Stochastic analysis, rough path analysis and fractional Brownian motions

 In this paper we show, by using dyadic approximations, the existence of a geometric rough path associated with a fractional Brownian motion with Hurst parameter greater than 1/4. Using the integral representation of fractional Brownian motions, we furthermore obtain a Skohorod integral representation of the geometric rough path we constructed. By the results in [Ly1], a stochastic integration theory may be established for fractional Brownian motions, and strong solutions and a Wong-Zakai type limit theorem for stochastic differential equations driven by fractional Brownian motions can be deduced accordingly. The method can actually be applied to a larger class of Gaussian processes with covariance functions satisfying a simple decay condition. Received: 11 May 2000 / Revised version: 20 March 2001 / Published online: 11 December 2001     

Qualitative properties of solutions of an integral equation associated with the Benjamin–Ono–Zakharov–Kuznetsov operator

We study qualitative properties of solutions of an integral equation associated the Benjamin–Ono–Zakharov–Kuznetsov operator. We establish the regularity of the positive solutions without the assumption of being in fractional Sobolev–Liouville spaces. Moreover we show that the solutions are axially symmetric. Furthermore we establish Lipschitz continuity and the decay rate of the solutions.     

Solvability of an implicit fractional integral equation via a measure of noncompactness argument

In this paper, we study the existence of solutions to an implicit functional equation involving a fractional integral with respect to a certain function, which generalizes the Riemann-Liouville fractional integral and the Hadamard fractional integral. We establish an existence result to such kind of equations using a generalized version of Darbo's theorem associated to a certain measure of noncompactness. Some examples are presented.     

Integral equations and initial value problems for nonlinear differential equations of fractional order

We discuss the solvability of integral equations associated with initial value problems for a nonlinear differential equation of fractional order. The differential operator is the Caputo fractional derivative and the inhomogeneous term depends on the fractional derivative of lower orders. We obtain the existence of at least one solution for integral equations using the Leray–Schauder Nonlinear Alternative for several types of initial value problems. In addition, using the Banach contraction principle, we establish sufficient conditions for unique solutions. Our approach in obtaining integral equations is the “reduction” of the fractional order of the integro-differential equations based on certain semigroup properties of the Caputo operator.     

Solution of nonlinear weakly singular Volterra integral equations using the fractional‐order Legendre functions and pseudospectral method

In this article, our main goal is to render an idea to convert a nonlinear weakly singular Volterra integral equation to a non‐singular one by new fractional‐order Legendre functions. The fractional‐order Legendre functions are generated by change of variable on well‐known shifted Legendre polynomials. We consider a general form of singular Volterra integral equation of the second kind. Then the fractional Legendre–Gauss–Lobatto quadratures formula eliminates the singularity of the kernel of the integral equation. Finally, the Legendre pseudospectral method reduces the solution of this problem to the solution of a system of algebraic equations. This method also can be utilized on fractional differential equations as well. The comparison of results of the presented method and other numerical solutions shows the efficiency and accuracy of this method. Also, the obtained maximum error between the results and exact solutions shows that using the present method leads to accurate results and fast convergence for solving nonlinear weakly singular Volterra integral equations. Copyright © 2015 John Wiley & Sons, Ltd.     

Generalized Abel type integral equations with localized fractional integrals and derivatives

Generalized Abel type integral equations with Gauss, Kummer's and Humbert's confluent hypergeometric functions in the kernel and generalized Abel type integral equations with localized fractional integrals are considered. The left-hand sides of these equations are inversed by using generalized fractional derivatives. Explicit solutions of the equations in the class of locally summable functions are obtained. They are represented in terms of hypergeometric functions. Asymptotic power exponential type expansions of the generalized and localized fractional integrals are obtained. The base solutions of the generalized Abel type integral equation are given in the form of asymptotic series.     

Periodic solutions of Weyl fractional order integral systems†

In this paper, we show the existence and uniqueness results for periodic solutions of Weyl fractional order integral systems. A numerical example is given to illustrate our theoretical results. Our results show that periodic orbits can be obtained by putting the periodic conditions to some certain fractional order integral systems. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

Solutions to BSDEs driven by both standard and fractional Brownian motions

The backward stochastic differential equations driven by both standard and fractional Brownian motions (or, in short, SFBSDE) are studied. A Wick-Itô stochastic integral for a fractional Brownian motion is adopted. The fractional Itô formula for the standard and fractional Brownian motions is provided. Introducing the concept of the quasi-conditional expectation, we study some its properties. Using the quasi-conditional expectation, we also discuss the existence and uniqueness of solutions to general SFBSDEs, where a fixed point principle is employed. Moreover, solutions to linear SFBSDEs are investigated. Finally, an explicit solution to a class of linear SFBSDEs is found.     

Approximate solutions of nonlinear fractional equations

Fractional exponential that are invariant under fractional derivatives, elementary and special fractional functions are introduced. Approximate solutions to fractional Burgers equation, by using the homotopy perturbation method, are obtained. Furthermore, real integral representations for some H-functions are found that may be very helpful in numerical computations.     

Existence of Solutions for Fractional Integro-differential Equations with Impulsive and Integral Conditions

This paper presents the existence of solutions for a class of Cauchy problems with integral condition for impulsive fractional integro-differential equations. Based on definition of solution for impulsive fractional integro- differential equations, the existence theorems of solutions of fractional differ- ential equation are obtained by applying fixed point methods. Finally, three examples are given to demonstrate the feasibility of the obtained results.     

应用多项式完全判别系统方法求解时空分数阶复Ginzburg-Landau方程

研究了时空分数阶复Ginzburg-Landau方程.首先通过分数阶复变换将时空分数阶复Ginzburg-Landau方程转化为一个常微分方程.然后将常微分方程化为初等积分形式.最后用多项式完全判别系统法求得一系列精确解,其中包含有孤立波解、有理函数解、三角函数周期解、Jacobi椭圆函数双周期解.     

Weak solutions for fractional differential equations in nonreflexive Banach spaces via Riemann‐Pettis integrals

The aim of this paper is to develop fractional calculus for vector‐valued functions using the weak Riemann integral. Also, we establish the existence of weak solutions for a class of fractional differential equations with fractional weak derivatives.     

Decay mild solutions of Hilfer fractional differential variational–hemivariational inequalities

The primary objective of this paper is to explore a decay mild solution governed by a class of dynamical systems, called Hilfer fractional differential variational–hemivariational inequality (HFDVHVI, for short), which is composed of a Hilfer fractional evolution differential inclusion and a variational–hemivariational inequality involving two history-dependent operators in the framework of spaces. Our first aim is to investigate the solvability of the mild solutions to (HFDVHVI) by means of fixed point principle. The second step of the paper is to study the existence of decay mild solutions to (HFDVHVI) via giving expression for the Mittag-Leffler function and the Wright function.     

Positive solutions for nonlinear Caputo fractional differential equations with integral boundary conditions

In this paper, we consider the properties of Green’s function for a class of nonlinear Caputo fractional differential equations with integral boundary conditions by constructing an available integral operator. By means of well-known fixed point theorems and lower and upper solutions method, some new existence and nonexistence criteria of single or multiple positive solutions for fractional differential equation boundary value problems are established. As applications, some interesting examples are presented to illustrate the main results.     

Weighted Sobolev L2 estimates for a class of Fourier integral operators

In this paper we develop elements of the global calculus of Fourier integral operators in ${{\mathbb R}^n}$ under minimal decay assumptions on phases and amplitudes. We also establish global weighted Sobolev L² estimates for a class of Fourier integral operators that appears in the analysis of global smoothing problems for dispersive partial differential equations. As an application, we exhibit a new type of weighted estimates for hyperbolic equations, where the decay of data in space is quantitatively translated into the time decay of solutions.     

Existence Results for Fractional Differential Equations with the Riesz-Caputo Derivative

In this paper, we apply some fixed point theorems to attain the existence of solutions for fractional differential equations with the space-time Riesz-Caputo derivative. We study the boundary value problems that the nonlinearity term $f$ is relevant to fractional integral and fractional derivative. In addition, the boundary conditions involve integral. Two examples are given to show the effectiveness of theoretical results.     

Approximation Technique for Fractional Evolution Equations with Nonlocal Integral Conditions

We consider a class of fractional evolution equations with nonlocal integral conditions in Banach spaces. New existence of mild solutions to such a problem are established using Schauder fixed-point theorem, diagonal argument and approximation techniques under the hypotheses that the nonlinear term is Carathéodory continuous and satisfies some weak growth condition, the nonlocal term depends on all the value of independent variable on the whole interval and satisfies some weak growth condition. This work may be viewed as an attempt to develop a general existence theory for fractional evolution equations with general nonlocal integral conditions. Finally, as a sample of application, the results are applied to a fractional parabolic partial differential equation with nonlocal integral condition. The results obtained in this paper essentially extend some existing results in this area.     

Flow of a generalized Maxwell fluid induced by a constantly accelerating plate between two side walls

The unsteady flow of a viscoelastic fluid with the fractional Maxwell model, induced by a constantly accelerating plate between two side walls perpendicular to the plate, is investigated by means of the integral transforms. Exact solutions for the velocity field are presented under integral and series forms in terms of the derivatives of generalized Mittag–Leffler functions. The corresponding solutions for Maxwell fluids are obtained as limiting cases for β → 1. In the absence of the side walls, all solutions that have been determined reduce to those corresponding to the motion over an infinite plate.      

Well–posedness for an Abstract Semilinear Integro–differential Fractional Problem

We consider an abstract second order semilinear integrodifferential equation involving fractional time derivatives of order between 1 and 2. Well–posedness is established under appropriate conditions on the initial data and the nonlinearity which is itself a fractional integral. These conditions will determine the exact underlying space where to look for solutions.