Other q-Fractional Calculi Annaby, Mahmoud H; Mansour, Zeinab S
Q-Fractional Calculus and Equations,
2012, 20120711, Volume:
2056
Book Chapter
Peer reviewed
In this chapter we investigate q-analogues of some known fractional operators. This chapter includes as well the fractional generalization of the Askey–Wilson operator introduced in (Ismail and ...Rahman, J. Approx. Theor. 114(2), 269–307, 2002). At the end of this chapter we introduce a fractional generalization of the q-difference operator introduced in (Rubin, J. Math. Anal. Appl. 212(2), 571–582, 1997).
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12.
Riemann–Liouville q-Fractional Calculi Annaby, Mahmoud H; Mansour, Zeinab S
Q-Fractional Calculus and Equations,
2012, 20120711, Volume:
2056
Book Chapter
Peer reviewed
In this chapter we investigate q-analogues of the classical fractional calculi. We study the q-Riemann–Liouville fractional integral operator introduced by Al-Salam (Proc. Am. Math. Soc. 17, 616–621, ...1966; Proc. Edinb. Math. Soc. 2(15), 135–140, 1966/1967) and by Agarwal (Proc. Camb. Phil. Soc. 66, 365–370, 1969). We give rigorous proofs of existence of the fractional q-integral and q-derivative. Therefore we establish a q-analogue of Abel’s integral equation and its solutions.
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13.
Fractional q-Difference Equations Annaby, Mahmoud H; Mansour, Zeinab S
Q-Fractional Calculus and Equations,
2012, 20120711, Volume:
2056
Book Chapter
Peer reviewed
As in the classical theory of ordinary fractional differential equations, q-difference equations of fractional order are divided into linear, nonlinear, homogeneous, and inhomogeneous equations with ...constant and variable coefficients. This chapter is devoted to certain problems of fractional q-difference equations based on the basic Riemann–Liouville fractional derivative and the basic Caputo fractional derivative. In this chapter, we investigate questions concerning the solvability of these equations in a certain space of functions. A special class of Cauchy type q-fractional problems is also developed at the end of this chapter.
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14.
q—Type Sampling Theorems Annaby, Mahmoud H.
Resultate der Mathematik,
11/2003, Volume:
44, Issue:
3-4
Journal Article
Peer reviewed
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15.
q-Sturm–Liouville Problems Annaby, Mahmoud H; Mansour, Zeinab S
Q-Fractional Calculus and Equations,
2012, 20120711, Volume:
2056
Book Chapter
Peer reviewed
In this chapter we introduce the study held by Annaby and Mansour in (J. Phys. A Math. Gen. 38(17), 3775–3797, 2005) of a self adjoint basic Sturm–Liouville eigenvalue problem in a Hilbert space. The ...last two sections of this chapter are about the q2-Fourier transform introduced by Rubin in (J. Math. Anal. Appl. 212(2), 571–582, 1997; Proc. Am. Math. Soc. 135(3), 777–785, 2007), when q lies in a proper subset of (0, 1) and the generalization of Rubin’s q2-Fourier transform, introduced in (Mansour, Generalizations of Rubin’s q2-fourier transform and q-difference operator, submitted, 2012) for any q ∈ (0, 1).
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This chapter includes analytic investigations on q-type Leibniz rules of q-Riemann–Liouville fractional operator introduced by Al-Salam and Verma in (Pac. J. Math. 60(2), 1–9, 1975). In this chapter, ...we provide a generalization of the Riemann–Liouville fractional q-Leibniz formula introduced by Agarwal in (Ganita 27(1–2), 25–32, 1976). Purohit (Kyungpook Math. J. 50(4), 473–482, 2010) introduced a Leibniz formula for Weyl q-fractional operator only when α is an integer. In this respect, We extend Purohit’s result for any \documentclass12pt{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}
$$\alpha \in \mathbb{R}$$
\end{document}. We end the chapter with deriving some q-series and formulae by applying the fractional Leibniz formula mentioned and derived earlier in the chapter.
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17.
q-Difference Equations Annaby, Mahmoud H; Mansour, Zeinab S
Q-Fractional Calculus and Equations,
2012, 20120711, Volume:
2056
Book Chapter
Peer reviewed
This chapter includes proofs of the existence and uniqueness of the solutions of first order systems of q-difference equations in a neighborhood of a point a, \documentclass12pt{minimal}
...\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}
$$a \geq 0$$
\end{document}. Then, as applications of the main results, we study linear q-difference equations as well as the q-type Wronskian. These results are mainly based on (Mansour, q-Difference Equations, Master’s thesis, Faculty of Science, Cairo University, Giza, Egypt, 2001). This chapter also includes a section on the asymptotics of zeros of some q-functions.
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18.
q-Mittag–Leffler Functions Annaby, Mahmoud H; Mansour, Zeinab S
Q-Fractional Calculus and Equations,
2012, 20120711, Volume:
2056
Book Chapter
Peer reviewed
The classical Mittag–Leffler function plays an important role in fractional differential equations. In this chapter we mention in brief the q-analogues of the Mittag–Leffler functions defined by ...mathematicians. We pay attention to a pair of q-analogues of the Mittag–Leffler function that may be considered as a generalization of the q-exponential functions eq(z) and Eq(z). We study their main properties and give a Mellin–Barnes integral representations and Hankel contour integral representation for them. As in the classical case we prove that the q-Mittag–Leffler functions are solutions of q-type Volterra integral equations. Finally, asymptotics of zeros of one of the pair of the q-Mittag–Leffler function will be given at the end of the chapter.
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We derive estimates for the truncation, amplitude and jitter type errors associated with Hermite-type interpolations at equidistant nodes of functions in Paley-Wiener spaces. We give pointwise and ...uniform estimates. Some examples and comparisons which indicate that applying Hermite interpolations would improve the methods that use the classical sampling theorem are given.
Using the topological equivalence between the Riemann sphere
S
and the extended complex plane
C
¯
=
C
∪
{
∞
}
, where
C
is the field of complex numbers, we establish 2D-bijective representations of ...3D point clouds. Points of 3D point clouds are mapped into the Riemann sphere
S
, and a stereographic projection is implemented to map the points into the complex plane
C
. The way the 3D objects are mapped into
S
may be varied for various applications. To prove the accuracy and efficiency of the proposed 2D representation of 3D objects, we apply this correspondence to 3D point cloud encryption. We utilize chaotic permutations, chaotic circuits, and Latin cubes in addition to the stereographic projection representation to construct our scheme. The permutation steps using chaotic maps and Latin cubes are carried out on the object data points in both
S
and
C
¯
, while the chaotic circuits are applied to 2D projections of the 3D objects. To the best of our knowledge, no earlier work employed stereographic projections for 3D object encryption. Experimental simulations of this method show high encryption strength and strong confusion and diffusion properties based on quantitative and statistical measures.
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