Legendre-Weighted Residual Methods for System of Fractional Order Differential Equations

The numerical approach for finding the solution of fractional order systems of boundary value problems (BPVs) is derived in this paper. The implementation of the weighted residuals such as Galerkin, Least Square, and Collocation methods are included for solving fractional order differential equations, which is broadened to acquire the approximate solutions of fractional order systems with differentiable polynomials, namely Legendre polynomials, as basis functions. The algorithm of the residual formulations of matrix form can be coded efficiently. The interpretation of Caputo fractional derivatives is employed here. We have demonstrated these methods numerically through a few examples of linear and nonlinear BVPs. The results in absolute errors show that the present method efficiently finds the numerical solutions of fractional order systems of differential equations.

Robust Fractional-Order Proportional-Integral Observer for Synchronization of Chaotic Fractional-Order Systems

In this paper, we propose a robust fractional-order proportional-integral(FOPI) observer for the synchronization of nonlinear fractional-order chaotic systems. The convergence of the observer is proved, and sufficient conditions are derived in terms of linear matrix inequalities(LMIs) approach by using an indirect Lyapunov method. The proposed FOPI observer is robust against Lipschitz additive nonlinear uncertainty. It is also compared to the fractional-order proportional(FOP) observer and its performance is illustrated through simulations done on the fractional-order chaotic Lorenz system.

GLOBAL SOLUTIONS OF SYSTEMS OF NONLINEAR IMPULSIVE VOLTERRA INTEGRAL EQUATIONS IN BANACH SPACES

The existence of solutions for systems of nonlinear impulsive Volterra integral equations on the infinite interval R+ with an infinite number of moments of impulse effect in Banach spaces is studied. Some existence theorems of extremal solutions are obtained, which extend the related results for this class of equations on a finite interval with a finite. number of moments of impulse effect. The results are demonstrated by means of an example of an infinite systems for impulsive integral equations.

ONE TYPE OF INTEGRALS FOR THE EQUATIONS OF MOTION OF HIGHER-ORDER NONHOLONOMIC SYSTEMS

This paper presents one type of integrals and its condition of existence for the equations of motion of higher-order nonholonomic systems, including l-order integral (generalized energy integral), 2-order integral and p-order integral (p>2)All of these integrals can be constructed by the Lagrangian function of the system. Two examples are given to illustrate the application of the suggested method.

Existence Theorem for a Nonlinear Functional Integral Equation and an Initial Value Problem of Fractional Order in L₁(R₊)

The aim of this paper is to study the existence of integrable solutions of a nonlinear functional integral equation in the space of Lebesgue integrable functions on unbounded interval, L1(R+). As an application we deduce the existence of solution of an initial value problem of fractional order that be studied only on a bounded interval. The main tools used are Schauder fixed point theorem, measure of weak noncompactness, superposition operator and fractional calculus.

EXISTENCE OF SOLUTIONS FOR MIXED MONOTONE IMPULSIVE VOLTERRA INTEGRAL EQUATIONS IN BANACH SPACES

This paper studies the existence of solutions for mixed monotone impulsive Volterra integral equations on the infinite interval R+ with an infinite number of moments of impulse effect in Banach spaces. By using the mixed monotone iterative technique and Monch fixed point theorem, Some existence theorems of solutions and coupled minimal and maximal quasisolutions are obtained. Finally, an example is worked out.

On Solving a System of Volterra Integral Equations with Relaxed Monte Carlo Method

A random simulation method was used for treatment of systems of Volterra integral equations of the second kind. Firstly, a linear algebra system was obtained by discretization using quadrature formula. Secondly, this algebra system was solved by using relaxed Monte Carlo method with importance sampling and numerical approximation solutions of the integral equations system were achieved. It is theoretically proved that the validity of relaxed Monte Carlo method is based on importance sampling to solve the integral equations system. Finally, some numerical examples from literatures are given to show the efficiency of the method.

A New Newton-Type Method with Third-Order for Solving Systems of Nonlinear Equations

In this paper, a new two-step Newton-type method with third-order convergence for solving systems of nonlinear equations is proposed. We construct the new method based on the integral interpolation of Newton’s method. Its cubic convergence and error equation are proved theoretically, and demonstrated numerically. Its application to systems of nonlinear equations and boundary-value problems of nonlinear ODEs are shown as well in the numerical examples.

Approximate Controllability of Fractional Order Retarded Semilinear Control Systems

In this paper, approximate controllability of fractional order retarded semilinear systems is studied when the nonlinear term satisfies the newly formulated bounded integral contractor-type conditions. We have shown the existence and uniqueness of the mild solution for the fractional order retarded semilinear systems using an iterative procedure approach. Finally, we obtain the approximate controllability results of the system under simple condition.

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 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.

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.

A COUPLED SYSTEM OF INTEGRAL EQUATIONS IN REFLEXIVE BANACH SPACES

We present an existence theorem for at least one weak solution for a coupled system of integral equations of Volterra type in a reflexive Banach spaces relative to the weak topology.

SOLUTIONS FOR A SYSTEM OF NONLINEAR RANDOM INTEGRAL AND DIFFERENTIAL EQUATIONS UNDER WEAK TOPOLOGY

In this paper, a Darbao type random fixed point theorem for a system of weak continuous random operators with random domain is first proved. When, by using the theorem, some existence criteria of random solutions for a systems of nonlinear random Volterra integral equations relative to the weak topology in Banach spaces are given. As applications, some existence theorems of weak random solutions for the random Cauchy problem of a system of nonlinear random differential equations are obtained, as well as the existence of extremal random solutions and random comparison results for these systems of random equations relative to weak topology in Banach spaces. The corresponding results of Szep, Mitchell-Smith, Cramer-Lakshmikantham, Lakshmikantham-Leela and Ding are improved and generalized by these theorems.

CLASSIFICATION OF POSITIVE SOLUTIONS TO ASYSTEM OF HARDY-SOBOLEV TYPE EQUATIONS

In this paper, we are concerned with the following Hardy-Sobolev type system{(-?)^(α/2) u(x) =v^q(x)/|y|^(t_2) (-?)α/2 v(x) =u^p(x)/|y|^(t_1),x =(y, z) ∈(R ~k\{0}) × R^(n-k),(0.1)where 0 < α < n, 0 < t_1, t_2 < min{α, k}, and 1 < p ≤ τ_1 :=(n+α-2t_1)/( n-α), 1 < q ≤ τ_2 :=(n+α-2 t_2)/( n-α).We first establish the equivalence of classical and weak solutions between PDE system(0.1)and the following integral equations(IE) system{u(x) =∫_( R^n) G_α(x, ξ)v^q(ξ)/|η|t^2 dξ v(x) =∫_(R^n) G_α(x, ξ)(u^p(ξ))/|η|^(t_1) dξ,(0.2)where Gα(x, ξ) =(c n,α)/(|x-ξ|^(n-α))is the Green's function of(-?)^(α/2) in R^n. Then, by the method of moving planes in the integral forms, in the critical case p = τ_1 and q = τ_2, we prove that each pair of nonnegative solutions(u, v) of(0.1) is radially symmetric and monotone decreasing about the origin in R^k and some point z0 in R^(n-k). In the subcritical case (n-t_1)/(p+1)+(n-t_2)/(q+1)> n-α,1 < p ≤ τ_1 and 1 < q ≤ τ_2, we derive the nonexistence of nontrivial nonnegative solutions for(0.1).

Fractional Modeling and Analysis of Coupled MR Damping System

The coupled magnetorheological (MR) damping system addressed in this paper contains rubber spring and magnetorheological damper. The device inherits the damping merits of both the rubber spring and the magnetorheological damper. Here a fractional-order constitutive equation is introduced to study the viscoelasticity of the combined damper. An introduction to the definitions of fractional calculus, and the transfer function representation of a fractional-order system are given. The fractional-order system model of a magnetorheological vibration platform is set up using fractional calculus, and the function of displacement is presented. It is indicated that the fractional-order constitutive equation and the transfer function are feasible and effective means for investigating of magnetorheological vibration device. © 2014 Chinese Association of Automation.

一类分数阶Lotka-Volterra系统的动力学研究

本文建立了一个新的分数阶Lotka-Volterra模型.首先,我们考虑所涉及的分数阶Lotka-Volterra模型的存在性、唯一性和非负性.其次,通过分析特征方程将时滞视为分岔参数,建立一个新的充分条件来保证系统的稳定性和Hopf分岔的出现.

Existence and Uniqueness for the Boundary Value Problems of Nonlinear Fractional Differential Equation

This paper studies the existence and uniqueness of solutions for a class of boundary value problems of nonlinear fractional order differential equations involving the Caputo fractional derivative by employing the Banach’s contraction principle and the Schauder’s fixed point theorem. In addition, an example is given to demonstrate the application of our main results.

基于El-Nabulsi分数阶模型的广义Birkhoff系统的积分因子方法

将积分因子方法应用于El-Nabulsi分数阶模型下广义Birkhoff系统.首先,给出了分数阶广义El-Nabulsi-Birkhoff方程的积分因子定义;其次,寻求El-Nabulsi分数阶模型下Birkhoff系统守恒量存在的必要条件,建立了相应的守恒定理;最后,讨论了该系统特例情况下的积分因子方法.