Symmetries and conserved quantities for fractional action-like Pfaffian variational problems

The fractional Pfaffian variational problems and the fractional Noether theory are studied under a fractional model presented by El-Nabulsi. Firstly, the fractional action-like Pfaffian variational problem is presented, the El-Nabulsi–Pfaff–Birkhoff–d’Alembert fractional principle is established, then the El-Nabulsi–Birkhoff fractional equations are derived; secondly, the definitions and criteria of the fractional Noether symmetric transformations are given, which are based on the invariance of El-Nabulsi–Pfaffian action under the infinitesimal transformations of group, then the inner relationship between a fractional Noether symmetry and a fractional conserved quantity is established; finally, two examples are given to illustrate the application of the results.     

Approximate symmetry group classification for a nonlinear fractional filtration equation of diffusion-wave type

A one-dimensional nonlinear fractional filtration equation with the Riemann–Liouville time-fractional derivative is proposed for modeling fluid flow through a porous medium. This equation is derived under an assumption that the fluid has a fractional equation of state in which the fluid density depends on the time-fractional derivative of pressure. The obtained equation belongs to the diffusion-wave type of equations. A case when the order of fractional differentiation is close to an integer number is considered, and a small parameter is introduced into the fractional filtration equation under consideration. An expansion of the Riemann–Liouville time-fractional derivative into the series with respect to this small parameter is obtained. Using this expansion, a first-order approximation of the derived fractional filtration equation is performed. Next, the problem of approximate Lie point symmetry group classification for this approximate nonlinear filtration equation with a small parameter is studied. It is shown that approximate symmetry groups admitted by different realizations of the approximate filtration equation have much more dimensions than the corresponding exact Lie point symmetry groups admitted by unperturbed fractional diffusion-wave equations. Obtained classification results permit to construct approximate invariant solutions for the considered nonlinear time-fractional filtration equations.     

On fractional Euler–Lagrange and Hamilton equations and the fractional generalization of total time derivative

Fractional mechanics describe both conservative and nonconservative systems. The fractional variational principles gained importance in studying the fractional mechanics and several versions are proposed. In classical mechanics, the equivalent Lagrangians play an important role because they admit the same Euler–Lagrange equations. By adding a total time derivative of a suitable function to a given classical Lagrangian or by multiplying with a constant, the Lagrangian we obtain are the same equations of motion. In this study, the fractional discrete Lagrangians which differs by a fractional derivative are analyzed within Riemann–Liouville fractional derivatives. As a consequence of applying this procedure, the classical results are reobtained as a special case. The fractional generalization of Faà di Bruno formula is used in order to obtain the concrete expression of the fractional Lagrangians which differs from a given fractional Lagrangian by adding a fractional derivative. The fractional Euler–Lagrange and Hamilton equations corresponding to the obtained fractional Lagrangians are investigated, and two examples are analyzed in detail.     

Correcting the initialization of models with fractional derivatives via history-dependent conditions

Fractional differential equations are more and more used in modeling memory(history-dependent,nonlocal,or hereditary) phenomena.Conventional initial values of fractional differential equations are define at a point,while recent works defin initial conditions over histories.We prove that the conventional initialization of fractional differential equations with a Riemann–Liouville derivative is wrong with a simple counter-example.The initial values were assumed to be arbitrarily given for a typical fractional differential equation,but we fin one of these values can only be zero.We show that fractional differential equations are of infinit dimensions,and the initial conditions,initial histories,are define as functions over intervals.We obtain the equivalent integral equation for Caputo case.With a simple fractional model of materials,we illustrate that the recovery behavior is correct with the initial creep history,but is wrong with initial values at the starting point of the recovery.We demonstrate the application of initial history by solving a forced fractional Lorenz system numerically.     

Noether symmetries and conserved quantities for Birkhoffian systems with time delay

The Noether symmetries and conserved quantities for Birkhoffian systems with time delay are proposed and studied. First, the Pfaff–Birkhoff principle with time delay is proposed, and Birkhoff’s equations with time delay are obtained. Second, based on the invariance of the Pfaff action with time delay under a group of infinitesimal transformations, the Noether symmetric transformations and the Noether quasisymmetric transformations of the system are defined, and the criteria of the Noether symmetries are established. Finally, the relationship between the symmetries and the conserved quantities are studied, and the Noether theorems for Birkhoffian systems with time delay are established. Some examples are given to illustrate the application of the results.     

Fractional-order Euler–Lagrange equations and formulation of Hamiltonian equations

This paper presents the fractional order Euler–Lagrange equations and the transversality conditions for fractional variational problems with fractional integral and fractional derivatives defined in the sense of Caputo and Riemann–Liouville. A fractional Hamiltonian formulation was developed and some illustrative examples were treated in detail.

     

The Boubaker polynomials and their application to solve fractional optimal control problems

In this paper, we focus on Boubaker polynomials in fractional calculus area and obtain the operational matrix of Caputo fractional derivative and the operational matrix of the Riemann–Liouville fractional integration for the first time. Also, a general formulation for the operational matrix of multiplication of these polynomials has been achieved to solve the nonlinear problems. Then, these matrices are applied to solve fractional optimal control problems directly. In fact, the functions of the problem are approximated by Boubaker polynomials with unknown coefficients in the constraint equations, performance index and conditions. Thus, a fractional optimal control problem converts to an optimization problem, which can then be solved easily. Convergence of the algorithm is proved. Numerical results are given for several test examples to demonstrate the applicability and efficiency of the method.     

A Hilbert Space Approach to Fractional Differential Equations

Journal of Dynamics and Differential Equations - We study fractional differential equations of Riemann–Liouville and Caputo type in Hilbert spaces. Using exponentially weighted spaces of...     

相对论Birkhoff系统的Noether理论

给出相对论Pfaff-Birkhoff原理和相对论Birkhoff方程。定义相对论Birkhoff系统的无限小变换生成元,根据在无限小变换下微分变分原理的不变性,得到相对论Birkhoff系统的Noether定理和Noether逆定理。最后出应用实例。     

Stability for nonlinear fractional order systems: an indirect approach

In this paper, we first verify that fractional order systems using Caputo’s or Riemann–Liouville’s derivative can be represented by the continuous frequency distributed model with initial value carefully allocated. Then, the relation of the stability between the fractional order system and its corresponding integer order system is discussed and it is proven that stability of integer order system implies the stability of its corresponding fractional order system under some mild conditions. Moreover, we extend the stability theorems to the finite-dimensional case since fractional order systems are always implemented by approximation. Some illustrative examples are finally provided to show the usage and effectiveness of the proposed stability theorems.     

Discrete optimal control for Birkhoffian systems

In this paper, we investigate a discrete variational optimal control for mechanical systems that admit a Birkhoffian representation. Instead of discretizing the original equations of motion, our research is based on a direct discretization of the Pfaff–Birkhoff–d’Alembert principle. The resulting discrete forced Birkhoffian equations then serve as constraints for the minimization of the objective functional. In this way, the optimal control problem is transformed into a finite-dimensional optimization problem, which can be solved by standard methods. This approach yields discrete dynamics, which is more faithful to the continuous equations of motion and consequently yields more accurate solutions to the optimal control problem which is to be approximated. We illustrate the method numerically by optimizing the control for the damped oscillator.     

A uniqueness criterion for fractional differential equations with Caputo derivative

We investigate the uniqueness of solutions to an initial value problem associated with a nonlinear fractional differential equation of order α∈(0,1). The differential operator is of Caputo type whereas the nonlinearity cannot be expressed as a Lipschitz function. Instead, the Riemann–Liouville derivative of this nonlinearity verifies a special inequality.     

Fractional generalized Hamiltonian equations and its integral invariants

In this paper, we present a new kind of fractional dynamical equations, i.e. the fractional generalized Hamiltonian equations, and study variation equations and the method of the construction of integral invariants of the system. Based on the definition of Riemann–Liouville fractional derivatives, fractional generalized Hamiltonian equations and its variation equations are established. Then, the relation between first integral and integral invariant of the system is studied, and it is proved that, using a first integral, we can construct an integral invariant of the system. As deductions of above results, a construction method of integral invariants of a traditional generalized Hamiltonian system are given. Further, one example of fractional generalized Hamiltonian system is given to illustrate the method and results of the application. Finally, we study the first integral and integral invariant of the Euler equation of a rigid body which rotates with respect to a fixed-point.     

Newtonian law with memory

In this study we analyzed the Newtonian equation with memory. One physical model possessing memory effect is analyzed in detail. The fractional generalization of this model is investigated and the exact solutions within Caputo and Riemann–Liouville fractional derivatives are reported.     

A General Formulation and Solution Scheme for Fractional Optimal Control Problems

Accurate modeling of many dynamic systems leads to a set of Fractional Differential Equations (FDEs). This paper presents a general formulation and a solution scheme for a class of Fractional Optimal Control Problems (FOCPs) for those systems. The fractional derivative is described in the Riemann–Liouville sense. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of FDEs. The Calculus of Variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler–Lagrange equations for the FOCP. The formulation presented and the resulting equations are very similar to those that appear in the classical optimal control theory. Thus, the present formulation essentially extends the classical control theory to fractional dynamic system. The formulation is used to derive the control equations for a quadratic linear fractional control problem. An approach similar to a variational virtual work coupled with the Lagrange multiplier technique is presented to find the approximate numerical solution of the resulting equations. Numerical solutions for two fractional systems, a time-invariant and a time-varying, are presented to demonstrate the feasibility of the method. It is shown that (1) the solutions converge as the number of approximating terms increase, and (2) the solutions approach to classical solutions as the order of the fractional derivatives approach to 1. The formulation presented is simple and can be extended to other FOCPs. It is hoped that the simplicity of this formulation will initiate a new interest in the area of optimal control of fractional systems.     

Synchronization of chaotic-type delayed neural networks and its application

Nonlinear Dynamics - In this paper, the authors analyze a time-fractional advection–diffusion equation, involving the Riemann–Liouville derivative, with a nonlinear source term. They...     

Invariant analysis and exact solutions of nonlinear time fractional Sharma–Tasso–Olver equation by Lie group analysis

This paper is concerned with the time fractional Sharma–Tasso–Olver (FSTO) equation, Lie point symmetries of the FSTO equation with the Riemann–Liouville derivatives are considered. By using the Lie group analysis method, the invariance properties of the FSTO equation are investigated. In the sense of point symmetry, the vector fields of the FSTO equation are presented. And then, the symmetry reductions are provided. By making use of the obtained Lie point symmetries, it is shown that this equation can transform into a nonlinear ordinary differential equation of fractional order with the new independent variable ξ=xt ^?α/3. The derivative is an Erdélyi–Kober derivative depending on a parameter α. At last, by means of the sub-equation method, some exact and explicit solutions to the FSTO equation are given.     

Conserved quantities and adiabatic invariants for fractional generalized Birkhoffian systems

Based on Riemann-Liouville fractional derivatives, conserved quantities and adiabatic invariants for fractional generalized Birkhoffian systems are investigated. Firstly, fractional generalized Birkhoff equations are obtained by studying fractional generalized Pfaff-Birkhoff principle. Secondly, the definition of fractional generalized quasi-symmetry is given, the criteria of fractional generalized quasi-symmetry and the corresponding conserved quantity are achieved for fractional generalized Birkhoffian systems. Thirdly, perturbation to symmetry and adiabatic invariants for disturbed fractional generalized Birkhoffian systems are presented. Finally, an example is given to illustrate the results.