Fractional Integral Formulas Research Articles

Some Fractional Integral and Derivative Formulas Revisited

In the most common literature about fractional calculus, we find that Dtαaft=It−αaft is assumed implicitly in the tables of fractional integrals and derivatives. However, this is not straightforward from the definitions of Itαaft and Dtαaft. In this sense, we prove that Dt0ft=It−α0ft is true for ft=tν−1logt, and ft=eλt, despite the fact that these derivations are highly non-trivial. Moreover, the corresponding formulas for Dtα−∞t−δ and Itα−∞t−δ found in the literature are incorrect; thus, we derive the correct ones, proving in turn that Dtα−∞t−δ=It−α−∞t−δ holds true.

Improved Numerical Approach for Bagley–Torvik Equation Using Fractional Integral Formula and Adams–Moulton Method

Abstract In this paper, we introduce a numerical technique for solving Bagley–Torvik equations which plays an outstanding role in fractional calculus. To handle the derivatives and fractional integral in the Bagley–Torvik equations, the Laplace transform is employed to convert the equations to fractional integration equations. The resulting integral equations are solved by implicit Adams–Moulton methods. Moreover, we show the analytic convergence order of the proposed technique through the convergence analysis, and the analysis is validated by the numerical experiments. Illustrative experiments also demonstrate the validity and efficiency of the proposed method by comparing it with other existing methods.

Pathway Fractional Integral Formula Involving Extended Mittag-leffler Function in the Kernel of Generalized Elzaki Transform

Pathway Fractional Integral Formula Involving Extended Mittag-leffler Function in the Kernel of Generalized Elzaki Transform

Multivariate Mittag-Leffler function and related fractional integral operators

<abstract><p>In this paper, we describe a new generalization of the multivariate Mittag-Leffler (M-L) function in terms of generalized Pochhammer symbol and study its properties. We provide a few differential and fractional integral formulas for the generalized multivariate M-L function. Furthermore, by using the generalized multivariate M-L function in the kernel, we present a new generalization of the fractional integral operator. Finally, we describe some fundamental characteristics of generalized fractional integrals.</p></abstract>

New fractional integral formulas and kinetic model associated with the hypergeometric superhyperbolic sine function

In the paper, we consider some fractional integral formulas in terms of the Riemann–Liouville, Erdélyi–Kober type, and Weyl fractional integral operators and present the general fractional kinetic model involving the hypergeometric superhyperbolic sine function via the Gauss hypergeometric series.

Pathway fractional integral formula involving an extended Mittag–Leffler function

Abstract This work discovers the Laplace transform using a generalized pathway fractional integral formula involving an extended Mittag-Leffler function in the kernel for various parameters. Our findings are fairly broad in scope. Some well-known and novel results can also be obtained here.

An application of a fuzzy system for solving time delay fractional optimal control problems with Atangana–Baleanu derivative

AbstractIn this article, a new approach based on fuzzy systems is used for solving time delay fractional order optimal control problems. The fractional derivatives are considered in the Atangana–Baleanu sense that is a new derivative with the nonsingular and nonlocal kernel. By means of the calculus of variations and the formula for fractional integration by parts, the necessary optimality conditions associated to the time delay problem is derived. In order to solve the obtained optimality system, the solution of the system is first approximated by fuzzy solutions with adjustable parameters. The optimality system is then reduced to an unconstrained optimization problem by using appropriate error function. A learning algorithm is also presented to achieve the parameters of these fuzzy solutions. The efficiency and accuracy of the proposed approach are assessed through some illustrative examples of the time delay fractional optimal control problems.

Fractional Hypergeometric Functions

The fractional calculus of special functions has significant importance and applications in various fields of science and engineering. Here, we aim to find the fractional integral and differential formulas of the extended hypergeometric-type functions by using the Marichev–Saigo–Maeda operators. All the outcomes presented here are of general attractiveness and can yield a number of previous works as special cases due to the high degree of symmetry of the involved functions.

Generalized Fractional Integrals Involving Product of a Generalized Mittag–Leffler Function and Two H-Functions

The objective of this research is to obtain some fractional integral formulas concerning products of the generalized Mittag–Leffler function and two H-functions. The resulting integral formulas are described in terms of the H-function of several variables. Moreover, we give some illustrative examples for the efficiency of the general approach of our results.

Application of Lane-Emden differential equation numerical method in fair value analysis of financial accounting

Abstract In order to study the fair value analysis of financial accounting, the Euler wavelet method is proposed to solve the numerical solutions of a class of Lane-Emden type differential equations with Dirichlet, Neumann and Neumann-Robin boundary conditions. The results show that the fractional integral formula of Euler wavelet function under the Riemann-Liouville fractional order definition and the L∞ and L2 errors of Haar wavelet are derived by the analytic form of Euler polynomial. By fixing M=4 and increasing the resolution scale k of Euler wavelet, a stable convergence solution can be obtained. The Lane-Emden equation with boundary conditions is transformed into algebraic equations by Euler wavelet collocation method, and the numerical results are compared with the results and exact solutions of other methods. The application advantages of fair value can be exerted through financial accounting to promote the transformation and upgrading of enterprises and realise the stable economic growth.

A basic study of a fractional integral operator with extended Mittag-Leffler kernel

<abstract><p>In this present paper, the basic properties of an extended Mittag-Leffler function are studied. We present some fractional integral and differential formulas of an extended Mittag-Leffler function. In addition, we introduce a new extension of Prabhakar type fractional integrals with an extended Mittag-Leffler function in the kernel. Also, we present certain basic properties of the generalized Prabhakar type fractional integrals.</p></abstract>

Unified fractional integral and derivative formulas, integral transforms of incomplete $$\tau $$-hypergeometric function

The objective of this article is to evaluate unified fractional integrals and derivative formulas involving the incomplete $$\tau $$ -hypergeometric function. These integrals and derivatives are further applied in proving theorems on Marichev–Saigo–Maeda operators of fractional integration and differentiation. The results are expressed in terms of the generalized Gauss hypergeometric functions (Fox–Wright function). Corresponding assertions in terms of Saigo, Erdelyi–Kober, Riemann–Liouville, and Weyl type of fractional integrals and derivatives are presented. Also, we develop their composition formula by applying the Beta and Laplace transforms. Further, we point out also their relevance.

Fractional Calculus involving (p, q)-Mathieu Type Series

Abstract Aim of the present paper is to establish fractional integral formulas by using fractional calculus operators involving the generalized (p, q)-Mathieu type series. Then, their composition formulas by using the integral transforms are introduced. Further, a new generalized form of the fractional kinetic equation involving the series is also developed. The solutions of fractional kinetic equations are presented in terms of the Mittag-Leffler function. The results established here are quite general in nature and capable of yielding both known and new results.

Some Results of Fractional Integral Involving I-Function and General Class of Polynomial

In the present paper, we have derived two multiplication theorems for I-function by using fractional integral formula $$ \mathop I\nolimits_{0,x}^{\alpha ,\beta ,\eta } f(x) $$ and $$ \mathop J\nolimits_{0,\infty }^{\alpha ,\beta ,\eta } f(x) $$ (Banerji and Choudhary in Proc Natl Acad Sci India Sect A Phys Sci 66:271–278, 1996). Further, we established some applications of the theorem in the form of particular case.

A Note on Pathway Fractional Integral Formulas Associated with the Incomplete H-Functions

Very recently Srivastava et al. (Russ J Math Phys 25(1):116–138, 2018) have introduced the incomplete H-functions and investigated their several interesting properties, for example, decomposition and reduction formulas, derivative formulas, and various integral transforms. They also pointed out potential applications of many of those incomplete special functions, which are specialized from the incomplete H-functions, involving (for example) probability theory. In this paper, we aim to establish two pathway fractional integral formulas involving the incomplete H-functions. Also our main results are indicated to reduce to yield some known identities. Further, among numerous special cases of our main results, some of them are explicitly demonstrated.

Fractional Integral and Derivative Formulas by Using Marichev-Saigo-Maeda Operators Involving the S-Function

We establish fractional integral and derivative formulas by using Marichev-Saigo-Maeda operators involving the S-function. The results are expressed in terms of the generalized Gauss hypergeometric functions. Corresponding assertions in terms of Saigo, Erdélyi-Kober, Riemann-Liouville, and Weyl type of fractional integrals and derivatives are presented. Also we develop their composition formula by applying the Beta and Laplace transforms. Further, we point out also their relevance.

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.

Generalized variational calculus in terms of multi-parameters involving Atangana-Baleanu's derivatives and application

In this paper, the generalized variational calculus in terms of multi-parameters involving Atangana-Baleanu's Derivatives are discussed. We consider the Hilfers generalized fractional derivative that in sense Atangana-Baleanu derivatives. We develop integration by parts formulas for the generalized fractional derivatives which are key to developing fractional variational calculus. It is shown that many derivatives used recently and their variational formulations can be obtained by setting different parameters to different values. The fractional Euler-Lagrange equations of fractional Lagrangians for constrained systems contains a fractional Hilfer-Atangana-Baleanu's derivatives with multi parameters are investigated. We also define fractional generalized momenta and provide fractional Hamiltonian formulations in terms of the new generalized derivatives. An example is presented to show applications of the formulations presented here. Some possible extensions of this research are also discussed. We present a general formulation and a solution scheme for a class of Fractional Optimal Control Problems (FOCPs) for those systems. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of FDEs. The calculus of variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler-Lagrange equations for the FOCP.