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In this paper, we introduce and study a highly scalable domain decomposition based solver for the fully implicit solution of subsurface flows in porous media with fractional derivative. In the proposed approach, a fully implicit scheme is proposed to relax the time step size by the stability condition, and the Newton–Krylov method is then applied to guarantee the nonlinear consistency of the resultant algebraic system. Moreover, to accelerate the convergence of the Newton iterations and be favorable for the scalability of the linear solver, the emphasis is placed on introducing the overlapping restricted additive Schwarz (RAS) preconditioners. Numerical experiments are presented to demonstrate the robustness and parallel scalability of the proposed fully implicit algorithm for some subsurface flow problems on a supercomputer. [ABSTRACT FROM AUTHOR]

In previous study on the flow of fractional viscoelastic fluid between two coaxial cylinders, due to the complexity of computation, researchers commonly assume that the flow area is a rectangular pipe. This paper studies two-dimensional magnetohydrodynamic (MHD) flow and heat transfer of fractional Oldroyd-B nanofluid between two coaxial cylinders with variable pressure. The boundary layer fractional cylindrical governing equations are firstly formulated and derived. Meanwhile, the numerical solutions are obtained by using the newly developed finite difference method combined with L1-algorithm, whose validity is confirmed by the comparison with the exact solution constructed. Results show that the fractional derivative parameters α 1 , β and exponent parameter δ related to the wall temperature have significant effects on velocity and temperature fields. Specifically, the velocity decreases with the augment of α 1 , but increases as β rises up. The temperature profiles intersect each other for different δ. Meanwhile, increasing δ leads to an increase in temperature near the wall of inner tube, but a decline near the outer cylinder, which demonstrates increasing exponent parameter can enhance heat transfer. [ABSTRACT FROM AUTHOR]

In this paper, we consider the approximate controllability for a class of fractional stochastic wave equations in the sense of Riemann–Liouville fractional derivative. Based on some special functions, we transform the fractional stochastic wave equations into fractional evolution equations. Further, we give an appropriate definition of mild solutions for this type of fractional stochastic wave equations. Moreover, with the aid of fixed point theory, we discuss the existence results of the mild solutions of the proposed problem. Finally, we establish the approximate controllability when the corresponding linear system is approximately controllable. [ABSTRACT FROM AUTHOR]

A fractional dual-phase-lag constitutive relation is proposed to describe the anomalous diffusion in comb model. A novel governing equation with the Dirac delta function is formulated and the highest order is 1+ α which corresponds to a diffusion-wave equation. Solutions are obtained analytically with Laplace and Fourier transforms. Dynamic characteristics for the spatial and temporal evolution of particle distribution and the mean square displacement versus time with the effects of different parameters such as the fractional parameters and the relaxation parameters are analyzed and discussed in detail. Results show that the wave characteristic becomes stronger for a larger fractional parameter, a smaller microscopic relaxation parameter or a larger macroscopic one. For a larger α , a smaller β , a larger macroscopic relaxation parameter or a smaller microscopic one, a novel oscillating distribution versus time is presented, and at this condition, the magnitude of mean square displacement is larger at the smaller time while larger at the larger time. Most important of all, the anomalous diffusion in comb model with a diffusion-wave equation corresponds to a subdiffusion behavior because of its special structure. [ABSTRACT FROM AUTHOR]

Lie-group is introduced for studying boundary layer flow and heat transfer of fractional viscoelastic MHD fluid over a stretching sheet. Fractional boundary layer equations, based on Riemann–Liouville operators, are reduced and solved numerically by Grünwald scheme approximation. Results show that the skin friction and thermal conductivity are strongly affected by magnetic field parameter, fractional derivative and wall stretching exponent. The bigger of the fractional order derivative leads to the faster velocity of viscoelastic fluids near the plate but not to hold near the outer flow. Skin friction increases with increase of magnetic field parameter M , while the heat transfer decreases. For wall stretching exponent parameter β = 1 . 0 , the velocity profile decreases with the increase of similarity variable η . However, for β = − 1 . 5 , the velocity profile increases initially and then decreases afterwards with the biggest velocity at the interior of boundary layer. [ABSTRACT FROM AUTHOR]

This article is devoted to the study of the existence and uniqueness of mild solution to time- and space-fractional stochastic Burgers’ equation perturbed by multiplicative white noise. The required results are obtained by stochastic analysis techniques, fractional calculus and semigroup theory. Moreover, we prove the regularity properties of mild solution to this generalized Burgers’ equation in the framework of Bochner spaces. [ABSTRACT FROM AUTHOR]

In this paper, we consider an inverse problem for a time fractional diffusion equation with inhomogeneous source to determine the initial data from the observation data provided at a later time. In general, this problem is ill-posed, therefore we construct a regularized solution using the filter regularization method in both cases: the deterministic case and random noise case. First, we propose both parameter choice rule methods, the a-priori and the a-posteriori methods. Then, we obtain the convergence rates and provide examples of filters. We also provide a numerical example to illustrate our results. [ABSTRACT FROM AUTHOR]

We consider the inverse problem of determination of the solution and a source term for a time fractional diffusion equation in two dimensional space. The time fractional derivative is the Hilfer derivative. A bi-orthogonal system of functions in L 2 ( Ω ) , obtained from the associated non-self-adjoint spectral problem and its adjoint problem, is used to prove the existence and uniqueness of the solution of the inverse problem. The stability of the solution of the inverse problem on the given data is proved. [ABSTRACT FROM AUTHOR]

In the present paper, we consider the problem on a transverse impact of a viscoelastic sphere upon a viscoelastic Bernoulli–Euler beam, the viscoelastic features of which are defined via the fractional derivative standard linear solid models. As this takes place, only Young’s time-dependent operators are preassigned, while the bulk moduli are considered to be constant values, since the bulk relaxation for the majority of materials is far less than the shear relaxation. Beam’s displacement subjected to the concentrated contact force is found by the method of expansion in terms of eigen (beam) functions. The contact force driven displacement of the impactor, which is the sum of the beam’s displacement at the place of contact and the indentation of the impactor into the target, is defined from the equation of motion of the material point with the mass equal to sphere’s mass. Within the contact domain, the contact force is defined by the modified Hertzian contact law with the time-dependent rigidity function. For decoding the viscoelastic operators involving in the problem under consideration, the algebra of Rabotnov’s fractional operators is employed. A nonlinear integro-differential equation is obtained either in terms of the contact force or in the local bearing of the target and impactor materials. Using the duration of contact as a small parameter, approximate analytical solutions have been found, which allow one to define the maximal contact force and the duration of contact for colliding bodies. [ABSTRACT FROM AUTHOR]

In this paper, we consider the inverse problem of determining a source in a time fractional diffusion equation where data are given at a fixed time. This problem is ill-posed, i.e., the solution (if it exists) does not depend continuously on the data. Using the method of truncated integration and the Fourier transform, we construct regularized solutions and derive explicitly error estimate. Two numerical examples are presented to illustrate the validity and effectiveness of our method. [ABSTRACT FROM AUTHOR]

In this paper we investigate the solution of generalized distributed order diffusion equations with composite time fractional derivative by using the Fourier–Laplace transform method. We represent solutions in terms of infinite series in Fox H -functions. The fractional and second moments are derived by using Mittag-Leffler functions. We observe decelerating anomalous subdiffusion in case of two composite time fractional derivatives. Generalized uniformly distributed order diffusion equation, as a model for strong anomalous behavior, is analyzed by using Tauberian theorem. Some previously obtained results are special cases of those presented in this paper. [ABSTRACT FROM AUTHOR]

Sephadex™ gel beads are commonly used to separate mixtures of similar molecules based on trapping and size exclusion from internal submicron diameter cavities. Water, as it freely moves through the porous gel and enclosed chambers of Sephadex™ beads, exhibits both normal (Gaussian) and anomalous (non-Gaussian) water diffusion. The apparent diffusion coefficient (ADC) of water in Sephadex™ gels can be measured using magnetic resonance imaging (MRI) by applying diffusion-weighted pulse sequences. This study investigates the relationship between the ADC of water and the complexity (i.e., size and number of cavities) of a series of Sephadex™ beads. We first classified the stochastic movement of water by using the solution to the space and time fractional diffusion equation to extract the ADC and the fractional time and space parameters ( α , β ), which are essentially the order of the respective fractional derivatives in Fick’s second law. From the perspective of the continuous time random walk (CTRW) model of anomalous diffusion, these parameters reflect waiting times (trapping) and jump increments (nano-flow) of the water in the gels. The observed MRI diffusion signal decay represents the Fourier transform of the diffusion propagator (i.e., the characteristic function of the stochastic process). In two series of Sephadex™ gel beads, we observed a strong inverse correlation between bead porosity (which is also responsible for molecular size exclusion) and the fractional order parameters; as the gels become more heterogeneous, the ADC decreases, both α and β are reduced and the diffusion exhibits anomalous (sub-diffusion) behavior. In addition, as a new measure for the structural complexity in Sephadex™ gel beads, we propose using the spectral and the cumulative spectral entropy that are derived from the observed characteristic function. We find that both measures of entropy increase with the porosity and tortuosity of the gel in a manner consistent with fractional order diffusional dynamics. [ABSTRACT FROM AUTHOR]

An exact discretization of fractional-order Laplacian for N -dimensional space is suggested. Particular solutions of fractional-order partial difference equations with the proposed discrete Laplace operators are suggested. [ABSTRACT FROM AUTHOR]

In this paper, we devote to the study of high order finite difference schemes for one- and two-dimensional time–space fractional sub-diffusion equations. A fourth order finite difference scheme is invoked for the spatial fractional derivatives, and the L 1 approximation is applied to the temporal fractional parts. For the two-dimensional case, an alternating direction implicit scheme based on L 1 approximation is proposed. The stability and convergence of the proposed methods are studied. Numerical experiments are performed to verify the effectiveness and accuracy of the proposed difference schemes. [ABSTRACT FROM AUTHOR]

The Grünwald formula is used to solve the two-dimensional distributed-order differential equations. Two alternating direction implicit (ADI) difference schemes are derived. It is proved that the schemes are unconditionally stable and convergent with the convergence orders O ( τ + h 1 2 + h 2 2 + Δ α 2 ) and O ( τ + h 1 4 + h 2 4 + Δ α 4 ) in H 1 norm, respectively, where τ , h 1 , h 2 and Δ α are the step sizes in time, space in x - and y -direction, and distributed order. The extrapolation method is applied to improve the approximate accuracy to the orders O ( τ 2 ∣ ln τ ∣ 2 + h 1 2 + h 2 2 + Δ α 2 ) and O ( τ 2 ∣ ln τ ∣ 2 + h 1 4 + h 2 4 + Δ α 4 ) , respectively. Several numerical examples are given to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract: The classical parabolic heat equation based on Fourier’s law implies infinite heat propagation speed. To remedy this physical flaw, the hyperbolic heat equation is used, but it may instead predict temperatures less than absolute zero. In recent years, fractional heat equations have been proposed as generalizations of heat equations of integer order. By simulating a 1D model problem of size on the order of a thermal energy carrier’s mean free path length, we have done a study of four fractional generalized Cattaneo equations from Compte and Metzler (1997) called GCE, GCE I, GCE II, and GCE III and also a fractional version of the parabolic heat equation. We have observed that when the fractional order is large enough, these equations give temperatures less than absolute zero. But if the fractional order is small enough, GCE I does not have this problem when the domain length is comparable to the mean free path length. With larger size, GCE I and GCE III also give non-oscillating solutions for both small and large values of the fractional order. [Copyright &y& Elsevier]

Abstract: The stochastic solution to a diffusion equations with polynomial coefficients is called a Pearson diffusion. If the first time derivative is replaced by a Caputo fractional derivative of order less than one, the stochastic solution is called a fractional Pearson diffusion. This paper develops an explicit formula for the covariance function of a fractional Pearson diffusion in steady state, in terms of Mittag-Leffler functions. That formula shows that fractional Pearson diffusions are long-range dependent, with a correlation that falls off like a power law, whose exponent equals the order of the fractional derivative. [Copyright &y& Elsevier]

Abstract: In this paper, the laser short-pulse heating of a solid surface is considered. The time fractional Cattaneo model is used as the heat conduction model and the corresponding fractional heat conduction equation with a volumetric heat source is built. The analytical solution for the temperature distribution is obtained using the Laplace transformation method. Finally, the numerical results are presented graphically for various values of model parameters, and their effects on the speed of heat conduction propagation are discussed. [Copyright &y& Elsevier]

Abstract: This paper describes our experimental testing of linear viscoelastic creep behaviours in Hami Melon. Experimental data shows that Hami Melon has complex viscoelastic property which cannot be well described by the standard model. Consequently, this study develops a fractional derivative model to describe such complex viscoelastic creep behaviours of Hami Melon. The analytical creep function of the proposed fractional linear viscoelastic models is derived via the Boltzmann superposition principle and discrete inverse Laplace transform. And then, such analytical solutions are used to fit the experimental viscoelastic data of Hami Melon. Our study shows that the present fractional linear viscoelastic model with merely three parameters is more efficient and accurate than the generalised Kelvin viscoelastic model of six parameters to describe the stress–strain constitutive relations of Hami Melon. It is noted that the present fractional model with adjustable parameters can also be used to describe creep damage. [Copyright &y& Elsevier]

Abstract: We investigate a backward problem for a time-fractional diffusion process in inhomogeneous media, which aims to determine the initial status of some physical field such as temperature for slow diffusion from its present measurement data. This problem is well-known to be ill-posed due to the rapid decay of the forward process. By using the eigenfunction expansion, we construct a new regularizing scheme with an explicit solution for the noisy input data with the number of truncation terms as a regularizing parameter. The convergence rate depending on the choice of strategy of the regularizing parameter is given based on the asymptotic behavior of the Mittag-Leffler function. Numerical implementations are presented to show the validity of the proposed scheme for several models. [Copyright &y& Elsevier]

Abstract: The electrostatics properties of composite materials with fractal geometry are studied in the framework of fractional calculus. This consideration establishes a link between fractal geometry of the media and fractional integro-differentiation. The fractional Maxwell equations are obtained, and methods of fractional calculus are employed to obtain analytical expressions of the electric field inside the fractal composite structures. [Copyright &y& Elsevier]

Abstract: In this paper, we first present a version of Filippov’s Theorem for fractional semilinear differential inclusions of the form, where is the Caputo fractional derivative and is a set-valued map. After several existence results, the compactness of solutions sets is also investigated. Some results from topological fixed point theory together with notions of measure of noncompactness are used. [Copyright &y& Elsevier]

Abstract: This paper deals with the global relative controllability of linear and nonlinear fractional dynamical systems with distributed delays in control for finite dimensional spaces. Sufficient conditions for controllability results are obtained using Schauder’s fixed point theorem and the controllability Grammian matrix which is defined by Mittag Leffler matrix function. An example is given to illustrate the theory. [Copyright &y& Elsevier]

Abstract: The aim of this manuscript is to analyze the existence of a periodic mild solution to the problem of the following nonlinear fractional differential equation where denotes the Riemann–Liouville fractional derivative. We obtained the expressions of the general solution for the linear fractional differential equation by making use of the Laplace and inverse Laplace transforms. By making use of the Banach contraction mapping principle and the Schaefer fixed point theorem, the existence results of one or at least one mild solution for a nonlinear fractional differential equation were given. [Copyright &y& Elsevier]

Abstract: Here, we study the univariate fractional quantitative approximation of real valued functions on a compact interval by quasi-interpolation sigmoidal and hyperbolic tangent neural network operators. These approximations are derived by establishing Jackson type inequalities involving the moduli of continuity of the right and left Caputo fractional derivatives of the engaged function. The approximations are pointwise and with respect to the uniform norm. The related feed-forward neural networks are with one hidden layer. Our fractional approximation results into higher order converges better than the ordinary ones. [Copyright &y& Elsevier]

Abstract: In the present work the Samarskii–Ionkin type non-local problem with Caputo fractional order differential operator is studied. The considered problem generalizes some previous known problems formulated for some fourth order parabolic equations. We prove the existence and uniqueness of a regular solution of the formulated problem applying the method of separation of variables. [Copyright &y& Elsevier]

Abstract: In this paper, viability results for nonlinear fractional differential equations with the Riemann–Liouville derivative are proved. We give a necessary condition for fractional viability of a locally closed set with respect to a nonlinear function. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce two new subclasses of p-valent analytic functions defined by means of fractional derivative of order . We obtained the various results including coefficient bounds and distortion inequalities for these function classes. Furthermore, we determine some inclusion relations for the -neighborhoods of a family of p-valent analytic functions with negative coefficients which is defined by means of a certain nonhomogeneous Cauchy–Euler differential equation. [Copyright &y& Elsevier]

Abstract: This paper presents an extension to the time integral characteristics method for estimation of parameters in fractional subdiffusion equations containing Riemann–Liouville and Caputo fractional time derivatives. The explicit representations of the fractional diffusion coefficient and order of fractional differentiation via a Laplace transform of the concentration field are obtained. A technique of optimal Laplace parameter determination by minimization of relative errors bounds is described. The effectivity of the proposed approach is illustrated by numerical example. [Copyright &y& Elsevier]

Abstract: The unsteady flow of an incompressible Maxwell fluid with fractional derivative induced by a sudden moved plate has been studied using Fourier sine and Laplace transforms. The obtained solutions for the velocity field and shear stress, written in terms of generalized functions, are presented as sum of the similar Newtonian solutions and the corresponding non-Newtonian contributions. The non-Newtonian contributions, as expected, tend to zero for . Furthermore, the solutions for ordinary Maxwell fluid, performing the same motion, are obtained as limiting cases of general solutions and verified by comparison with previously known results. Finally, the influence of the material and the fractional parameters on the fluid motion, as well as a comparison among fractional Maxwell, ordinary Maxwell and Newtonian fluids is also analyzed by graphical illustrations. [Copyright &y& Elsevier]

Abstract: A second-order abstract problem involving derivatives of non-integer order in the nonlinearity as well as in the nonlocal conditions is considered. This problem covers many situations arising in real world phenomena and extends the case of integer order to the case of non-integer order. We establish the existence of mild solutions. Under extra appropriate sufficient assumptions on the nonlinearity and on the initial data these mild solutions are shown to be classical. [Copyright &y& Elsevier]

Abstract: We investigate the existence of nontrivial solutions for a multi-point boundary value problem for fractional differential equations. Under certain growth conditions on the nonlinearity, several sufficient conditions for the existence of nontrivial solution are obtained by using Leray–Schauder nonlinear alternative. As an application, some examples to illustrate our results are given. [Copyright &y& Elsevier]

Abstract: This paper presents the necessary and sufficient optimality conditions for problems of the fractional calculus of variations with a Lagrangian depending on the free end-points. The fractional derivatives are defined in the sense of Caputo. [Copyright &y& Elsevier]

Abstract: A Newmark-diffusive scheme is presented for the time-domain solution of dynamic systems containing fractional derivatives. This scheme combines a classical Newmark time-integration method used to solve second-order mechanical systems (obtained for example after finite element discretization), with a diffusive representation based on the transformation of the fractional operator into a diagonal system of linear differential equations, which can be seen as internal memory variables. The focus is given on the algorithm implementation into a finite element framework, the strategies for choosing diffusive parameters, and applications to beam structures with a fractional Zener model. [Copyright &y& Elsevier]

Abstract: This paper introduces three new operators and presents some of their properties. It defines a new class of variational problems (called Generalized Variational Problems, or GVPs) in terms of these operators and derives Euler–Lagrange equations for this class of problems. It is demonstrated that the left and the right fractional Riemann–Liouville integrals, and the left and the right fractional Riemann–Liouville, Caputo, Riesz–Riemann–Liouville and Riesz–Caputo derivatives are special cases of these operators, and they are obtained by considering a special kernel. Further, the Euler–Lagrange equations developed for functional defined in terms of the left and the right fractional Riemann–Liouville, Caputo, Riesz–Riemann–Liouville and Riesz–Caputo derivatives are special cases of the Euler–Lagrange equations developed here. Examples are considered to demonstrate the applications of the new operators and the new Euler–Lagrange equations. The concepts of adjoint differential operators and adjoint differential equations defined in terms of the new operators are introduced. A new class of generalized Lagrangian, Hamiltonian, and action principles are presented. In special cases, these formulations lead to fractional adjoint differential operators and adjoint differential equations, and fractional Lagrangian, Hamiltonian, and action principle. Thus, the new operators introduce a generalized approach to many problems in classical mechanics in general and variational calculus in particular. Possible extensions of the subject and the concepts discussed here are also outlined. [Copyright &y& Elsevier]

Abstract: The alternative system initial conditions versus the derivative initial conditions is focused in this paper. It is shown that Riemann–Liouville and Caputo initial conditions result from the corresponding derivative and not necessarily from the system at hand. To setup the correct system initialization, a formulation generalizing the integer order approach is presented. This is based on a generalization to the fractional environment of the well known jump formula. The obtained scheme is very general and does not depend on any transform. Besides, it can also be used in the time variant case. The Riemann–Liouville and Caputo initial conditions are interpreted in terms of this general framework and deduced equations where they are correct. [Copyright &y& Elsevier]

Abstract: This paper makes an attempt to develop a fractal derivative model of anomalous diffusion. We also derive the fundamental solution of the fractal derivative equation for anomalous diffusion, which characterizes a clear power law. This new model is compared with the corresponding fractional derivative model in terms of computational efficiency, diffusion velocity, and heavy tail property. The merits and distinctions of these two models of anomalous diffusion are then summarized. [Copyright &y& Elsevier]

Abstract: This paper deals with a viscoelastic model of the hamstring muscle group. The model includes fractional derivatives of stretching force and elongation, as well as restrictions on the coefficients that follow from the Clausius–Duhem inequality. On the basis of a ramp-and-hold type of experiment, four rheological parameters have been calculated by numerical treatment ab initio. Riemann–Liouville fractional derivatives were approximated numerically using the Grünwald–Letnikov definition. Obtained results were verified by use of the Laplace transform method. The stretching force in time domain involves Mittag-Leffler-type function. [Copyright &y& Elsevier]

Abstract: Synchronization of chaotic and uncertain Duffing–Holmes system has been done using the sliding mode control strategy. Regarding the synchronization task as a control problem, fractional order mathematics is used to express the system and sliding mode for synchronization. It has been shown that, not only the performance of the proposed method is satisfying with an acceptable level of control signal, but also a rather simple stability analysis is performed. The latter is usually a complicated task for uncertain nonlinear chaotic systems. [Copyright &y& Elsevier]

Abstract: Riemann’s conjecture is understood as a consequence of the -fractional order differential dynamics which seeds a constructive approach to the zeta function. A morphism between different representations of the dynamics reveals a hidden correlation between a phase angle introduced by the real term of the power and the complex part . [Copyright &y& Elsevier]

Abstract: In this paper, we first present an impulsive version of the Filippov–Ważewski theorem and a continuous version of the Filippov theorem for fractional differential inclusions of the form where denotes the Caputo fractional derivative, and is a set-valued map. The functions characterize the jump of the solutions at impulse points . Additional existence results are obtained under both convexity and nonconvexity conditions on the multivalued right-hand side. The proofs rely on the nonlinear alternative of Leray–Schauder type, a Bressan–Colombo selection theorem, and Covitz and Nadler’s fixed point theorem for multivalued contractions. The compactness of the solution set is also investigated. Finally, some geometric properties of solution sets, sets, acyclicity and contractibility, corresponding to Aronszajn–Browder–Gupta type results, are obtained. We also consider the impulsive fractional differential equations and where is a single map. Finally, we extend the existence result for impulsive fractional differential inclusions with periodic conditions, where is a multivalued map. The study of the above problems use an approach based on the topological degree combined with a Poincaré operator. [Copyright &y& Elsevier]

Optimal upper bounds are given for the deviation of a value of a multivariate function of a fractional space from its average, over convex and compact subsets of . In particular we work over rectangles, balls and spherical shells. These bounds involve the supremum and norms of related multivariate fractional derivatives of the function involved. The inequalities produced are sharp, namely they are attained. This work has been motivated by the works of Ostrowski [A. Ostrowski, Über die Absolutabweichung einer differentiebaren Funcktion von ihrem Integralmittelwert, Commentarii Mathematici Helvetici 10 (1938) 226–227], 1938, and of the author [G.A. Anastassiou, Fractional Ostrowski type inequalities, Communications in Applied Analysis 7 (2) (2003) 203–208], 2003. [Copyright &y& Elsevier]

Abstract: The paper gives some results and improves the derivation of the fractional Taylor''s series of nondifferentiable functions obtained recently in the form f (χ + h) = E α (h α D χ α)f(χ), 0 α ≤ 1, where E α is the Mittag-Leffier function. Here, one defines fractional derivative as the limit of fractional difference, and by this way one can circumvent the problem which arises with the definition of the fractional derivative of constant using Riemann-Liouville definition. As a result, a modified Riemann-Liouville definition is proposed, which is fully consistent with the fractional difference definition and avoids any reference to the derivative of order greater than the considered one''s. In order to support this F-Taylor series, one shows how its first term can be obtained directly in the form of a mean value formula. The fractional derivative of the Dirac delta function is obtained together with the fractional Taylor''s series of multivariate functions. The relation with irreversibility of time and symmetry breaking is exhibited, and to some extent, this F-Taylor''s series generalizes the fractional mean value formula obtained a few years ago by Kolwantar. [Copyright &y& Elsevier]

Abstract: By making use of certain familiar integral operators, the authors introduce and investigate several new subclasses of starlike, convex, close-to-convex, and quasi-convex functions. Among other results presented here, the authors establish a number of inclusion relationships associated with some of these integral operators. Some of the results established in this paper would provide extensions of those given in earlier works. [Copyright &y& Elsevier]

Abstract: A large variety of very general, but basic Lp (1 < p < ∞) form, Opial type inequalities [1] is established involving generalized fractional derivatives [2,3] of two functions in different orders and powers. The above rely on a generalization of Taylor''s formula for generalized fractional derivatives [2]. From the developed results derive several other concrete results of special interest. The sharpness of inequalities is established there. Finally, applications of some of these special inequalities are given in establishing uniqueness of solution and in giving upper bounds to solutions of initial value problems involving a very general system of two fractional differential equations. Also, upper bounds to various fractional derivatives of the solutions that are involved in the above systems are presented. [Copyright &y& Elsevier]

We present a general method for solving game problems of approach for dynamical systems with Volterra evolution. This method is based on the method of resolving functions and uses the apparatus of the theory of set-valued mappings. In more detail, we study game problems for systems with Riemann-Liouville fractional derivatives and regularized derivatives of Dshrbashyan-Nersesyan (fractal games) on the basis of the generalized matrix functions of Mittag-Leffler introduced here. [Copyright &y& Elsevier]