Lecture Notes in Mathematics

q-Fractional Calculus and Equations

Authors: Annaby, Mahmoud H., Mansour, Zeinab S.

  • First detailed rigorous study of q-calculi
  • First detailed rigorous study of q-difference equations
  • First detailed rigorous study of q-fractional calculi and equations
  • Proofs of many classical unproved results are given
  • Illustrative examples and figures helps readers to digest the new approaches
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About this book

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.

Reviews

From the reviews:

“This monograph briefly introduces q-calculus … . The book is carefully and well written. Each chapter is introduced by an informative abstract. The bibliography is extensive and useful, and useful tables of formulas appear in appendices. This monograph is of interest to people who want to learn to do research in q-fractional calculus as well as to people currently doing research in q-fractional calculus.” (P. W. Eloe, Mathematical Reviews, April, 2013)

Table of contents (9 chapters)

  • Preliminaries

    Annaby, Mahmoud H. (et al.)

    Pages 1-39

  • q-Difference Equations

    Annaby, Mahmoud H. (et al.)

    Pages 41-71

  • q-Sturm–Liouville Problems

    Annaby, Mahmoud H. (et al.)

    Pages 73-105

  • Riemann–Liouville q-Fractional Calculi

    Annaby, Mahmoud H. (et al.)

    Pages 107-146

  • Other q-Fractional Calculi

    Annaby, Mahmoud H. (et al.)

    Pages 147-173

Buy this book

eBook $69.99
price for USA (gross)
  • ISBN 978-3-642-30898-7
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $89.95
price for USA
  • ISBN 978-3-642-30897-0
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
q-Fractional Calculus and Equations
Authors
Series Title
Lecture Notes in Mathematics
Series Volume
2056
Copyright
2012
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Springer-Verlag Berlin Heidelberg
eBook ISBN
978-3-642-30898-7
DOI
10.1007/978-3-642-30898-7
Softcover ISBN
978-3-642-30897-0
Series ISSN
0075-8434
Edition Number
1
Number of Pages
XIX, 318
Number of Illustrations and Tables
6 b/w illustrations
Topics