Solvability of a fractional boundary value problem with fractional integral condition

Using Banach contraction principle and Leray-Schauder nonlinear alternative we establish sufficient conditions for the existence and uniqueness of solutions for boundary value problems for fractional differential equations with fractional integral condition, involving the Caputo fractional derivative. Some examples are given to illustrate our results.     

Dual characterization of fractional capacity via solution of fractional p-Laplace equation

In terms of weak solutions of the fractional p-Laplace equation with measure data, this paper offers a dual characterization for the fractional Sobolev capacity on bounded domain. In addition, two further results are given: one is an equivalent estimate for the fractional Sobolev capacity; the other is the removability of sets of zero capacity and its relation to solutions of the fractional p-Laplace equation.     

Approximate controllability of fractional neutral stochastic evolution equations in Hilbert spaces with fractional Brownian motion

In this paper, the approximate controllability for Sobolev-type fractional neutral stochastic evolution equations with fractional stochastic nonlocal conditions and fractional Brownian motion in a Hilbert space are studied. The results are obtained by using semigroup theory, fractional calculus, stochastic integrals for fractional Brownian motion, Banach's fixed point theorem, and methods adopted directly from deterministic control problems for the main results. Finally, an example is given to illustrate the application of our result.     

Singular fractional differential equations

Fractional differential equations have received increasing attention during recent years since the behavior of many physical systems can be properly described using the fractional order system theory. By fractional analog for Duhamel principle we give the existence-uniqueness result for linear and nonlinear time fractional evolution equations with singularities in corresponding norm in extended Colombeau algebra of generalized functions. In order to find the explicit solutions we use integral representation of the solution obtained via Laplace and Fourier transforms in succession and their inverses. We deal with some nonlinear models with singularities appearing in viscoelasticity and in anomalous processes, extending the results in viscoelasticity, continuum random walk, seismology, continuum mechanics and many other branches of life and science. The main task is finding existence-uniqueness results like in the case of evolution equations with entire derivatives. By examining the fractional evolution equations it turns out that they lead to till now known results from the evolution equations with entire derivatives in limiting case. They give more, behavior of the solution when order of derivatives are inside the intervals of entire points. In this way we can follow the influence of the operators generated by entire derivative in many fractional time evolution PDEs especially with singular initial data, and non-Lipschitz's nonlinear term. Apart from evolution equations we prove also an existence-uniqueness result for an initial value problem with singularities for linear and nonlinear fractional elliptic equation of Helmholtz type and fractional order α, where 1 < Re(α) ≤ 2, with respect to the one variable from R ₊. As a framework, we employ also Colombeau algebra of generalized functions containing fractional derivatives and operations among them in order to deal with the fractional equations with singularities. We apply the same techniques to the fractional Laplace and Poisson equation linear and nonlinear ones. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analytical approximate solutions of Riesz fractional diffusion equation and Riesz fractional advection–dispersion equation involving nonlocal space fractional derivatives

In this paper, we consider the analytical solutions of fractional partial differential equations (PDEs) with Riesz space fractional derivatives on a finite domain. Here we considered two types of fractional PDEs with Riesz space fractional derivatives such as Riesz fractional diffusion equation (RFDE) and Riesz fractional advection–dispersion equation (RFADE). The RFDE is obtained from the standard diffusion equation by replacing the second‐order space derivative with the Riesz fractional derivative of order α∈(1,2]. The RFADE is obtained from the standard advection–dispersion equation by replacing the first‐order and second‐order space derivatives with the Riesz fractional derivatives of order β∈(0,1] and of order α∈(1,2] respectively. Here the analytic solutions of both the RFDE and RFADE are derived by using modified homotopy analysis method with Fourier transform. Then, we analyze the results by numerical simulations, which demonstrate the simplicity and effectiveness of the present method. Here the space fractional derivatives are defined as Riesz fractional derivatives. Copyright © 2015 John Wiley & Sons, Ltd.     

Anti-periodic fractional boundary value problems for nonlinear differential equations of fractional order

In this paper, the existence of solutions of an anti-periodic fractional boundary value problem for nonlinear fractional differential equations is investigated. The contraction mapping principle and Leray-Schauder’s fixed point theorem are applied to establish the results.     

Global attractivity for nonlinear fractional differential equations

We present some results for the global attractivity of solutions for fractional differential equations involving Riemann-Liouville fractional calculus. The results are obtained by employing Krasnoselskii’s fixed point theorem. Similar results for fractional differential equations involving Caputo fractional derivative are also obtained by using the classical Schauder’s fixed point theorem. Several examples are given to illustrate our main results.     

Anti-periodic BVP of fractional order with fractional impulsive conditions and variable parameter

In this paper, by using two fixed point theorems, we discuss two classes of anti-periodic BVP for fractional differential equations with fractional impulsive conditions and variable parameter. The existence of solutions is obtained. Examples are also given to illustrate our theoretical results.     

About fractional quantization and fractional variational principles

In this paper, a new method of finding the fractional Euler–Lagrange equations within Caputo derivative is proposed by making use of the fractional generalization of the classical Faá di Bruno formula. The fractional Euler–Lagrange and the fractional Hamilton equations are obtained within the 1 + 1 field formalism. One illustrative example is analyzed.     

Duality for nonlinear fractional programs

Summary This paper develops duality results for nonlinear fractional programming problems. This is accomplished by using some known results connecting the solutions of a nonlinear fractional program with those of a suitably defined parametric convex program.     

关于分数k-因子临界图与分数k-可扩图的若干结果

一个简单图G, 如果对于V(G)的任意k元子集S, 子图G-S都包含分数完美匹配, 那么称G为分数k-因子临界图. 如果图G的每个k-匹配M都包含在一个分数完美匹配中, 那么称图G为分数k-可扩图. 给出一个图是分数k-因子临界图和分数k-可扩图的充分条件, 并给出一个图是分数k-因子临界图的充分必要条件.     

Fractional variational integrators for fractional variational problems

In this paper, the fractional variational integrators for a class of fractional variational problems are developed. The fractional discrete Euler-Lagrange equation is obtained. Based on the Grünwald-Letnikov method and Diethelm’s fractional backward differences, some fractional variational integrators are presented and the fractional variational errors are discussed. Some numerical examples are presented to illustrate these results.     

Few fractional order derivatives and their computations

This work presents an introductory development of fractional order derivatives and their computations. Historical development of fractional calculus is discussed. This paper presents how to obtain computational results of fractional order derivatives for some elementary functions. Computational results are illustrated in tabular and graphical forms. A few observations regarding integer and fractional order derivatives are mentioned in the conclusion.     

Oscillatory solutions of fractional integro-differential equations

We give necessary conditions to get oscillatory solutions of a class of fractional order neutral differential equations with continuously distributed delay by means of the fractional derivative with respect to a given function. In particular, oscillatory solutions of the considered fractional equations with Caputo and Hadamard type of fractional derivatives are established. Some explicit examples are given to illustrate the main results.     

A higher dimensional fractional Borel‐Pompeiu formula and a related hypercomplex fractional operator calculus

In this paper, we develop a fractional integro‐differential operator calculus for Clifford algebra‐valued functions. To do that, we introduce fractional analogues of the Teodorescu and Cauchy‐Bitsadze operators, and we investigate some of their mapping properties. As a main result, we prove a fractional Borel‐Pompeiu formula based on a fractional Stokes formula. This tool in hand allows us to present a Hodge‐type decomposition for the fractional Dirac operator. Our results exhibit an amazing duality relation between left and right operators and between Caputo and Riemann‐Liouville fractional derivatives. We round off this paper by presenting a direct application to the resolution of boundary value problems related to Laplace operators of fractional order.     

A note on fractional derivatives and fractional powers of operators

Definitions of fractional derivatives and fractional powers of positive operators are considered. The connection of fractional derivatives with fractional powers of positive operators is presented. The formula for fractional difference derivative is obtained.     

Stochastic integration for tempered fractional Brownian motion

Tempered fractional Brownian motion is obtained when the power law kernel in the moving average representation of a fractional Brownian motion is multiplied by an exponential tempering factor. This paper develops the theory of stochastic integrals for tempered fractional Brownian motion. Along the way, we develop some basic results on tempered fractional calculus.     

The finite element method for fractional diffusion with spectral fractional Laplacian

This paper focuses on the finite element method for Caputo-type parabolic equation with spectral fractional Laplacian, where the time derivative is in the sense of Caputo with order in (0,1) and the spatial derivative is the spectral fractional Laplacian. The time discretization is based on the Hadamard finite-part integral (or the finite-part integral in the sense of Hadamard), where the piecewise linear interpolation polynomials are used. The spatial fractional Laplacian is lifted to the local spacial derivative by using the Caffarelli–Silvestre extension, where the finite element method is used. Full-discretization scheme is constructed. The convergence and error estimates are obtained. Finally, numerical experiments are presented which support the theoretical results.     

Functions of finite fractional variation and their applications to fractional impulsive equations

We introduce a notion of a function of finite fractional variation and characterize such functions together with their weak σ-additive fractional derivatives. Next, we use these functions to study differential equations of fractional order, containing a σ-additive term—we prove existence and uniqueness of a solution as well as derive a Cauchy formula for the solution. We apply these results to impulsive equations, i.e. equations containing the Dirac measures.     

A new fractional finite volume method for solving the fractional diffusion equation

The inherent heterogeneities of many geophysical systems often gives rise to fast and slow pathways to water and chemical movement. One approach to model solute transport through such media is by fractional diffusion equations with a space–time dependent variable coefficient. In this paper, a two-sided space fractional diffusion model with a space–time dependent variable coefficient and a nonlinear source term subject to zero Dirichlet boundary conditions is considered.Some finite volume methods to solve a fractional differential equation with a constant dispersion coefficient have been proposed. The spatial discretisation employs fractionally-shifted Grünwald formulas to discretise the Riemann–Liouville fractional derivatives at control volume faces in terms of function values at the nodes. However, these finite volume methods have not been extended to two-dimensional and three-dimensional problems in a natural manner. In this paper, a new weighted fractional finite volume method with a nonlocal operator (using nodal basis functions) for solving this two-sided space fractional diffusion equation is proposed. Some numerical results for the Crank–Nicholson fractional finite volume method are given to show the stability, consistency and convergence of our computational approach. This novel simulation technique provides excellent tools for practical problems even when a complex transition zone is involved. This technique can be extend to two-dimensional and three-dimensional problems with complex regions.