Cosmological Models with Fractional Derivatives and Fractional Action Functional

Cosmological models of a scalar field with dynamical equations containing fractional derivatives or derived from the Einstein-Hilbert action of fractional order, are constructed. A number of exact solutions to those equations of fractional cosmological models in both eases is given.

Fractional Order Proportional Integral Derivative Controller Design and Simulation for Bioengineering Systems

This paper deals with the study of fractional order system tuning method based on Factional Order Proportional Integral Derivative( FOPID) controller in allusion to the nonlinear characteristics and fractional order mathematical model of bioengineering systems. The main contents include the design of FOPID controller and the simulation for bioengineering systems. The simulation results show that the tuning method of fractional order system based on the FOPID controller outperforms the fractional order system based on Fractional Order Proportional Integral( FOPI) controller. As it can enhance control character and improve the robustness of the system.

NON-EXISTENCE FOR FRACTIONALLY DAMPED FRACTIONAL DIFFERENTIAL PROBLEMS

In this paper, we are concerned with a fractional differential inequality containing a lower order fractional derivative and a polynomial source term in the right hand side. A non-existence of non-trivial global solutions result is proved in an appropriate space by means of the test-function method. The range of blow up is found to depend only on the lower order derivative. This is in line with the well-known fact for an internally weakly damped wave equation that solutions will converge to solutions of the parabolic part.

Existence of Solution for a Coupled System of Fractional Integro-Differential Equations on an Unbounded Domain

We present the existence of solution for a coupled system of fractional integro-differential equations. The differential operator is taken in the Caputo fractional sense. We combine the diagonalization method with Arzela-Ascoli theorem to show a fixed point theorem of Schauder.

Research on Stick-Slip Vibration Suppression Method of Drill String Based on Machine Learning Optimization

During the drilling process,stick-slip vibration of the drill string is mainly caused by the nonlinear friction gen-erated by the contact between the drill bit and the rock.To eliminate the fatigue wear of downhole drilling tools caused by stick-slip vibrations,the Fractional-Order Proportional-Integral-Derivative(FOPID)controller is used to suppress stick-slip vibrations in the drill string.Although the FOPID controller can effectively suppress the drill string stick-slip vibration,its structure isflexible and parameter setting is complicated,so it needs to use the cor-responding machine learning algorithm for parameter optimization.Based on the principle of torsional vibration,a simplified model of multi-degree-of-freedom drill string is established and its block diagram is designed.The continuous nonlinear friction generated by cutting rock is described by the LuGre friction model.The adaptive learning strategy of genetic algorithm(GA),particle swarm optimization(PSO)and particle swarm optimization improved(IPSO)by arithmetic optimization(AOA)is used to optimize and adjust the controller parameters,and the drill string stick-slip vibration is suppressed to the greatest extent.The results show that:When slight drill string stick-slip vibration occurs,the FOPID controller optimized by machine learning algorithm has a good effect on suppressing drill string stick-slip vibration.However,the FOPID controller cannot get the drill string system which has fallen into serious stick-slip vibration(stuck pipe)out of trouble,and the machine learning algorithm is required to mark a large amount of data on adjacent Wells to train the model.Set a reasonable range of drilling parameters(weight on bit/drive torque)in advance to avoid severe stick-slip vibration(stuck pipe)in the drill string system.

Asymptotic Formulas of the Solutions and the Trace Formulas for the Polynomial Pencil of the Sturm-Liouville Operators

This work studies the asymptotic formulas for the solutions of the Sturm-Liouville equation with the polynomial dependence in the spectral parameter. Using these asymptotic formulas it is proved some trace formulas for the eigenvalues of a simple boundary problem generated in a finite interval by the considered Sturm-Liouville equation.

Control of a 3-RRR Planar Parallel Robot Using Fractional Order PID Controller

3-RRR planar parallel robots are utilized for solving precise material-handling problems in industrial automation applications.Thus,robust and stable control is required to deliver high accuracy in comparison to the state of the art.The operation of the mechanism is achieved based on three revolute(3-RRR)joints which are geometrically designed using an open-loop spatial robotic platform.The inverse kinematic model of the system is derived and analyzed by using the geometric structure with three revolute joints.The main variables in our design are the platform base positions,the geometry of the joint angles,and links of the 3-RRR planar parallel robot.These variables are calcula ted based on Cayley-Menger determinants and bilateration to det ermine the final position of the platform when moving and placing objects.Additionally,a proposed fractional order proportional integral derivative(FOPID)is optimized using the bat optimization algorithm to control the path tracking of the center of the 3-RRR planar parallel robot.The design is compared with the state of the art and simulated using the Matlab environment to validate the effectiveness of the proposed controller.Furthermore,real-time implementation has been tested to prove that the design performance is practical.

Blow-up of Solutions for a Time-space Fractional Evolution System

In this paper, firstly, we study the local existence and uniqueness of mild solutions for fractional evolution systems with nonlocal in time nonlinearity. Then, we claim that such a mild solution is weak solution of this system. Finally, we prove a blow-up result under some conditions.

Mild Solutions of a Class of Conformable Fractional Differential Equations with Nonlocal Conditions

This paper deals with the existence,uniqueness and continuous dependence of mild solutions for a class of conformable fractional differential equations with nonlocal initial conditions.The results are obtained by means of the classical fixed point theorems combined with the theory of cosine family of linear operators.

EXISTENCE AND UNIQUENESS OF POSITIVE SOLUTIONS TO A SYSTEM OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS WITH DELAY

Using a fixed point method, in this paper we discuss the existence and uniqueness of positive solutions to a class system of nonlinear fractional differential equations with delay and obtain some new results.

Application of Fractional Order PID Controller for AGC Under Deregulated Environment

In this paper, a fractional order proportional integral derivative （FOPID） controller for multiarea automatic generation control （AGC） scheme has been designed. FOPID controller has five parameters and provides two additional degrees of flexibility in comparison to a proportional integral derivative （PID） controller. The optimal values of parameters of FOPID controller have been determined using Big Bang Big Crunch （BBBC） search algorithm. The designed controller regulates real power output of generators to achieve the best dynamic response of frequency and tie-line power on a load perturbation. The complete scheme for designing of the controllers has been developed and demonstrated on multiarea deregulated power system. The performance of the designed FOPID controllers has been compared with the optimally tuned PID controllers. It is observed from the results that the FOPID controller shows a considerable improvement in the performance as compared to the conventional PID controller.

A Unified FastMemory-Saving Time-SteppingMethod for Fractional Operators and Its Applications

Time-dependent fractional partial differential equations typically require huge amounts of memory and computational time,especially for long-time integration,which taxes computational resources heavily for high-dimensional problems.Here,we first analyze existing numerical methods of sum-of-exponentials for approximating the kernel function in constant-order fractional operators,and identify the current pitfalls of such methods.In order to overcome the pitfalls,an improved sum-of-exponentials is developed and verified.We also present several sumof-exponentials for the approximation of the kernel function in variable-order fractional operators.Subsequently,based on the sum-of-exponentials,we propose a unified framework for fast time-stepping methods for fractional integral and derivative operators of constant and variable orders.We test the fast method based on several benchmark problems,including fractional initial value problems,the time-fractional Allen-Cahn equation in two and three spatial dimensions,and the Schr¨odinger equation with nonreflecting boundary conditions,demonstrating the efficiency and robustness of the proposed method.The results show that the present fast method significantly reduces the storage and computational cost especially for long-time integration problems.

EXISTENCE AND UNIQUENESS OF THE SOLUTION FOR A CLASS OF NONLINEAR FRACTIONAL DIFFERENTIAL SYSTEM

In this paper we prove the existence and uniqueness of the solution for a class of nonlinear fractional differential system, and investigate the dependence of the solution on the orderαi.

CAUCHY PROBLEM FOR FRACTIONAL BEAM-LIKE EQUATIONS

In one space-and in one time -dimension a beam-like equation is solved, where the second time derivative is replaced by the α- fractional time derivative, 1 〈 α ≤ 2. The solution is given in closed form in terms of the Mttag-Leffler functions in two parameters.