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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

Two optimal control problems are studied for linear stationary systems of fractional order with lumped variables whose dynamics is described by equations with Hadamard derivative, a minimum-norm control problem and a time-optimal problem with a constraint on the norm of the control. The setting of the problem with nonlocal initial conditions is considered. Admissible controls are sought in the class of functions p-integrable on an interval for some p. The main approach to the study is based on the moment method. The well-posedness and solvability of the moment problem are substantiated. For several special cases, the optimal control problems under consideration are solved analytically. An analogy between the obtained results and known results for systems of integer and fractional order described by equations with Caputo and Riemann–Liouville derivatives is specified.

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

We study the shape derivative of the strongly singular volume integral operator that describes time-harmonic electromagnetic scattering from homogeneous medium. We show the existence and a representation of the derivative, and we deduce a characterization of the shape derivative of the solution to the diffraction problem as a solution to a volume integral equation of the second kind.

The purpose of the current investigation is to generalize the concept of fractional derivative in the sense of Caputo–Fabrizio derivative (CF-derivative) for interval-valued function under uncertainty. The reason to choose this new approach is originated from the non-singularity property of the kernel that is critical to interpret the memory aftermath of the system, which was not precisely illustrated in the previous definitions. We study the properties of CF-derivative for interval-valued functions under generalized Hukuhara-differentiability. Then, the fractional differential equations under this notion are presented in details. We also study three real-world systems such as the falling body problem, Basset and Decay problem under interval-valued CF-differentiability. Our cases involve a demonstration that this new notion is accurately applicable for the mechanical and viscoelastic models based on the interval CF-derivative equations.

Fractional calculus is a powerful and effective tool for modelling nonlinear systems. In this paper, we introduce a class of new fractional derivative named general conformable fractional derivative (GCFD) to describe the physical world. The GCFD is generalized from the concept of conformable fractional derivative (CFD) proposed by Khalil. We point out that the term $$t^{1-\alpha }$$ in CFD definition is not essential and it is only a kind of “fractional conformable function”. We also give physical and geometrical interpretations of GCFD which thus indicate potential applications in physics and engineering. It is easy to demonstrate that CFD is a special case of GCFD, then to the authors’ knowledge, so far we first give the physical and geometrical interpretations of CFD. The above work is done by a new framework named Extended Gâteaux derivative and Linear Extended Gâteaux derivative which are natural extensions of Gâteaux derivative. As an application, we discuss a scheme for solving fractional differential equations of GCFD.

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.

The aim of this work is to study the non-local dynamic behavior of triple pendulum-type systems. We use the Euler-Lagrange and the Hamiltonian formalisms to obtain the dynamic models, based on the Riemann-Liouville, Liouville-Caputo, Caputo-Fabrizio and Atangana-Baleanu-Caputo fractional derivative definitions. In these representations, an auxiliary parameter σ is introduced, to define the equations in a fractal temporal geometry, which provides an entire new family of solutions for the dynamic behavior of the pendulum-type systems. The phase diagrams allow to visualize the effect of considering the fractional order approach, the classical behavior is recovered when the order of the fractional derivative is 1.

The model of nonlinear heat was generalized using the new trend of derivative with fractional order. The new definition of derivative with fractional order has no singular kernel thus allows a description of the variation on time or space from the lower to the upper boundaries within the space/time interval which the investigation is taken place for a given model. In detail, we presented the analysis of unique and existence of a solution for the nonlinear fractional equation. We present the derivation of a special solution using an iterative method.

Recently, Atangana and Baleanu proposed a derivative with fractional order to answer some outstanding questions that were posed by many researchers within the field of fractional calculus. Their derivative has a non-singular and nonlocal kernel. In this paper, we presented further relationship of their derivatives with some integral transform operators. New results are presented. We applied this derivative to a simple nonlinear system. We show in detail the existence and uniqueness of the system solutions of the fractional system. We obtain a chaotic behavior which was not obtained by local derivative.

In present paper we proved that the operator generated by the differential expression of second order with fractional derivative in lower terms, does not generate associated functions and that the system of eigenfunctions of this operator forms a basis in L2(0, 1).

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 suggest a new approach to the qualitative analysis of dynamic properties of solutions of differential equations with the Hukuhara derivative with the use of geometric properties of convex bodies, the comparison method, and the Chaplygin–Wazewski method for approximate integration of ordinary differential equations. We obtain estimates for the area of solutions for a class of nonlinear differential equations with the Hukuhara derivative.

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.

Exercises involving the calculation of the derivative of piecewise defined functions are common in calculus, with the aim of consolidating beginners’ knowledge of applying the definition of the derivative. In such exercises, the piecewise function is commonly made up of two smooth pieces joined together at one point. A strategy which avoids using the definition of the derivative is to find the derivative function of each smooth piece and check whether these functions agree at the chosen point. Showing that this strategy works together with investigating discontinuities of the derivative is usually beyond a calculus course. However, we shall show that elementary arguments can be used to clarify the calculation and behaviour of the derivative for piecewise functions.

Some necessary and sufficient conditions for the existence of the ℍ-derivative of a quaternion function are given in terms of its differential.

We use a generalization of the Hukuhara difference for closed intervals on the real line to develop a theory of the fractional calculus for interval-valued functions. The properties of Riemann–Liouville fractional integral, Riemann–Liouville fractional derivative and Caputo fractional derivative for interval-valued functions are investigated. Several examples are presented to illustrate the concepts and results.

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 ) .

The estimation of derivatives using floating point numbers is usually accomplished using finite differences. In this chapter we present the finite difference approximation to the derivative as a finite version of the definition of a derivative. Using Taylor series we can show the accuracy of an approximation to the derivative and predict how the error behaves as the difference parameter goes to zero. We also discuss second-derivatives and the finite difference version of the diffusion operator. Richardson extrapolation is presented as a means to get highly accurate derivative estimates by manipulating the error terms in the Taylor series. Finally, we demonstrate that complex arithmetic enables highly-accurate derivative estimates without finite differences via the complex step approximation.

In the paper, the necessary and sufficient conditions for the existence of the left-sided Hadamard-type fractional derivative are derived. Next, an analogous result for the right-sided Hadamard-type derivative is formulated.

This paper further extends the generalized covariant derivative from the first covariant derivative to the second one on curved surfaces. Through the linear transformation between the first generalized covariant derivative and the second one, the second covariant differential transformation group is set up. Under this transformation group, the second class of differential invariants and integral invariants on curved surfaces is made clear. Besides, the symmetric structure of the tensor analysis on curved surfaces are revealed.

This paper extends the classical covariant derivative to the generalized covariant derivative on curved surfaces. The basement for the extension is similar to the previous paper, i.e., the axiom of the covariant form invariability. Based on the generalized covariant derivative, a covariant differential transformation group with orthogonal duality is set up. Through such orthogonal duality, tensor analysis on curved surfaces is simplified intensively. Under the covariant differential transformation group, the differential invariabilities and integral invariabilities are constructed on curved surfaces.

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.

The generalization of summation-by-parts operators for the first derivative of Del Rey Fernandez, Boom, and Zingg [J. Comput. Phys., 266 (2014), pp. 214--239] is extended to approximations of second derivatives with a constant or variable coefficient. This enables the construction of second-derivative operators with one or more of the following characteristics: (i) nonrepeating interior point operators, (ii) nonuniform nodal distributions, and (iii) exclusion of one or both boundary nodes. Definitions are proposed that give rise to generalized summation-by-parts operators that result in consistent, conservative, and stable discretizations of partial differential equations with or without mixed derivatives. It is proven that approximations to the second derivative with a variable coefficient can be constructed using the constituent matrices of the constant-coefficient operator. Moreover, for operators with a repeating interior point operator, a decomposition is proposed that makes the application of such o...

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.