q -Fractional Calculus and Equations

  • Mahmoud H. Annaby
  • Zeinab S. Mansour

Part of the Lecture Notes in Mathematics book series (LNM, volume 2056)

Table of contents

  1. Front Matter
    Pages i-xix
  2. Mahmoud H. Annaby, Zeinab S. Mansour
    Pages 1-39
  3. Mahmoud H. Annaby, Zeinab S. Mansour
    Pages 41-71
  4. Mahmoud H. Annaby, Zeinab S. Mansour
    Pages 73-105
  5. Mahmoud H. Annaby, Zeinab S. Mansour
    Pages 107-146
  6. Mahmoud H. Annaby, Zeinab S. Mansour
    Pages 147-173
  7. Mahmoud H. Annaby, Zeinab S. Mansour
    Pages 175-199
  8. Mahmoud H. Annaby, Zeinab S. Mansour
    Pages 201-222
  9. Mahmoud H. Annaby, Zeinab S. Mansour
    Pages 223-270
  10. Mahmoud H. Annaby, Zeinab S. Mansour
    Pages 271-293
  11. Back Matter
    Pages 295-318

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.


33D15, 26A33, 30C15, 39A13, 39A70 Basic Hypergeometric functions One variable calculus Zeros of analytics functions q$-difference equations

Authors and affiliations

  • Mahmoud H. Annaby
    • 1
  • Zeinab S. Mansour
    • 2
  1. 1.Faculty of Science, Department of MathematicsCairo UniversityGizaEgypt
  2. 2.Faculty of Science, Department of MathematicsKing Saud UniversityRiyadhSaudi Arabia

Bibliographic information

  • DOI
  • Copyright Information Springer-Verlag Berlin Heidelberg 2012
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-642-30897-0
  • Online ISBN 978-3-642-30898-7
  • Series Print ISSN 0075-8434
  • Series Online ISSN 1617-9692
  • Buy this book on publisher's site
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