This paper considers the Riemann–Liouville fractional operator as a tool to reduce linear ordinary equations with variable coefficients to simpler problems, avoiding the singularities of the original equation. The main result is that this technique allow us to obtain an extension of the classical integral representation of the special functions related with the original differential equations. ...

and Applied Analysis 3 Definition 2.2. Let ρ t t − 1 be the backward jump operator. Then i the nabla left fractional sum of order α > 0 starting from a is defined by ∇−α a f t 1 Γ α t ∑ s a 1 ( t − ρ s α−1f s , t ∈ Na 1 2.4 ii the nabla right fractional sum of order α > 0 ending at b is defined by b∇−αf t 1 Γ α b−1 ∑ s t ( s − ρ t α−1f s 1 Γ α b−1 ∑ s t σ s − t α−1f s , t ∈b−1 N. 2.5 We want to...

We obtain a representation for the norm of the composition operator Cφ on the Hardy space H 2 whenever φ is a linear-fractional mapping of the form φ(z) = b/(cz + d). The representation shows that, for such mappings φ, the norm of Cφ always exceeds the essential norm of Cφ. Moreover, it shows that a formula obtained by Cowen for the norms of composition operators induced by mappings of the form...

For more details as regards semigroup theory of operators, see [1]. Fractional differential equations have been widely applied in many important areas, including thermodynamics, porous media, plasma dynamics, cosmic rays, continuum mechanics, electrodynamics, quantummechanics, biological systems and prime number theory [2, 3]. In particular, the fractional diffusion equations have been successf...

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...

The present paper deals with the wavelet transform of fractional integral operator (the RiemannLiouville operators) on Boehmian spaces. By virtue of the existing relation between the wavelet transform and the Fourier transform, we obtained integrable Boehmians defined on the Boehmian space for the wavelet transform of fractional integrals.

This paper considers various aspects of the initial value problem for fractional order differential equations. The main contribution of this paper is to use the solutions to known spatially distributed systems to demonstrate that fractional differintegral operators require an initial condition term that is time-varying due to past distributed storage of information.