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This paper aims to demonstrate that, beyond the small world of Riemann–Liouville and Caputo derivatives, there is a vast and rich world with many derivatives suitable for specific problems and various theoretical frameworks to develop, corresponding to different paths taken. The notions of time and scale sequences are introduced, and general associated basic derivatives, namely, right/stretching and left/shrinking, are defined. A general framework for fractional derivative definitions is reviewed and applied to obtain both known and new fractional-order derivatives. Several fractional derivatives are considered, mainly Liouville, Hadamard, Euler, bilinear, tempered, q-derivative, and Hahn. [ABSTRACT FROM AUTHOR]

In this research article, we study the coupled system of nonlinear Langevin fractional q-difference equations associated with two different fractional orders in Banach Space. The existence, uniqueness, and stability in the sense of Ulam are established for the proposed system. Our approach is based on the technique of measure of noncompactness combined with Mönch fixed point theorem, the implementation Banach contraction principle fixed point theorem, and the employment of Urs’s stability approach. Two examples illustrating the effectiveness of the theoretical results are presented. [ABSTRACT FROM AUTHOR]

The intent of this work is to create and study certain fundamental characteristics of new generalizations of the M-function using q-calculus. We establish its characteristics, like the convergence condition, recurrence relation, integral representations, q-derivative formulae, q-Laplace and q-Sumudu transformations, and image formulas for the Caputo fractional q-derivative and the Hilfer fractional q-derivative. Furthermore, we analyze some specific situations to demonstrate of our key findings.

Abstract In this work, a proposed system of fractional boundary value problems is investigated concerning its unbounded solutions’ existence for a class of nonlinear fractional q-difference equations in the context of the Riemann–Liouville fractional q-derivative on an infinite interval. The system’s solution is formulated with the help of Green’s function. A compactness criterion is established in a special space. All the obtained results of uniqueness and existence are investigated with the help of fixed-point theorems. Some essential examples are illustrated to support our main outcomes.

Abstract In this work, we consider a generalized quantum fractional Sturm–Liouville–Langevin difference problem with terminal boundary conditions. The relevant results rely on Mönch’s fixed point theorem along with a theoretical method by terms of Kuratowski measure of noncompactness (MNC) and the Banach contraction principle (BCP). Furthermore, two dynamical notions of Ulam–Hyers (UH) and generalized Ulam–Hyers (GUH) stability are addressed for solutions of the supposed Sturm–Liouville–Langevin quantum boundary value problem (q-FBVP). Two examples are presented to show the validity and also the effectiveness of theoretical results. In the last part of the paper, we conclude our exposition with some final remarks and observations.

In this work, a proposed system of fractional boundary value problems is investigated concerning its unbounded solutions' existence for a class of nonlinear fractional q-difference equations in the context of the Riemann–Liouville fractional q-derivative on an infinite interval. The system's solution is formulated with the help of Green's function. A compactness criterion is established in a special space. All the obtained results of uniqueness and existence are investigated with the help of fixed-point theorems. Some essential examples are illustrated to support our main outcomes. [ABSTRACT FROM AUTHOR]

In this research collection, we estimate the existence of the unique solution for the boundary value problem of nonlinear fractional q -difference equation having the given form c D q ζ v (t) − h (t , v (t)) = 0 , 0 ≤ t ≤ 1 , α 1 v (0) + β 1 D q v (0) = v (η 1) , α 2 v (1) − β 2 D q v (1) = v (η 2) , where 1 < ζ ≤ 2 , (η 1 , η 2) ∈ (0 , 1) 2 , α i , β i ∈ ℝ (i = 1 , 2) , h ∈ C ([ 0 , 1 ] × ℝ , ℝ) and c D q ζ represents the Caputo-type nonclassical q -derivative of order ζ. We use well-known principal of Banach contraction, and Leray–Schauder nonlinear alternative to vindicate the unique solution existence of the given problem. Regarding the applications, some examples are solved to justify our outcomes. [ABSTRACT FROM AUTHOR]

In this work, we consider a generalized quantum fractional Sturm–Liouville–Langevin difference problem with terminal boundary conditions. The relevant results rely on Mönch's fixed point theorem along with a theoretical method by terms of Kuratowski measure of noncompactness (MNC) and the Banach contraction principle (BCP). Furthermore, two dynamical notions of Ulam–Hyers (UH) and generalized Ulam–Hyers (GUH) stability are addressed for solutions of the supposed Sturm–Liouville–Langevin quantum boundary value problem (q-FBVP). Two examples are presented to show the validity and also the effectiveness of theoretical results. In the last part of the paper, we conclude our exposition with some final remarks and observations. [ABSTRACT FROM AUTHOR]

The intent of this work is to create and study certain fundamental characteristics of new generalizations of the M-function using q -calculus. We establish its characteristics, like the convergence condition, recurrence relation, integral representations, q -derivative formulae, q -Laplace and q -Sumudu transformations, and image formulas for the Caputo fractional q -derivative and the Hilfer fractional q -derivative. Furthermore, we analyze some specific situations to demonstrate of our key findings. [ABSTRACT FROM AUTHOR]

A fractional q-derivative operator is introduced and some of its properties have been proved. Next, a fractional q-differential equation of Gauss type is introduced and solved by means of a power series method. Finally, q-extensions of some classical q-orthogonal polynomials are introduced and some of the main properties of the newly defined functions are given. [ABSTRACT FROM AUTHOR]

Motivated by the success of the applications of the Mittag-Leffler function in the problems of physics, biology, engineering and applied sciences, we propose here a q-extension of certain generalizations of Mittag-Leffler function. With the aid of q-Riemann-Liouville fractional integral operator, q-Kober fractional integral operator and fractional q-differential operator of arbitrary order, we study certain properties including the q-integro-differential equations of this proposed function. [ABSTRACT FROM AUTHOR]

In this paper, starting from the historical developments of fractional calculus, certain results regarding fractional calculus have been discussed. These results have been further used to establish transformation formulae for ordinary hypergeometric series as well as for q-hypergeometric series. [ABSTRACT FROM AUTHOR]

In this paper, locally asymptotic stability of q-fractional order nonlinear dynamical systems is introduced and studied. The sufficient conditions for local stability of such dynamical systems are obtained. Also, useful definitions of fractional order q-integrals and q-derivatives are recalled. Finally, a q-fractional order nonlinear dynamical model is considered [ABSTRACT FROM AUTHOR]

This paper is devoted to fractional q-derivative of special functions. To begin with the theorem on term by term q-fractional differentiation has been derived. The result is an extension of an earlier result due to Yadav and Purohit [8] As a special case, of fractional q-differentiation of Generalized Miller-Ross function has been obtained. [ABSTRACT FROM AUTHOR]