On boundary values of fractional resolvent families

Boundary values of analytic fractional resolvent families are deduced via the approximation of fractional powers of operators, and square root reductions of fractional resolvent families are given as well.     

Boundary value problems for differential equations with fractional derivatives

This paper is devoted to the spectral analysis of the operators generated by differential equations of second order with fractional derivatives in lower terms and boundary conditions of Sturm–Liouville type.     

A semigroup approach to fractional powers

We say that A has fractional powers {A _t } _t≥0 if there exists a nondegenerate C-regularized semigroup {W(t)} _t≥0 such that A=C ⁻¹ W(1); then A _t ≡C ⁻¹ W(t). We show that this generalizes the usual definition of fractional powers for nonnegative operators, and enables many operators with spectrum containing the entire unit disc to have fractional powers. Our definition gives clear, simple proofs of the basic properties of fractional powers. We show that, for nonnegative operators, the fractional powers with the property that, if A is of type θ, then A _t is of type t θ, whenever t θ<π, are unique. More generally, for injective G∈B(X) commuting with A, we show that an operator A of G-regularized type θ has a unique family of fractional powers with the property that A _t is of G-regularized type t θ whenever t θ<π. This leads to a construction of fractional powers of operators with polynomially bounded resolvent outside of an appropriate sector. We show that an operator is of regularized type if and only if it has exponentially bounded regularized imaginary powers. This work was done while the second author was visiting Ohio University, with funding from Universitat de València. He would like to thank Ohio University and Professor deLaubenfels for their hospitality and support.     

A note on immersions of domains of fractional powers of certain sectorial operators in Sobolev spaces

In this note we give a proof of a result on immersions of domains of fractional powers of certain sectorial operators associated to strongly elliptic operators in Sobolev spaces; such immersions preserve information on fractional derivatives. We also briefly comment on the application of this result to a problem of optimal control of mosquito populations.     

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.     

The representation of fractional powers of coercive differential operators

In this paper, under the definition of fractional powers by Straub, we will give the representation of fractional powers of coercive differential operators by using pseudo differential operators.     

Quantitative approximation by fractional smooth general singular operators

In this article we study the fractional smooth general singular integral operators on the real line, regarding their convergence to the unit operator with fractional rates in the uniform norm. The related established inequalities involve the higher order moduli of smoothness of the associated right and left Caputo fractional derivatives of the engaged function. Furthermore we produce a fractional Voronovskaya type result giving the fractional asymptotic expansion of the basic error of our approximation.We finish with applications to fractional trigonometric singular integral operators. Our operators are not in general positive.     

Hamilton-Jacobi fractional mechanics

As a continuation of Rabei et al. work [Eqab M. Rabei, Khaled I. Nawafleh, Raed S. Hijjawi, Sami I. Muslih, Dumitru Baleanu, The Hamilton formalism with fractional derivatives, J. Math. Anal. Appl. 327 (2007) 891-897], the Hamilton-Jacobi partial differential equation is generalized to be applicable for systems containing fractional derivatives. The Hamilton-Jacobi function in configuration space is obtained in a similar manner to the usual mechanics. Two problems are considered to demonstrate the application of the formalism. The result is found to be in exact agreement with Agrawal's formalism.     

An algorithmic representation of fractional powers of positive operators

We are concerned with a new representation of fractional powers of operators by a series using exclusively contractive operators or operators with uniformly bounded powers. This representation is suitable for the construction of efficient approximations for which the error estimates are given and the convergence is investigated. The rate of convergence turns out to be exponential for bounded operators and polynomial for unbounded operators     

The Hamilton formalism with fractional derivatives

Recently the traditional calculus of variations has been extended to be applicable for systems containing fractional derivatives. In this paper the passage from the Lagrangian containing fractional derivatives to the Hamiltonian is achieved. The Hamilton's equations of motion are obtained in a similar manner to the usual mechanics. In addition, the classical fields with fractional derivatives are investigated using Hamiltonian formalism. Two discrete problems and one continuous are considered to demonstrate the application of the formalism, the results are obtained to be in exact agreement with Agrawal's formalism.     

Computing eigenelements of boundary value problems with fractional derivatives

Variational iteration method has been successfully implemented to handle linear and nonlinear differential equations. The main property of the method is its flexibility and ability to solve nonlinear equations accurately and conveniently. In this paper, first, a general framework of the variational iteration method is presented for analytic treatment of differential equations of fractional order where the fractional derivatives are described in Caputo sense. Second, the new framework is used to compute approximate eigenvalues and the corresponding eigenfunctions for boundary value problems with fractional derivatives. Numerical examples are tested to show the pertinent features of this method. This approach provides a new way to investigate eigenvalue problems with fractional order derivatives.     

Numerical solution of fractional oscillator equation

We focus on a numerical scheme applied for a fractional oscillator equation in a finite time interval. This type of equation includes a complex form of left- and right-sided fractional derivatives. Its analytical solution is represented by a series of left and right fractional integrals and therefore is difficult in practical calculations. Here we elaborated two numerical schemes being dependent on a fractional order of the equation. The results of numerical calculations are compared with analytical solutions. Then we illustrate convergence and stability of our schemes.     

On a generalization of the operators of fractional integration and differentiation

Some general fractional integral operators are studied including those of RIEMANN-LIOUVILLE, HADAMARD and others. They are used to solve a generalized ABEL equation.     

Sharp weighted bounds for fractional integral operators

The relationship between the operator norms of fractional integral operators acting on weighted Lebesgue spaces and the constant of the weights is investigated. Sharp bounds are obtained for both the fractional integral operators and the associated fractional maximal functions. As an application improved Sobolev inequalities are obtained. Some of the techniques used include a sharp off-diagonal version of the extrapolation theorem of Rubio de Francia and characterizations of two-weight norm inequalities.     

Positive solutions to singular fractional differential system with coupled boundary conditions

By using the fixed point theory in cone and constructing some available integral operators together with approximating technique, the existence of positive solution for a singular nonlinear semipositone fractional differential system with coupled boundary conditions is established. Two examples are then given to demonstrate the validity of our main results.     

A note on the identity operators of fractional calculus

The application of an identity operator for Saigo’s fractional calculus operators is shown by evaluating the limit of an indeterminate form. Its special case yields the result which has been used as an infinitesimal generator in the semigroup theory. Also, an identity operator for the recently introduced multi-dimensional fractional operators (due to Srivastava and Raina [8]) is discussed.     

Interpolation series for fractional derivatives and iterates of functions and operators

For a class of complex valued functions on the real line a fractional derivative is defined which is an entire function of exponential type of the order. It is shown that these derivatives can be found by a Newton interpolation series. For a class of linear operators, a fractional derivative for their resolvents also is defined. These fractional derivatives and the fractional iterates of these operators are related and both can be found by a Newton interpolation series on the nth-order iterates of the operators.     

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)].     

Approximate analytical solutions of fractional gas dynamic equations

In this article, differential transform method (DTM) has been successfully applied to obtain the approximate analytical solutions of the nonlinear homogeneous and non-homogeneous gas dynamic equations, shock wave equation and shallow water equations with fractional order time derivatives. The true beauty of the article is manifested in its emphatic application of Caputo fractional order time derivative on the classical equations with the achievement of the highly accurate solutions by the known series solutions and even for more complicated nonlinear fractional partial differential equations (PDEs). The method is really capable of reducing the size of the computational work besides being effective and convenient for solving fractional nonlinear equations. Numerical results for different particular cases of the equations are depicted through graphs.     

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.