Fractional Derivative Formulas Research Articles

Certain generalized fractional calculus formulas and integral transforms involving (p,q)-Mathieu-type series

In this paper, we establish sixteen interesting generalized fractional integral and derivative formulas including their composition formulas by using certain integral transforms involving generalized (p,q)-Mathieu-type series.

Fractional calculus of generalized p-k-Mittag-Leffler function using Marichev–Saigo–Maeda operators

In this paper, we establish fractional integral and derivative formulas involving the generalized p-k-Mittag-Leffler function by using Marichev–Saigo–Maeda type fractional integral and derivative operators. We also consider some special cases of derived results by considering specific values of the parameters of the generalized p-k-Mittag-Leffler function to give the application of our main results.

Extended Type k-Mittag–Leffler Function and Its Applications

In this paper, we introduce the extended type k-Mittag–Leffler function of two variables and find its relation with other special functions. We also establish fractional integral and derivative formulas of the extended type k-Mittag–Leffler function of two variables to give the applications of our main results by using Marichev–Saigo–Maeda type operators with the kernel $$F_{3}$$ . Besides, we obtain certain interesting particular cases of derived results considering specific values of the parameters.

A new Riemann\u2013Liouville type fractional derivative operator and its application in generating functions

Here, the concept of a new and interesting Riemann–Liouville type fractional derivative operator is exploited. Treatment of a fractional derivative operator has been made associated with the extended Appell hypergeometric functions of two variables and Lauricella hypergeometric function of three variables. With a view on analytic properties and application of new Riemann–Liouville type fractional derivative operator, we have obtained new fractional derivative formulas for some familiar functions and for Mellin transformation formulas. For the sake of justification of our new operator, we have established some presumably new generating functions for an extended hypergeometric function using the new definition of fractional derivative operator.

Non-linear boundary value problems involving Caputo derivatives of complex fractional order

We study approximate solutions of CDtβy(t)=f(t,y(t)), separately, for β ∈ (0, 1) and β ∈ (1, 2) with different boundary data, where CDtβ is the Caputo fractional derivative of complex-order. For this purpose we use the expansion formula for such fractional derivatives and prove the existence and the uniqueness of approximate solutions under certain conditions and their convergence to the original solutions.

Certain fractional calculus formulas involving extended generalized Mathieu series

We establish fractional integral and derivative formulas by using fractional calculus operators involving the extended generalized Mathieu series. Next, we develop their composition formulas by applying the integral transforms. Finally, we discuss special cases.

Optimal Error Analysis of a FEM for Fractional Diffusion Problems by Energy Arguments

In this article, the piecewise-linear finite element method (FEM) is applied to approximate the solution of time-fractional diffusion equations on bounded convex domains. Standard energy arguments do not provide satisfactory results for such a problem due to the low regularity of its exact solution. Using a delicate energy analysis, a priori optimal error bounds in $$L^2(\varOmega )$$ -, $$H^1(\varOmega )$$ -norms, and a quasi-optimal bound in $$L^{\infty }(\varOmega )$$ -norm are derived for the semidiscrete FEM for cases with smooth and nonsmooth initial data. The main tool of our analysis is based on a repeated use of an integral operator and use of a $$t^m$$ type of weights to take care of the singular behavior of the continuous solution at $$t=0$$ . The generalized Leibniz formula for fractional derivatives is found to play a key role in our analysis. Numerical experiments are presented to illustrate some of the theoretical results.

A family of the incomplete hypergeometric functions and associated integral transform and fractional derivative formulas

Recently, Srivastava et al. [Integral Transforms Spec. Funct. 23 (2012), 659-683] introduced the incomplete Pochhammer symbols that led to a natural generalization and decomposition of a class of hypergeometric and other related functions as well as to certain potentially useful closed-form representations of definite and improper integrals of various special functions of applied mathematics and mathematical physics. In the present paper, our aim is to establish several formulas involving integral transforms and fractional derivatives of this family of incomplete hypergeometric functions. As corollaries and consequences, many interesting results are shown to follow from our main results.

Integral transform and fractional derivative formulas involving the extended generalized hypergeometric functions and probability distributions

In the present paper, our aim is to establish several formulas involving integral transforms, fractional derivatives, and a certain family of extended generalized hypergeometric functions. As corollaries and consequences, many interesting results are shown to follow from our main results. A probability density function involving the extended generalized hypergeometric function is introduced, and its properties are studied. The corresponding properties of some of the classical probability distributions and their associated probability density functions are easily derivable as special cases of our general results. Copyright © 2016 John Wiley & Sons, Ltd.

FRACTIONAL CALCULUS OPERATORS OF GENERALIZED CONFLUENT HYPERGEOMETRIC FUNCTION

In the present paper, the authors establish some fractional integral and fractional derivative formulas involving a generalized confluent type hypergeometric function introduced by Parmer [6]. AMS Subject Classification: 26A33, 33B15, 33C05, 33C20, 44A10, 44A20, 33B99, 33C99, 33D90

Stability Analysis of a Fractional Order Food Chain Model

We provide an analytical proof of the local stability contained in the region of coexistence of the three species of a tritrophic fractional--order food chain model. An expansion formula for fractional derivatives given as in form of a series involving function and moments of its k-thderivative is used. The tritrophic fractional--order food chain system is converted to a system of ordinary differential equation of order 3M. Also stability analysis is studied by using the fractional Routh--Hurwitz stability conditions in origin. Numerical results show that the presented method is easy to implement and accurateto differential equations of fractional order.

Euler–Lagrange Equations for Lagrangians Containing Complex-order Fractional Derivatives

Two variational problems of finding the Euler---Lagrange equations corresponding to Lagrangians containing fractional derivatives of real- and complex-order are considered. The first one is the unconstrained variational problem, while the second one is the fractional optimal control problem. The expansion formula for fractional derivatives of complex-order is derived in order to approximate the fractional derivative appearing in the Lagrangian. As a consequence, a sequence of approximated Euler---Lagrange equations is obtained. It is shown that the sequence of approximated Euler---Lagrange equations converges to the original one in the weak sense as well as that the sequence of the minimal values of approximated action integrals tends to the minimal value of the original one.

Extended Riemann-Liouville fractional derivative operator and its applications

Many authors have introduced and investigated certain extended fractional derivative operators. The main object of this paper is to give an extension of the Riemann-Liouville fractional derivative operator with the extended Beta function given by Srivastava et al. [22] and investigate its various (potentially) useful and (presumably) new properties and formulas, for example, integral representations, Mellin transforms, generating functions, and the extended fractional derivative formulas for some familiar functions. c

Certain unified fractional integrals and derivatives for a product of Aleph function and a general class of multivariable polynomials

Saigo and Maeda (Transform Methods and Special Functions, Varna, Bulgaria, pp. 386-400, 1996) introduced and investigated certain generalized fractional integral and derivative operators involving the Appell function . Here we aim at presenting four unified fractional integral and derivative formulas of Saigo and Maeda type, which are involved in a product of ℵ-function and a general class of multivariable polynomials. The main results, being of general nature, are shown to be some unification and extension of many known formulas given, for example, by Saigo and Maeda (Transform Methods and Special Functions, Varna, Bulgaria, pp. 386-400, 1996), Saxena et al. (Kuwait J. Sci. Eng. 35(1A):1-20, 2008), Srivastava and Garg (Rev. Roum. Phys. 32:685-692, 1987), Srivastava et al. (J. Math. Anal. Appl.193:373-389, 1995) and so on. Our main results are also shown to be further specialized to yield a large number of known and (presumably) new formulas involving, for instance, Saigo fractional calculus operators, several special functions such as H-function, I-function, Mittag-Leffler function, generalized Wright hypergeometric function, generalized Bessel-Maitland function. MSC:26A33, 33E20, 33C45, 33C60, 33C70.

Boundary fractional differential equation in a complex domain

We discuss univalent solutions of boundary fractional differential equations in a complex domain. The fractional operators are taken in the sense of the Srivastava-Owa calculus in the unit disk. The existence of subsolutions and supersolutions (maximal and minimal) is established. The existence of a unique univalent solution is imposed. Applications are constructed by making use of a transformation formula for fractional derivatives as well as generalized fractional derivatives.

Theory Analysis of Left-Handed Grünwald-Letnikov Formula with0<α<1to Detect and Locate Singularities

We study fractional-order derivatives of left-handed Grünwald-Letnikov formula with0<α<1to detect and locate singularities in theory. The widely used four types of ideal singularities are analyzed by deducing their fractional derivative formula. The local extrema of fractional derivatives are used to locate the singularities. Theory analysis indicates that fractional-order derivatives of left-handed Grünwald-Letnikov formula with0<α<1can detect and locate four types of ideal singularities correctly, which shows better performance than classical 1-order derivatives in theory.

A note on some new series of special functions

Recently, Brychkov [On some new series of special functions. Appl Math Comput. 2007;187:101–104] obtained several finite and infinite series identities involving special functions by making use of an operator related to the Riemann–Liouville fractional calculus operator. In this paper, we present several identities involving the Jacobi polynomials, the generalized Laguerre function and the first Appell's function. These relations are obtained by using a fractional calculus operator related to the Riemann–Liouville operator and a new transformation formula [Tremblay R, Gaboury S, Fugère B-J. A new transformation formula for fractional derivatives with applications. Integral Transforms Spec Funct. 2013;24(3):172–186].

An expansion formula for fractional derivatives of variable order

Abstract In this work we extend our previous results and derive an expansion formula for fractional derivatives of variable order. The formula is used to determine fractional derivatives of variable order of two elementary functions. Also we propose a constitutive equation describing a solidifying material and determine the corresponding stress relaxation function.

Expansion formula for fractional derivatives in variational problems

We modify the expansion formula introduced in [T.M. Atanacković, B. Stanković, An expansion formula for fractional derivatives and its applications, Fract. Calc. Appl. Anal. 7 (3) (2004) 365–378] for the left Riemann–Liouville fractional derivative in order to apply it to various problems involving fractional derivatives. As a result we obtain a new form of the fractional integration by parts formula, with the benefit of a useful approximation for the right Riemann–Liouville fractional derivative, and derive a consequence of the fractional integral inequality ∫0Ty⋅0Dtαydt≥0. Further, we use this expansion formula to transform fractional optimization (minimization of a functional involving fractional derivatives) to the standard constrained optimization problem. It is shown that when the number of terms in the approximation tends to infinity, solutions to the Euler–Lagrange equations of the transformed problem converge, in a weak sense, to solutions of the original fractional Euler–Lagrange equations. An illustrative example is treated numerically.

On the Bagley–Torvik Equation

We propose a new method for finding solution of Bagley–Torvik equation based on recently derived expansion formula for fractional derivatives. The case of nonlinear equations of Bagley–Torvik type is also discussed.