Scharfetter–Gummel discretization method is essentially a special case of our scheme.In addition,the consistency and convergence of the two core algorithms are proved,with three numerical examples designed to demonstrate the accuracy and computational performance of this solver.Finally,we validate the Fr-DD model for a steady-state organic field effect transistor(OFET)by fitting the simulated transconductance and output curves to the experimental data.

Positive Solutions for Systems of Coupled Fractional Boundary Value Problems

We investigate the existence and nonexistence of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations with coupled integral boundary conditions which contain some positive constants.

Qualitative Analysis of a Fractional Pandemic Spread Model of the Novel Coronavirus (COVID-19)

In this study,we classify the genera of COVID-19 and provide brief information about the root of the spread and the transmission from animal(natural host)to humans.We establish a model of fractional-order differential equations to discuss the spread of the infection from the natural host to the intermediate one,and from the intermediate one to the human host.At the same time,we focus on the potential spillover of bat-borne coronaviruses.We consider the local stability of the co-existing critical point of the model by using the Routh–Hurwitz Criteria.Moreover,we analyze the existence and uniqueness of the constructed initial value problem.We focus on the control parameters to decrease the outbreak from pandemic form to the epidemic by using both strong and weak Allee Effect at time t.Furthermore,the discretization process shows that the system undergoes Neimark–Sacker Bifurcation under specific conditions.Finally,we conduct a series of numerical simulations to enhance the theoretical findings.

A collocation method for numerical solutions of fractional-order logistic population model

In this study, a collocation technique is presented for approximate solution of the fractional-order logistic population model. Actually, we develop the Bessel collocation method by using the fractional derivative in the Caputo sense to obtain the approximate solutions of this model problem. By means of the fractional derivative in the Caputo sense, the collocation points, the Bessel functions of the first kind, the method transforms the model problem into a system of nonlinear algebraic equations. Numerical applications are given to demonstrate efficiency and accuracy of the method. In applications, the reliability of the scheme is shown by the error function based on the accuracy of the approximate solution.