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The purpose of this present effort is to define a new fractional differential operator Tzβ,τ,γ, involving Srivastava–Owa fractional derivative operator. Further, we investigate some geometric properties such as univalency, starlikeness, convexity for their normalization, we also study boundedness and compactness of analytic and univalent functions on weighted μ-Bloch space for this operator. The method in this study is based on the generalized hypergeometric function. Keywords: Analytic functions, Univalent functions, Srivastava–Owa fractional derivative operator, Generalized differential operator, Weighted μ-Bloch space, Convolution (or Hadamard product)

In recent years, non-Newtonian fluids have been widely applied in a number of engineering applications. One particular subclass of non-Newtonian fluids is the generalized Oldroyd-B fluids with fractional derivative constitutive equations, which can be used to approximate the response of many dilute polymeric liquids. Different to the general time fractional diffusion equation, the constitutive equation not only has a multi-term time derivative but also possesses a special time fractional operator on the spatial derivative, which is challenging to approximate. The literature reported on the numerical solution of this model is extremely sparse. In this paper, we will consider the finite difference method for its discretisation and propose a new scheme to approximate the time fractional derivative. Then we derive an implicit finite difference scheme and establish some new theoretical analysis of the stability and convergence. Furthermore, we present a numerical scheme to improve the time order. Finally, we present two numerical examples to show the effectiveness of our method and apply it to solve the generalized Oldroyd-B fluid model. Keywords Finite difference method Generalized Oldroyd-B fluid Fractional diffusion equation Multi-term time derivative Caputo fractional derivative

A collocation method based on multiquadric radial basis functions is proposed for numerical solution of fractional differential equations. The fractional derivative is sense of Caputo–Fabrizio derivative. An efficient error bound of the method is also introduced in the $$L_2$$ norm, using properties of native spaces. We test this approach for two examples. The obtained numerical results confirm theoretical prediction of the convergence of this method.

Positive homogeneous functions on R of a negative degree are characterized by a new counterpart of the Euler’s homogeneous function theorem using quantum calculus and replacing the classical derivative operator by Jackson derivative. As application we start by characterizing the harmonic functions associated to Jackson derivative. Then, the solution of the Cauchy problem associated to the analogue of the Euler operator is given. Using this solution we study the associated ν-potential. Its Markovianity property is treated. Keywords: Homogeneous functions, Euler’s theorem, Quantum calculus, Cauchy problem, Markovian semigroups

In this paper, we establish an energy estimate for the second derivative of 4 -convex functions. Such kinds of estimates for the first derivative of 2 -convex (convex) functions were obtained by Hussain, Pečarić and Shashivili [5].

In the paper, we derive an equivalent definition of the generalized Riemann-Liouville derivative (derivative in the Hilfer sense). Next, we show that the new formula for such derivative is very useful in practical applications.

Made available in DSpace on 2018-12-11T16:53:26Z (GMT). No. of bitstreams: 0 Previous issue date: 2018-05-01 Eigenfunctions associated with Riemann–Liouville and Caputo fractional differential operators are obtained by imposing a restriction on the fractional derivative parameter. Those eigenfunctions can be used to express the analytical solution of some linear sequential fractional differential equations. As a first application, we discuss analytical solutions for the so-called fractional Helmholtz equation with one variable, obtained from the standard equation in one dimension by replacing the integer order derivative by the Riemann–Liouville fractional derivative. A second application consists of an initial value problem for a fractional wave equation in two dimensions in which the integer order partial derivative with respect to the time variable is replaced by the Caputo fractional derivative. The classical Mittag-Leffler functions are important in the theory of fractional calculus because they emerge as solutions of fractional differential equations. Starting with the solution of a specific fractional differential equation in terms of these functions, we find a way to express the exponential function in terms of classical Mittag-Leffler functions. A remarkable characteristic of this relation is that it is true for any value of the parameter n appearing in the definition of the functions, i.e., we have an infinite family of different expressions for ex in terms of classical Mittag-Leffler functions. Departamento de Bioprocessos e Biotecnologia FCA-UNESP, Rua José Barbosa de Barros 1780 Departamento de Matemática Aplicada IMECC-UNICAMP Departamento de Matemática Faculdade de Ciências UNESP, Av. Eng. Luiz Edmundo Carrijo Coube, 14-01 Bairro: Vargem Limpa Departamento de Bioprocessos e Biotecnologia FCA-UNESP, Rua José Barbosa de Barros 1780 Departamento de Matemática Faculdade de Ciências UNESP, Av. Eng. Luiz Edmundo Carrijo Coube, 14-01 Bairro: Vargem Limpa

In terms of the derivative operator and Baily's F 1 2 ( 1 2 ) -series identity, two kinds of summation formulae involving generalized harmonic numbers are established.

The existence of solutions for a coupled system of time-fractional differential equations including continuous functions and the Caputo-Fabrizio fractional derivative is examined. After that we investigated a coupled system of time-fractional differential inclusions including compact- and convex-valuedL1-Caratheodory multifunctions and the Caputo-Fabrizio fractional derivative.

In this work, certain classes of admissible functions are considered. Some strong differential subordination and superordination properties of analytic functions associated with new generalized derivative operator $\mathfrak{B}^{\mu,q,s}_{\lambda_{1},\lambda_{2},\ell,d}$ are investigated. New strong differential sandwich-type results associated with the generalized derivative operator are also given.

The fractional order derivative is a famous subject and it has been studied about three centuries. There are not any studies on the physical and geometrical interpretation of fractional order derivative. The aim of this study is to interpret the geometrical meaning of the fractional order derivatives of any function. We used a new definition for fractional order derivative to interpret it. At this aim, simple and easily understandable subjects distance, velocity and acceleration were used to depict the experimental results and interpretations. The important contribution of this paper is that when the order of fractional order derivative approaches to 1, the result of derivative process approaches to classical derivative and operator is linear; otherwise, the result of derivative process is different from classical derivative and applied operator is non-linear.

In this paper, we consider two types of space-time fractional diffusion equations(STFDE) on a finite domain. The equation can be obtained from the standard diffusion equation by replacing the second order space derivative by a Riemann-Liouville fractional derivative of order s (1

In this paper the notion of Lie derivative of a tensor fieldT of type (1,1) of real hypersurfaces in complex space forms with re-spect to the generalized Tanaka-Webster connection is introduced andis called generalized Tanaka-Webster Lie derivative. Furthermore, threedimensional real hypersurfaces in non-flat complex space forms whosegeneralized Tanaka-Webster Lie derivative of 1) shape operator, 2) struc-ture Jacobi operator coincides with the covariant derivative of them withrespect to any vector field X orthogonal to ξ are studied. 1. IntroductionA complex space form is an n-dimensional Kahler manifold of constant holo-morphic sectional curvature c. A complete and simply connected complex spaceform is analytically isometric to a complex projective space CP n if c > 0, or toa complex Euclidean space C n if c = 0, or to a complex hyperbolic space CH n if c < 0. The complex projective and complex hyperbolic spaces are callednon-flat complex space forms, since c 6= 0 and the symbol M

In this paper, on the basis of the theory of degenerate semigroups of operators and the contraction mapping theorem, we prove the local unique solvability of initial problems for a class of first-order linear differential operator equations with memory and with degenerate operator multiplying the derivative. The resulting abstract results are used to study initial boundary-value problems for partial integro-differential equations not solvable with respect to the time derivative.

The concept of fractional order derivative can be found in extensive range of many different subject areas. For this reason, the concept of fractional order derivative should be examined. After giving different methods mostly used in engineering and scientific applications, the omissions or errors of these methods will be discussed in this study. The mostly used methods are Euler, Riemann-Liouville and Caputo which are fractional order derivatives. The applications of these methods to constant and identity functions will be given in this study. Obtained results demonstrated that all of three methods have errors and deficiencies. In fact, the obtained results demonstrated that the methods given as fractional order derivatives are curve fitting or curve approximation methods. In this paper, we redefined fractional order derivative by using classical derivative definition and L'Hospital method, since classical derivative definition concluded in indefinite limit such as 0/0. The obtained definition is same as classical derivative definition in case of fractional order is equal to 1. This implies that definitions and theorems in this paper sound and complete.

We prove necessary and sufficient conditions for complete continuity of the Frechet derivative at a point and the asymptotic derivative (in the case of their existence). For any measure of noncompactness ψ , we introduce two new classes of operators: a locally strongly ψ -condensing operator at a point and a strongly ψ -condensing at infinity on spheres. The new classes contain all completely continuous operators and some noncondensing operators. In addition, we extend some results of M.A. Krasnosel'skii on bifurcation points to a class of vector fields more general than completely continuous.

By using Bismut’s approach to the Malliavin calculus with jumps, we establish a derivative formula of Bismut–Elworthy–Li’s type for SDEs driven by multiplicative Levy noises, whose Levy measure satisfies some order conditions. In particular, α -stable-like noises are allowed. Moreover, we also obtain the sharp gradient estimate in short time for the corresponding transition semigroup provided α ∈ ( 1 , 2 ) .

In this paper, we consider the shape identification problem of a body immersed in the incompressible fluid governed by Stokes-Oseen equations. Based on the domain derivative method, we derive the explicit representation of the derivative of solution with respect to the boundary. Then, according to the boundary parametrization technique, we propose a regularized Gauss-Newton algorithm for the shape inverse problem. Finally, numerical examples indicate that the iterative algorithm is feasible and effective for the practical purpose.

We characterize the difference of derivative weighted composition operators on the Bergman space in the unit disk and determine when linear-fractional derivative weighted composition operators belong to the same component of the space of derivative weighted composition operators on the Bergman space under the operator norm topology.

In this paper, we introduce a subclass of analytic functions by using the subordination concept between this function and generalized derivative operator. Some interesting properties of this class are obtained. Article DOI: https://dx.doi.org/10.20319/mijst.2016.s11.318324 This work is licensed under the Creative Commons Attribution-Non-commercial 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc/4.0/ or send a letter to Creative Commons, PO Box 1866, Mountain View, CA 94042, USA.

For a standard tool in econometrics, matrix calculus, an approach is illustrated in this note that is unusual in that context, a coordinate-free approach. It can help to eliminate the persistent use of non-standard conventions. The Kronecker product and its use can be better understood. The complications and pitfalls of defining twice differentiability by partial derivatives are avoided. Its use is demonstrated, for example by giving a coordinate-free determination of the derivative of the determinant. A vector-space calculation of the derivative of the determinant function.All conventions of matrix calculus arise naturally in the vector-space approach.The vector-space approach clarifies the role of the Kronecker product in matrix calculus.The vector-space approach gives a quick access to the v -space in interior point methods.

Let be a polynomial of degree n and for a complex number , let denote the polar derivative of the polynomial with respect to . In this paper, first we extend as well as generalize the result proved by Dewan and Mir [Inter. Jour. Math. and Math. Sci., 16 (2005), 2641-2645] to polar derivative. Besides, another result due to Dewan et al. [J. Math. Anal. Appl. 269 (2002), 489-499] is also extended to polar derivative.

The object of the present paper is to investigate various argument results of analytic and multivalent functions which are defined by using a certain fractional derivative operator. Some interesting applications are also considered.

International audience; We investigate the 1D Riemann-Liouville fractional derivative focusing on the connections with fractional Sobolev spaces, the space BV of functions of bounded variation, whose derivatives are not functions but measures andthe space SBV, say the space of bounded variation functions whose derivative has no Cantor part. We prove that SBV is included in W^{s,1} $ for every s \in (0,1) while the result remains open for BV. We study examples and address open questions.

The key purpose of the present work is to examine a fractional giving up smoking model pertaining to a new fractional derivative with non-singular kernel. The numerical simulations are conducted with the aid of an iterative technique. The existence of the solution is discussed by employing the fixed point postulate, and the uniqueness of the solution is also proved. The effect of various parameters is shown graphically. The numerical results for the smoking model associated with the new fractional derivative are compared with numerical results for a smoking model pertaining to the standard derivative and Caputo fractional derivative.

We introduce and investigate a subclass of analytic and bi-univalent functions defined by a fractional derivative operator in the open unit disk. Using the Faber polynomial expansions, we obtain upper bounds for the coefficients of functions belonging to this class.

In this paper, the stabilization problem of a autonomous linear time invariant fractional order (LTI-FO) switched system with different derivative order in subsystems is outlined. First, necessary and sufficient condition for stability of an LTI-FO switched system with different derivative order in subsystems based on the convex analysis and linear matrix inequality (LMI) for two subsystems is presented and proved. Also, sufficient condition for stability of an LTI-FO switched system with different derivative order in subsystems for more than two subsystems is proved. Then a sliding sector is designed for each subsystem of the LTI-FO switched system. Finally, a switching control law is designed to switch the LTI-FO switched system among subsystems to ensure the decrease of the norm of the switched system. Simulation results are given to show the effectiveness of the proposed variable structure controller.

In this paper, we present some inequalities for the non-tan-gential derivative of f(z). For the function f(z) = z + b p+1 z p+1 +b p+2 z p+2 + ··· defined in the unit disc, with ℜ f ′ (z)λf ′ (z)+1−λ > β,0 ≤ β β, 0 ≤ β < 1, 0 ≤ λ < 1.Consider the functiong(z) =

The generalization of the n-dimensional cube, an n-dimensional chain, the exterior derivative and the integral of a differential n-form on it are introduced and investigated. The analogue of Stokes theorem for the differential space is given., 16 pages

In this paper, fuzzy nth-order derivative for n in N is introduced. To do this, nth-order derivation under generalized Hukuhara derivative here in discussed. Calculations on the fuzzy nth-order derivative on fuzzy functions and their relationships, in general, are introduced. Then, the fuzzy nth-order differential equations is solved, for n in N.

An analog of the Oleinik-Hopf normal derivative lemma for the Laplace operator on a polyhedral set is considered.

Fractional derivatives can be used to model time delays in a diffusion process. When the order of the fractional derivative is distributed over the unit interval, it is useful for modeling a mixture of delay sources. In some special cases distributed order derivative can be used to model ultra-slow diffusion. We extend the results of Baeumer and Meerschaert (3) in the single order fractional derivative case to distributed order fractional derivative case. In particular, we develop the strong analytic solutions of distributed order fractional Cauchy problems.

The Malliavin derivative operator for the Poisson process introduced by Carlen and Pardoux [Differential calculus and integration by parts on a Poisson space, in Stochastics, Algebra and Analysis in Classical and Quantum Dynamics, S. Albeverio et al. (eds), Kluwer, Dordrecht, 1990, pp. 63–73] is extended to Levy processes. It is a true derivative operator (in the sense that it satisfies the chain rule), and we deduce a sufficient condition for the absolute continuity of functionals of the Levy process. As an application, we analyse the absolute continuity of the law of the solution of some stochastic differential equations with jumps.

If a function can be explicitly expressed, then one can easily compute its Caputo derivative by the known methods. If a function cannot be explicitly expressed but it satisfies a differential equation, how to seek Caputo derivative of such a function has not yet been investigated. In this paper, we propose a numerical algorithm for computing the Caputo derivative of a function defined by a classical (integer-order) differential equation. By the properties of Caputo derivative derived in this paper, we can change the original typical differential system into an equivalent Caputo-type differential system. Numerical examples are given to support the derived numerical method.

In the present paper, by approximating the derivatives in the Newton-Steffensen third-order method by central difference quotient, we obtain a new modification of this method free from derivatives. We prove that the method obtained preserves their order of convergence, without calculating any derivative. Finally, numerical tests confirm that our method give the better performance as compare to the other well known derivative free Steffensen type methods.

A Cauchy-type nonlinear problem for a class of fractional differential equations involving sequential derivatives is considered. Some properties and composition identities are derived. The equivalence with the associated integral equation is established. The existence and uniqueness of global solutions in the space of Lebesgue integrable functions are proved.

It has been pointed out that the derivative chains rules in fractional differential calculus via fractional calculus are not quite satisfactory as far as they can yield different results which depend upon how the formula is applied, that is to say depending upon where is the considered function and where is the function of function. The purpose of the present short note is to display some comments (which might be clarifying to some readers) on the matter. This feature is basically related to the non-commutativity of fractional derivative on the one hand, and furthermore, it is very close to the physical significance of the systems under consideration on the other hand, in such a manner that everything is right so. As an example, it is shown that the trivial first order system may have several fractional modelling depending upon the way by which it is observed. This suggests some rules to construct the fractional models of standard dynamical systems, in as meaningful a model as possible. It might happen that this pitfall comes from the feature that a function which is continuous everywhere, but is nowhere differentiable, exhibits random-like features.

By using the Malliavin calculus and solving a control problem, Bismut type derivative formulae are established for a class of degenerate diffusion semigroups with non-linear drifts. As applications, explicit gradient estimates and Harnack inequalities are derived.

The topological derivative measures the sensitivity of a given shape functional with respect to an infinitesimal singular domain perturbation. According to the literature, the topological derivative has been fully developed for a wide range of one single physical phenomenon modeled by partial differential equations. In addition, the topological asymptotic analysis associated to multi-physics problems has been reported in the literature only on the level of mathematical analysis of singularly perturbed geometrical domains. In this work, we present the topological derivative in its closed form for the total potential mechanical energy associated to a thermo-mechanical semi-coupled system, when a small circular inclusion is introduced at an arbitrary point of the domain. In particular, we consider the linear elasticity system (modeled by the Navier equation) coupled with the steady-state heat conduction problem (modeled by the Laplace equation). The mechanical coupling term comes out from the thermal stress induced by the temperature field. Since this term is non-local, we introduce a non-standard adjoint state, which allows to obtain a closed form for the topological derivative. Finally, we provide a full mathematical justification for the derived formulas and develop precise estimates for the remainders of the topological asymptotic expansion. Fil: Giusti, Sebastian Miguel. Universidad Tecnológica Nacional. Facultad Regional Córdoba; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina Fil: Novotny, A. A.. Laboratorio Nacional de Computacao Cientifica; Brasil Fil: Muñoz Rivera, J. E.. Laboratorio Nacional de Computacao Cientifica; Brasil Fil: Esparta Rodriguez, J. E.. Laboratorio Nacional de Computacao Cientifica; Brasil

In this paper we prove a derivative formula of Bismut–Elworthy–Li’s type as well as a gradient estimate for stochastic differential equations driven by α -stable noises, where α ∈ ( 0 , 2 ) . As an application, the strong Feller property for stochastic partial differential equations driven by subordinated cylindrical Brownian motions is presented.

In this paper, first derivative of smooth function is defined by the optimal solution of a special optimization problem. In the next step, by using this optimization problem for nonsmooth function, we obtain an approximation for first derivative of nonsmooth function which it is called generalized first derivative. We then extend it to define generalized second derivative for nonsmooth function. Finally, we show the efficiency of our approach by evaluating derivative and generalized first and second derivative of some smooth and nonsmooth functions, respectively.

The aim of this paper is to study certain problems of calculus of variations, that are dependent upon a Lagrange function on a Caputo-type fractional derivative. This type of fractional operator is a generalization of the Caputo and the Caputo--Hadamard fractional derivatives, that are dependent on a real parameter ro. Sufficient and necessary conditions of the first and second order are presented. The cases of integral and holonomic constraints are also considered., Comment: This is a preprint of a paper whose final and definite form will be published in Journal of Optimization Theory and Applications