Formulation of Euler-Lagrange equations for fractional variational problems

This paper presents extensions to traditional calculus of variations for systems containing fractional derivatives. The fractional derivative is described in the Riemann-Liouville sense. Specifically, we consider two problems, the simplest fractional variational problem and the fractional variational problem of Lagrange. Results of the first problem are extended to problems containing multiple fractional derivatives and unknown functions. For the second problem, we also present a Lagrange type multiplier rule. For both problems, we develop the Euler-Lagrange type necessary conditions which must be satisfied for the given functional to be extremum. Two problems are considered to demonstrate the application of the formulation. The formulation presented and the resulting equations are very similar to those that appear in the field of classical calculus of variations.     

Sharp bounds for singular values of fractional integral operators

From the results of Dostanic [M.R. Dostanic, Asymptotic behavior of the singular values of fractional integral operators, J. Math. Anal. Appl. 175 (1993) 380-391] and V? and Gorenflo [Kim Tuan V?, R. Gorenflo, Singular values of fractional and Volterra integral operators, in: Inverse Problems and Applications to Geophysics, Industry, Medicine and Technology, Ho Chi Minh City, 1995, Ho Chi Minh City Math. Soc., Ho Chi Minh City, 1995, pp. 174-185] it is known that the jth singular value of the fractional integral operator of order α>0 is approximately (πj)^−α for all large j. In this note we refine this result by obtaining sharp bounds for the singular values and use these bounds to show that the jth singular value is (πj)^−α[1+O(j⁻¹)].     

The Fractional Calculus for Some Stochastic Processes

Abstract

The integration and differentiation of fractional orders are well known concepts for deterministic functions (see Miller, K.S.; Ross, B. An Introduction to Fractional Calculus and Fractional Differential Equations; John Wiley: New York, 1993; I. Podlubny and Ahmed M.A. El-Sayed, On two definitions of fractional calculus Slovak Academy of Sciences Institute of experimental Phys. UEF-03-96 ISBN 80-7099-252-2, 1996; Podlubny, I. Fractional Differential Equations; Acad. Press: San Diego – New York, London etc. 1999; Samko, S.G.; Kilbas, A.A.; Marichev, O. Integral and derivatives of the fractional orders and some of their applications. Nauka i Teknika Minisk 1983). In earlier work, we have studied the fractional calculus for mean square continuous stochastic processes. In this work, we shall study the mean square (m.s.) fractional calculus for stochastic processes which are m.s. Riemann-integrable and prove some its properties.     

A general finite element formulation for fractional variational problems

This paper presents a general finite element formulation for a class of Fractional Variational Problems (FVPs). The fractional derivative is defined in the Riemann-Liouville sense. For FVPs the Euler-Lagrange and the transversality conditions are developed. In the Fractional Finite Element Formulation (FFEF) presented here, the domain of the equations is divided into several elements, and the functional is approximated in terms of nodal variables. Minimization of this functional leads to a set of algebraic equations which are solved using a numerical scheme. Three examples are considered to show the performance of the algorithm. Results show that as the number of discretization is increased, the numerical solutions approach the analytical solutions, and as the order of the derivative approaches an integer value, the solution for the integer order system is recovered. For unspecified boundary conditions, the numerical solutions satisfy the transversality conditions. This indicates that for the class of problems considered, the numerical solutions can be obtained directly from the functional, and there is no need to solve the fractional Euler-Lagrange equations. Thus, the formulation extends the traditional finite element approach to FVPs.     

Variational methods for the fractional Sturm–Liouville problem

This article is devoted to the regular fractional Sturm–Liouville eigenvalue problem. By applying the methods of fractional variational analysis, we prove the existence of a countable set of orthogonal solutions and corresponding eigenvalues. Moreover, we formulate two results showing that the lowest eigenvalue is the minimum value for a certain variational functional.     

Global monotonic solutions of multi term fractional differential equations

Based on the Leray-Schauder principle, a fixed point theorem is established to study the existence of a global monotonic solution for some multi term differential equations of fractional type. Some existence result for the inclusion problem is proved.     

Operational solution of fractional differential equations

The main goal of this paper is to solve fractional differential equations by means of an operational calculus. Our calculus is based on a modified shift operator which acts on an abstract space of formal Laurent series. We adopt Weyl’s definition of derivatives of fractional order.     

Solving fractional integral equations by the Haar wavelet method

Haar wavelets for the solution of fractional integral equations are applied. Fractional Volterra and Fredholm integral equations are considered. The proposed method also is used for analysing fractional harmonic vibrations. The efficiency of the method is demonstrated by three numerical examples.     

Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations

Haar wavelet operational matrix has been widely applied in system analysis, system identification, optimal control and numerical solution of integral and differential equations. In the present paper we derive the Haar wavelet operational matrix of the fractional order integration, and use it to solve the fractional order differential equations including the Bagley-Torvik, Ricatti and composite fractional oscillation equations. The results obtained are in good agreement with the existing ones in open literatures and it is shown that the technique introduced here is robust and easy to apply.     

Set-valued system of fractional differential equations with hysteresis

In this paper, we investigate the global solvability in L₁(0,1) of a set-valued system of nonlinear fractional differential equations with hysteresis. Some existence theorems for both single and multivalued systems are proved.     

Identification of constitutive parameters for fractional viscoelasticity

This paper develops a numerical model to identify constitutive parameters in the fractional viscoelastic field. An explicit semi-analytical numerical model and a finite difference (FD) method based numerical model are derived for solving the direct homogenous and regionally inhomogeneous fractional viscoelastic problems, respectively. A continuous ant colony optimization (ACO) algorithm is employed to solve the inverse problem of identification. The feasibility of the proposed approach is illustrated via the numerical verification of a two-dimensional identification problem formulated by the fractional Kelvin–Voigt model, and the noisy data and regional inhomogeneity etc. are taken into account.     

Subordination and superordination for univalent solutions for fractional differential equations

In this article, we establish the existence and uniqueness of univalent solution for fractional differential equation. Moreover, we illustrate some properties of this solution containing differential and integral subordination properties.     

Decay of potential vortex for a viscoelastic fluid with fractional Maxwell model

This paper is concerned with the exact analytic solutions for the velocity field and the associated tangential stress corresponding to a potential vortex for a fractional Maxwell fluid. The fractional calculus approach is taken into account in the constitutive relationship of a non-Newtonian fluid model. Exact analytic solutions are obtained by using the Hankel transform and the discrete Laplace transform of sequential fractional derivatives. The solutions for a Maxwell fluid appear as the limiting cases of our general solutions by setting α=1$\alpha =1$. The influence of fractional coefficient on the decay of vortex velocity is also analyzed by graphical illustrations.     

On the nonlinear Hammerstein integral equations in Banach spaces and application to the boundary value problem of fractional order

In this paper, we study the existence of solutions of the operator equations p+λGfx=x in the Banach space C[I,E]. It is assumed the vector-valued function f is nonlinear Pettis-integrable. Some additional assumptions imposed on f are expressed in terms of a weak measure of noncompactness. To encompass the full scope of the paper, we investigate the existence of pseudo-solutions for the nonlinear boundary value problem of fractional type

under the Pettis integrability assumption imposed on f.     

Comments on “Modeling fractional stochastic systems as non-random fractional dynamics driven Brownian motions”

Some results presented in the paper “Modeling fractional stochastic systems as non-random fractional dynamics driven Brownian motions” [I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999] are discussed in this paper. The slightly modified Grünwald-Letnikov derivative proposed there is used to deduce some interesting results that are in contradiction with those proposed in the referred paper.     

General necessary conditions for infinite horizon fractional variational problems

We consider infinite horizon fractional variational problems, where the fractional derivative is defined in the sense of Caputo. Necessary optimality conditions for higher-order variational problems and optimal control problems are obtained. Transversality conditions are obtained in the case state functions are free at the initial time.     

A fractional-order model of HIV infection with drug therapy effect

In this paper, the fractional-order model that describes HIV infection of CD4⁺ T cells with therapy effect is given. Generalized Euler Method (GEM) is employed to get numerical solution of such problem. The fractional derivatives are described in the Caputo sense.     

Box Dimension of Weyl Fractional Integral of Continuous Functions with Bounded Variation

We know that the Box dimension of f(x) ∈ C~1[0,1] is 1. In this paper, we prove that the Box dimension of continuous functions with bounded variation is still1. Furthermore, Box dimension of Weyl fractional integral of above function is also 1.