On recent developments in the theory of abstract differential equations with fractional derivatives

This note is motivated from some recent papers treating the problem of the existence of a solution for abstract differential equations with fractional derivatives. We show that the existence results in [Agarwal et al. (2009) [1], Belmekki and Benchohra (2010) [2], Darwish et al. (2009) [3], Hu et al. (2009) [4], Mophou and N’Guérékata (2009) [6] and [7], Mophou (2010) [8] and [9], Muslim (2009) [10], Pandey et al. (2009) [11], Rashid and El-Qaderi (2009) [12] and Tai and Wang (2009) [13]] are incorrect since the considered variation of constant formulas is not appropriate. In this note, we also consider a different approach to treat a general class of abstract fractional differential equations.     

Fractional stochastic differential equation with discontinuous diffusion

In this article, we study a class of stochastic differential equations driven by a fractional Brownian motion with H > 1/2 and a discontinuous coefficient in the diffusion. We prove existence and uniqueness for the solution of these equations. This is a first step to define a fractional version of the skew Brownian motion.     

Solutions for a generalized fractional anomalous diffusion equation

In this paper, we investigate the solutions for a generalized fractional diffusion equation that extends some known diffusion equations by taking a spatial time-dependent diffusion coefficient and an external force into account, which subjects to the natural boundaries and the generic initial condition. We obtain explicit analytical expressions for the probability distribution and study the relation between our solutions and those obtained within the maximum entropy principle by using the Tsallis entropy.     

Computation of the fractional Fourier transform

In this paper we make a critical comparison of some programs for the digital computation of the fractional Fourier transform that are freely available and we describe our own implementation that filters the best out of the existing ones. Two types of transforms are considered: first, the fast approximate fractional Fourier transform algorithm for which two algorithms are available. The method is described in [H.M. Ozaktas, M.A. Kutay, G. Bozda i, IEEE Trans. Signal Process. 44 (1996) 2141–2150]. There are two implementations: one is written by A.M. Kutay, the other is part of package written by J. O'Neill. Second, the discrete fractional Fourier transform algorithm described in the master thesis by Ç. Candan [Bilkent University, 1998] and an algorithm described by S.C. Pei, M.H. Yeh, and C.C. Tseng [IEEE Trans. Signal Process. 47 (1999) 1335–1348].     

Fractional set systems with few disjoint pairs

Ahlswede (1980) [1] and Frankl (1977) [5] independently found a result about the structure of set systems with few disjoint pairs. Bollobás and Leader (2003) [3] gave an alternate proof by generalizing to fractional set systems and noting that the optimal fractional set systems are {0,1}-valued. In this paper we show that this technique does not extend to t-intersecting families. We find optimal fractional set systems for some infinite classes of parameters, and we point out that they are strictly better than the corresponding {0,1}-valued fractional set systems. We prove some results about the structure of an optimal fractional set system, which we use to produce an algorithm for finding such systems. The run time of the algorithm is polynomial in the size of the ground set.     

Entropy solution theory for fractional degenerate convection–diffusion equations

We study a class of degenerate convection-diffusion equations with a fractional non-linear diffusion term. This class is a new, but natural, generalization of local degenerate convection-diffusion equations, and include anomalous diffusion equations, fractional conservation laws, fractional porous medium equations, and new fractional degenerate equations as special cases. We define weak entropy solutions and prove well-posedness under weak regularity assumptions on the solutions, e.g. uniqueness is obtained in the class of bounded integrable solutions. Then we introduce a new monotone conservative numerical scheme and prove convergence toward the entropy solution in the class of bounded integrable BV functions. The well-posedness results are then extended to non-local terms based on general Lévy operators, connections to some fully non-linear HJB equations are established, and finally, some numerical experiments are included to give the reader an idea about the qualitative behavior of solutions of these new equations.     

On evolutionary equations with material laws containing fractional integrals

A well‐posedness result for a time‐shift invariant class of evolutionary operator equations involving material laws with fractional time‐integrals of order α ? ]0, 1[ is considered. The fractional derivatives are defined via a function calculus for the (time‐)derivative established as a normal operator in a suitable L² type space. Employing causality, we show that the fractional derivatives thus obtained coincide with the Riemann‐Liouville fractional derivative. We exemplify our results by applications to a fractional Fokker‐Planck equation, equations describing super‐diffusion and sub‐diffusion processes, and a Kelvin‐Voigt type model in fractional visco‐elasticity. Moreover, we elaborate a suitable perspective to deal with initial boundary value problems. Copyright © 2014 John Wiley & Sons, Ltd.     

On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order

In this paper, we prove the existence and uniqueness of solutions for a four-point nonlocal boundary value problem of nonlinear integro-differential equations of fractional order q ∈ (1, 2] by applying some standard fixed point theorems. An illustrative example is also presented.     

Similarity solutions to nonlinear heat conduction and Burgers/Korteweg–deVries fractional equations

We analyze self-similar solutions to a nonlinear fractional diffusion equation and fractional Burgers/Korteweg–deVries equation in one spatial variable. By using Lie-group scaling transformation, we determined the similarity solutions. After the introduction of the similarity variables, both problems are reduced to ordinary nonlinear fractional differential equations. In two special cases exact solutions to the ordinary fractional differential equation, which is derived from the diffusion equation, are presented. In several other cases the ordinary fractional differential equations are solved numerically, for several values of governing parameters. In formulating the numerical procedure, we use special representation of a fractional derivative that is recently obtained.     

Local existence and non-explosion of solutions for stochastic fractional partial differential equations driven by multiplicative noise

In this paper we prove the local existence and uniqueness of solutions for a class of stochastic fractional partial differential equations driven by multiplicative noise. We also establish that for this class of equations adding linear multiplicative noise provides a regularizing effect: the solutions will not blow up with high probability if the initial data is sufficiently small, or if the noise coefficient is sufficiently large. As applications our main results are applied to various types of SPDE such as stochastic reaction–diffusion equations, stochastic fractional Burgers equation, stochastic fractional Navier–Stokes equation, stochastic quasi-geostrophic equations and stochastic surface growth PDE.     

On the fractional differential equations with uncertainty

This paper is based on the concept of fuzzy differential equations of fractional order introduced by Agarwal et al. [R.P. Agarwal, V. Lakshmikantham, J.J. Nieto, On the concept of solution for fractional differential equations with uncertainty, Nonlinear Anal. 72 (2010) 2859-2862]. Using this concept, we prove some results on the existence and uniqueness of solutions of fuzzy fractional differential equations.     

PROPERTIES OF FRACTIONAL κ-FACTORS OF GRAPHS

In this paper the properties of some maximum fractional [0, κ]-factors of graphs are presented. And consequently some results on fractional matchings and fractional 1-factors are generalized and a characterization of fractional κ-factors is obtained.     

The existence of mild solutions for impulsive fractional partial differential equations

This paper is concerned with the existence of mild solutions for a class of impulsive fractional partial semilinear differential equations. Some errors in Mophou (2010) [2] are corrected, and some previous results are generalized.     

FRACTIONAL DIFFUSION EQUATIONS WITH INTERNAL DEGREES OF FREEDOM

We present a generalization of the linear one-dimensional diffusion equation by com-bining the fractional derivatives and the internal degrees of freedom. The solutions areconstructed from those of the scalar fractional diffusion equation. We analyze the in-terpolation between the standard diffusion and wave equations defined by the fractionalderivatives. Our main result is that we can define a diffusion process depending on theinternal degrees of freedom associated to the system.     

Homotopy perturbation method for a limit case Stefan problem governed by fractional diffusion equation

The work presents a mathematical model describing the time fractional anomalous-diffusion process of a generalized Stefan problem which is a limit case of a shoreline problem. In this model, the governing equations include a fractional time derivative of order 0 < α ? 1 and variable latent heat. The approximate solution of the problem is obtained by homotopy perturbation method. The results thus obtained are compared graphically with the exact solutions. A brief sensitivity study is also performed.     

Stock exchange fractional dynamics defined as fractional exponential growth driven by (usual) Gaussian white noise. Application to fractional Black-Scholes equations

Stock exchange dynamics of fractional order are usually modeled as a non-random exponential growth process driven by a fractional Brownian motion. Here we propose to use rather a non-random fractional growth driven by a (standard) Brownian motion. The key is the Taylor’s series of fractional order where E_α(.) denotes the Mittag-Leffler function, and is the so-called modified Riemann-Liouville fractional derivative which we introduced recently to remove the effects of the non-zero initial value of the function under consideration. Various models of fractional dynamics for stock exchange are proposed, and their solutions are obtained. Mainly, the Itô’s lemma of fractional order is illustrated in the special case of a fractional growth with white noise. Prospects for the Merton’s optimal portfolio are outlined, the path probability density of fractional stock exchange dynamics is obtained, and two fractional Black-Scholes equations are derived. This approach avoids using fractional Brownian motion and thus is of some help to circumvent the mathematical difficulties so involved.     

Uniform persistence for a two-species ratio-dependent predator-prey system with diffusion and variable time delays

A two-species ratio-dependent predator-prey diffusion model with variable time delays is investigated. By constructing some suitable Lyapunov functionals and utilizing some analysis techniques, we obtain the permanence of this system. Our results generalize and improve the results of [10] and [11].     

PROPERTIES OF FRACTIONAL k-FACTORS OF GRAPHS

In this paper the properties of some maximum fractional [0, k]-factors of graphs are presented. And consequently some results on fractional matchings and fractional 1-factors are generalized and a characterization of fractional k-factors is obtained.