and Applied Analysis 3 2. Preliminaries In this section, we present definitions and properties of generalized fractional operators. As particular cases, by choosing appropriate kernels, these operators are reduced to standard fractional integrals and fractional derivatives. Other nonstandard kernels can also be considered as particular cases. For more on the subject of generalized fractional ca...

In this paper we extend the umbral calculus, developed to deal with difference equations on uniform lattices, to q-difference equations. We show that many of the properties considered for shift invariant difference operators satisfying the umbral calculus can be implemented to the case of the q-difference operators. This qumbral calculus can be used to provide solutions to linear q-difference e...

where D , D , and D are the standard Riemann-Liouville fractional derivatives, I and I are the Riemann-Liouville fractional integrals, and 0 < γ < 1 < β < 2 < α < 3, ν,ω > 0, 0 < η, ξ < 1, k ∈R, f ∈ C([0, 1]×R×R,R), g ∈ C([0, 1]×R,R). The p-Laplacian operator is defined as φp(t) = |t|p–2t, p > 1, and (φp) = φq, 1 p + 1 q = 1. The study of boundary value problems in the setting of fractional cal...

In recent years, many works on the subject of fractional calculus contain interesting accounts of the theory and applications of fractional calculus operators in a number of areas of mathematical analysis ( such as ordinary and partial differential equations, integral equations, summation of series, etc.). The main object of this paper is to construct multivariable extension of Jacobi polynomia...

This report is aimed at the engineering and/or scientific professional who wishes to learn about Fractional Calculus and its possible applications in his/her field(s) of study. The intent is to first expose the reader to the concepts, applicable definitions, and execution of fractional calculus (including a discussion of notation, operators, and fractional order differential equations), and sec...

In this paper we aim to construct an abstract model of a differential operator with fractional integro-differential composition in final terms, where modeling is understood as interpretation concrete operators terms the infinitesimal generator corresponding semigroup. We study such Kipriyanov operator, Riesz potential, difference operator. Along this, consider transforms m-accretive generalizat...

In this paper, we introduce a new class of functions which are analytic and univalent with negative coefficients defined by using a certain fractional calculus and fractional calculus integral operators. Characterization property,the results on modified Hadamard product and integrals transforms are discussed. Further, distortion theorem and radii of starlikeness and convexity are also determine...

We define a class of discrete operators that, in particular, include the delta and nabla fractional operators. Moreover, we prove fundamental theorem calculus for these