Generalized Pathway Fractional Integral Formulas Involving Extended Multi-Index Mittag-Leffler Function in Kernel of SUM Transform

The fractional integral and differential operators involving the family of special functions have found significant importance and applications in various fields of mathematics and engineering. The goal of this chapter is to find the fractional integral and differential formulas (also known as composition formulas) involving the extended incomplete generalized hypergeometric functions by using the generalized fractional calculus operators (the Marichev–Saigo–Maeda operators). After that, we established their image formulas by using the integral transforms like: Beta transform, Laplace transform and Whittaker transform. Moreover, the reduction formulas are also considered as special cases of our main findings associated with the well-known Saigo fractional integral and differential operators, Erdelyi-Kober fractional integral and differential operators, Riemann-Liouville fractional integral and differential operators and the Weyl fractional calculus operators.

The purpose of this paper is to develop some new recurrence relations for the two parametric Mittag-Leffler function. Then, we consider some applications of those recurrence relations. Firstly, we express many of the two parametric Mittag-Leffler functions in terms of elementary functions by combining suitable pairings of certain specific instances of those recurrence relations. Secondly, by applying Riemann–Liouville fractional integral and differential operators to one of those recurrence relations, we establish four new relations among the Fox–Wright functions, certain particular cases of which exhibit four relations among the generalized hypergeometric functions. Finally, we raise several relevant issues for further research.

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.

In this paper, our aim is to establish some mean value inequalities for the Fox–Wright functions, such as Turan-type inequalities, Lazarevic and Wilker-type inequalities. As applications we derive some new type inequalities for hypergeometric functions and the four-parametric Mittag–Leffler functions. Furthermore, we prove the monotonicity of ratios for sections of series of Fox–Wright functions. The results are also closely connected with Turan-type inequalities. Moreover, some other type inequalities are also presented. At the end of the paper, some problems are stated which may be of interest for further research.

The Fox–Wright functions and Laguerre multiplier sequences

<abstract><p>Motivated by the recent investigations of several authors, the main aim of this article is to derive several functional inequalities for a class of functions related to the incomplete Fox-Wright functions that were introduced and studied recently. Moreover, new functional bounds for functions related to the Fox-Wright function are deduced. Furthermore, a class of completely monotonic functions related to the Fox-Wright function is given. The main mathematical tools used to obtain some of the main results are the monotonicity patterns and the Mellin transform for certain functions related to the two-parameter Mittag-Leffler function. Several potential applications for this incomplete special function are mentioned.</p></abstract>

This paper presents three direct methods based on Grünwald–Letnikov, trapezoidal and Simpson fractional integral formulas to solve fractional optimal control problems (FOCPs). At first, the fractional integral form of FOCP is considered, then the fractional integral is approximated by Grünwald–Letnikov, trapezoidal and Simpson formulas in a matrix approach. Thereafter, the performance index is approximated either by trapezoidal or Simpson quadrature. As a result, FOCPs are reduced to nonlinear programming problems, which can be solved by many well-developed algorithms. To improve the efficiency of the presented method, the gradient of the objective function and the Jacobian of constraints are prepared in closed forms. It is pointed out that the implementation of the methods is simple and, due to the fact that there is no need to derive necessary conditions, the methods can be simply and quickly used to solve a wide class of FOCPs. The efficiency and reliability of the presented methods are assessed by ample numerical tests involving a free final time with path constraint FOCP, a bang-bang FOCP and an optimal control of a fractional-order HIV-immune system.

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.

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.

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 chapter, the necessary and sufficient optimality conditions for systems involving Atangana–Baleanu’s derivatives are discussed. The fractional Euler–Lagrange equations of fractional Lagrangians for constrained systems that contains a fractional Atangana–Baleanu’s derivatives are investigated. The fractional contains both the fractional derivatives and the fractional integrals in the sense of Atangana–Baleanu. We present a general formulation and a solution scheme for a class of Fractional Optimal Control Problems (FOCPs) for those systems. 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.

<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>

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.

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.

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.

In this paper, we obtain formulas of fractional integration (of Marichev– Saigo–Maeda type) of the generalized multi-index Mittag-Leffler functions Eγ,κ[(αj,βj)m; z] generalizing 2m-parametric Mittag-Leffler functions studied by Saxena and Nishimoto (J. Fract. Calc.37 (2010] 43–52). Some interesting special cases of our main results are considered too.