Fractional differential equations of motion in terms of combined Riemann-Liouville derivatives

In this paper, we focus on studying the fractional variational principle and the differential equations of motion for a fractional mechanical system. A combined Riemann-Liouville fractional derivative operator is defined, and a fractional Hamilton principle under this definition is established. The fractional Lagrange equations and the fractional Hamilton canonical equations are derived from the fractional Hamilton principle. A number of special cases are given, showing the universality of our conclusions. At the end of the paper, an example is given to illustrate the application of the results.

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.

An inverse problem to estimate an unknown order of a Riemann-Liouville fractional derivative for a fractional Stokes' first problem for a heated generalized second grade fluid

In this paper,we propose a numerical method to estimate the unknown order of a Riemann-Liouville fractional derivative for a fractional Stokes＇ first problem for a heated generalized second grade fluid.The implicit numerical method is employed to solve the direct problem.For the inverse problem,we first obtain the fractional sensitivity equation by means of the digamma function,and then we propose an efficient numerical method,that is,the Levenberg-Marquardt algorithm based on a fractional derivative,to estimate the unknown order of a Riemann-Liouville fractional derivative.In order to demonstrate the effectiveness of the proposed numerical method,two cases in which the measurement values contain random measurement error or not are considered.The computational results demonstrate that the proposed numerical method could efficiently obtain the optimal estimation of the unknown order of a RiemannLiouville fractional derivative for a fractional Stokes＇ first problem for a heated generalized second grade fluid.

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.

A note on the definition of fractional derivatives applied in rheology

It is known that there exist obivious differences between the two most commonly used definitions of fractional derivatives-Riemann-Liouville （R-L） definition and Caputo definition. The multiple definitions of fractional derivatives in fractional calculus have hindered the application of fractional calculus in rheology. In this paper, we clarify that the R-L definition and Caputo definition are both rheologically imperfect with the help of mechanical analogues of the fractional element model （Scott-Blair model）. We also clarify that to make them perfect rheologically, the lower terminals of both definitions should be put to ∞. We further prove that the R-L definition with lower terminal a →∞ and the Caputo definition with lower terminal a →∞ are equivalent in the differentiation of functions that are smooth enough and functions that have finite number of singular points. Thus we can define the fractional derivatives in rheology as the R-L derivatives with lower terminal a →∞ （or, equivalently, the Caputo derivatives with lower terminal a →∞） not only for steady-state processes, but also for transient processes. Based on the above definition, the problems of composition rules of fractional operators and the initial conditions for fractional differential equations are discussed, respectively. As an example we study a fractional oscillator with Scott-Blair model and give an exact solution of this equation under given initial conditions.

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.

On the definition of fractional derivatives in rheology

During the last two decades fractional calculus has been increasingly applied to physics, especially to rheology.It is well known that there are obivious differences between Riemann-Liouville (R-L) definition and Caputo definition,which are the two most commonly used definitions of fractional derivatives.The multiple definitions of fractional derivatives have hindered the application of fractional calculus in rheology.In this paper,we clarify that the R-L definition and Caputo definition are both Theologically unreasonable with the help of the mechanical analogues of the fractional element model.We also find that to make them more reasonable Theologically,the lower terminals of both definitions should be put to—∞.We further prove that the R-L definition with lower terminal—∞and the Caputo definition with lower terminal—∞are equivalent in the differentiation of functions that are smooth enough and functions that have finite number of singular points.Thus we can define the fractional derivatives in rheology as the R-L derivatives with lower terminal—∞(or,equivalently,the Caputo derivatives with lower terminal—∞) not only for steady-state processes,but also for transient processes.

An Approach to Differential Geometry of Fractional Order via Modified Riemann-Liouville Derivative

In order to cope with some difficulties due to the fact that the derivative of a constant is not zero with the commonly accepted Riemann-Liouville definition of fractional derivative, one （Jumarie） has proposed recently an alternative referred to as （local） modified Riemann-Liouville definition, which directly, provides a Taylor＇s series of fractional order for non differentiable functions. We examine here in which way this calculus can be used as a framework for a differential geometry of fractional or- der. One will examine successively implicit function, manifold, length of curves, radius of curvature, Christoffel coefficients, velocity, acceleration. One outlines the application of this framework to La- grange optimization in mechanics, and one concludes with some considerations on a possible fractional extension of the pseudo-geodesic of thespecial relativity and of the Lorentz transformation.

Implicit finite difference method for fractional percolation equation with Dirichlet and fractional boundary conditions

An implicit finite difference method is developed for a one-dimensional frac- tional percolation equation （FPE） with the Dirichlet and fractional boundary conditions. The stability and convergence are discussed for two special cases, i.e., a continued seep- age flow with a monotone percolation coefficient and a seepage flow with the fractional Neumann boundary condition. The accuracy and efficiency of the method are checked with two numerical examples.

Exact solutions for nonlinear partial fractional differential equations

′In this article, we use the fractional complex transformation to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations. We use the improved （G′/G）-expansion function method to calculate the exact solutions to the time- and space-fractional derivative foam drainage equation and the time- and space-fractional derivative nonlinear KdV equation. This method is efficient and powerful for solving wide classes of nonlinear evolution fractional order equations.

New analytical exact solutions of time fractional KdV KZK equation by Kudryashov methods

In this paper, new exact solutions of the time fractional KdV-Khokhlov-Zabolotskaya-Kuznetsov （KdV-KZK） equa- tion are obtained by the classical Kudryashov method and modified Kudryashov method respectively. For this purpose, the modified Riemann-Liouville derivative is used to convert the nonlinear time fractional KdV-KZK equation into the non- linear ordinary differential equation. In the present analysis, the classical Kudryashov method and modified Kudryashov method are both used successively to compute the analytical solutions of the time fractional KdV-KZK equation. As a result, new exact solutions involving the symmetrical Fibonacci function, hyperbolic function and exponential function are obtained for the first time. The methods under consideration are reliable and efficient, and can be used as an alternative to establish new exact solutions of different types of fractional differential equations arising from mathematical physics. The obtained results are exhibited graphically in order to demonstrate the efficiencies and applicabilities of these proposed methods of solving the nonlinear time fractional KdV-KZK equation.

Fractional Birkhoffian Dynamics Based on Quasi-fractional Dynamics Models and Its Lie Symmetry

In order to investigate the dynamic behavior of non-conservative systems,the Lie symmetries and conserved quantities of fractional Birkhoffian dynamics based on quasi-fractional dynamics model are proposed and studied.The quasi-fractional dynamics model here refers to the variational problem based on the definition of RiemannLiouville fractional integral(RLFI),the variational problem based on the definition of extended exponentially fractional integral(EEFI),and the variational problem based on the definition of fractional integral extended by periodic laws(FIEPL).First,the fractional Pfaff-Birkhoff principles based on quasi-fractional dynamics models are established,and the corresponding Birkhoff’s equations and the determining equations of Lie symmetry are obtained.Second,for fractional Birkhoffian systems based on quasi-fractional models,the conditions and forms of conserved quantities are given,and Lie symmetry theorems are proved.The Pfaff-Birkhoff principles,Birkhoff’s equations and Lie symmetry theorems of quasi-fractional Birkhoffian systems and classical Birkhoffian systems are special cases of this article.Finally,some examples are given.

Fractional Versions of the Fundamental Theorem of Calculus

The concept of fractional integral in the Riemann-Liouville, Liouville, Weyl and Riesz sense is presented. Some properties involving the particular Riemann-Liouville integral are mentioned. By means of this concept we present the fractional derivatives, specifically, the Riemann-Liouville, Liouville, Caputo, Weyl and Riesz versions are discussed. The so-called fundamental theorem of fractional calculus is presented and discussed in all these different versions.

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.

Application of the Improved Kudryashov Method to Solve the Fractional Nonlinear Partial Differential Equations

Our purpose of this paper is to apply the improved Kudryashov method for solving various types of nonlinear fractional partial differential equations. As an application, the time-space fractional Korteweg-de Vries-Burger (KdV-Burger) equation is solved using this method and we get some new travelling wave solutions. To acquire our purpose a complex transformation has been also used to reduce nonlinear fractional partial differential equations to nonlinear ordinary differential equations of integer order, in the sense of the Jumarie’s modified Riemann-Liouville derivative. Afterwards, the improved Kudryashov method is implemented and we get our required reliable solutions where the results are justified by mathematical software Maple-13.

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.

Variational iteration method for solving time-fractional diffusion equations in porous the medium

The variational iteration method is successfully extended to the case of solving fractional differential equations, and the Lagrange multiplier of the method is identified in a more accurate way. Some diffusion models with fractional derivatives are investigated analytically, and the results show the efficiency of the new Lagrange multiplier for fractional differential equations of arbitrary order.

On the Approximate Solution of Fractional Logistic Differential Equation Using Operational Matrices of Bernstein Polynomials

In this paper, operational matrices of Bernstein polynomials (BPs) are presented for solving the non-linear fractional Logistic differential equation (FLDE). The fractional derivative is described in the Riemann-Liouville sense. The operational matrices for the fractional integration in the Riemann-Liouville sense and the product are used to reduce FLDE to the solution of non-linear system of algebraic equations using Newton iteration method. Numerical results are introduced to satisfy the accuracy and the applicability of the proposed method.

Pseudo-Spectral Method for Space Fractional Diffusion Equation

This paper presents a numerical scheme for space fractional diffusion equations (SFDEs) based on pseudo-spectral method. In this approach, using the Guass-Lobatto nodes, the unknown function is approximated by orthogonal polynomials or interpolation polynomials. Then, by using pseudo-spectral method, the SFDE is reduced to a system of ordinary differential equations for time variable t. The high order Runge-Kutta scheme can be used to solve the system. So, a high order numerical scheme is derived. Numerical examples illustrate that the results obtained by this method agree well with the analytical solutions.