Skip to main content

2012 | Buch

q -Fractional Calculus and Equations

verfasst von: Mahmoud H. Annaby, Zeinab S. Mansour

Verlag: Springer Berlin Heidelberg

Buchreihe : Lecture Notes in Mathematics

insite
SUCHEN

Über dieses Buch

This nine-chapter monograph introduces a rigorous investigation of q-difference operators in standard and fractional settings. It starts with elementary calculus of q-differences and integration of Jackson’s type before turning to q-difference equations. The existence and uniqueness theorems are derived using successive approximations, leading to systems of equations with retarded arguments. Regular q-Sturm–Liouville theory is also introduced; Green’s function is constructed and the eigenfunction expansion theorem is given. The monograph also discusses some integral equations of Volterra and Abel type, as introductory material for the study of fractional q-calculi. Hence fractional q-calculi of the types Riemann–Liouville; Grünwald–Letnikov; Caputo; Erdélyi–Kober and Weyl are defined analytically. Fractional q-Leibniz rules with applications in q-series are also obtained with rigorous proofs of the formal results of Al-Salam-Verma, which remained unproved for decades. In working towards the investigation of q-fractional difference equations; families of q-Mittag-Leffler functions are defined and their properties are investigated, especially the q-Mellin–Barnes integral and Hankel contour integral representation of the q-Mittag-Leffler functions under consideration, the distribution, asymptotic and reality of their zeros, establishing q-counterparts of Wiman’s results. Fractional q-difference equations are studied; existence and uniqueness theorems are given and classes of Cauchy-type problems are completely solved in terms of families of q-Mittag-Leffler functions. Among many q-analogs of classical results and concepts, q-Laplace, q-Mellin and q2-Fourier transforms are studied and their applications are investigated.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Preliminaries
Abstract
This chapter includes definitions and properties of Jackson q-difference and q-integral operators, q-gamma and q-beta functions and finally q-analogues of Laplace and Mellin integral transforms.
Mahmoud H. Annaby, Zeinab S. Mansour
Chapter 2. q-Difference Equations
Abstract
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, \(a \geq 0\). 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.
Mahmoud H. Annaby, Zeinab S. Mansour
Chapter 3. q-Sturm–Liouville Problems
Abstract
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 q 2-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 q 2-Fourier transform, introduced in (Mansour, Generalizations of Rubin’s q 2-fourier transform and q-difference operator, submitted, 2012) for any q ∈ (0, 1).
Mahmoud H. Annaby, Zeinab S. Mansour
Chapter 4. Riemann–Liouville q-Fractional Calculi
Abstract
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.
Mahmoud H. Annaby, Zeinab S. Mansour
Chapter 5. Other q-Fractional Calculi
Abstract
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).
Mahmoud H. Annaby, Zeinab S. Mansour
Chapter 6. Fractional q-Leibniz Rule and Applications
Abstract
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 \(\alpha \in \mathbb{R}\). We end the chapter with deriving some q-series and formulae by applying the fractional Leibniz formula mentioned and derived earlier in the chapter.
Mahmoud H. Annaby, Zeinab S. Mansour
Chapter 7. q-Mittag–Leffler Functions
Abstract
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 e q (z) and E q (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.
Mahmoud H. Annaby, Zeinab S. Mansour
Chapter 8. Fractional q-Difference Equations
Abstract
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.
Mahmoud H. Annaby, Zeinab S. Mansour
Chapter 9. q-Integral Transforms for Solving Fractional q-Difference Equations
Abstract
Integral transforms like Laplace, Mellin, and Fourier transforms are used in finding explicit solutions for linear differential equations, linear fractional differential equations, and diffusion equations. See for example (Kilbas et al., Theory and Applications of Fractional Differential Equations, Elsevier, London, first edition, 2006; Mainardi, Appl. Math. Lett. 9(6), 23–28, 1996; Nikolova and Boyadjiev, Fract. Calc. Appl. Anal. 13(1), 57–67, 2010; Wyss, J. Math. Phys. 27(11), 2782–2785, 1986). This chapter is devoted to the use of the q-Laplace, q-Mellin, and q 2-Fourier transforms to find explicit solutions of certain linear q-difference equations, linear fractional q-difference equations, and certain fractional q-diffusion equations.
Mahmoud H. Annaby, Zeinab S. Mansour
Backmatter
Metadaten
Titel
q -Fractional Calculus and Equations
verfasst von
Mahmoud H. Annaby
Zeinab S. Mansour
Copyright-Jahr
2012
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-30898-7
Print ISBN
978-3-642-30897-0
DOI
https://doi.org/10.1007/978-3-642-30898-7