Authors:
First detailed rigorous study of qcalculi
First detailed rigorous study of qdifference equations
First detailed rigorous study of qfractional calculi and equations
Proofs of many classical unproved results are given
Illustrative examples and figures helps readers to digest the new approaches
Includes supplementary material: sn.pub/extras
Includes supplementary material: sn.pub/extras
Part of the book series: Lecture Notes in Mathematics (LNM, volume 2056)
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Table of contents (9 chapters)

Front Matter

Back Matter
About this book
This ninechapter monograph introduces a rigorous investigation of qdifference operators in standard and fractional settings. It starts with elementary calculus of qdifferences and integration of Jackson’s type before turning to qdifference equations. The existence and uniqueness theorems are derived using successive approximations, leading to systems of equations with retarded arguments. Regular qSturm–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 qcalculi. Hence fractional qcalculi of the types Riemann–Liouville; Grünwald–Letnikov; Caputo; Erdélyi–Kober and Weyl are defined analytically. Fractional qLeibniz rules with applications in qseries are also obtained with rigorous proofs of the formal results of AlSalamVerma, which remained unproved for decades. In working towards the investigation of qfractional difference equations; families of qMittagLeffler functions are defined and their properties are investigated, especially the qMellin–Barnes integral and Hankel contour integral representation of the qMittagLeffler functions under consideration, the distribution, asymptotic and reality of their zeros, establishing qcounterparts of Wiman’s results. Fractional qdifference equations are studied; existence and uniqueness theorems are given and classes of Cauchytype problems are completely solved in terms of families of qMittagLeffler functions. Among many qanalogs of classical results and concepts, qLaplace, qMellin and q^{2}Fourier transforms are studied and their applications are investigated.
Keywords
 33D15, 26A33, 30C15, 39A13, 39A70
 Basic Hypergeometric functions
 One variable calculus
 Zeros of analytics functions
 q$difference equations
Reviews
From the reviews:
“This monograph briefly introduces qcalculus … . 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 qfractional calculus as well as to people currently doing research in qfractional calculus.” (P. W. Eloe, Mathematical Reviews, April, 2013)Authors and Affiliations

Faculty of Science, Department of Mathematics, Cairo University, Giza, Egypt
Mahmoud H. Annaby

Faculty of Science, Department of Mathematics, King Saud University, Riyadh, Saudi Arabia
Zeinab S. Mansour
Bibliographic Information
Book Title: qFractional Calculus and Equations
Authors: Mahmoud H. Annaby, Zeinab S. Mansour
Series Title: Lecture Notes in Mathematics
DOI: https://doi.org/10.1007/9783642308987
Publisher: Springer Berlin, Heidelberg
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: SpringerVerlag Berlin Heidelberg 2012
Softcover ISBN: 9783642308970Published: 26 August 2012
eBook ISBN: 9783642308987Published: 27 August 2012
Series ISSN: 00758434
Series EISSN: 16179692
Edition Number: 1
Number of Pages: XIX, 318
Number of Illustrations: 6 b/w illustrations
Topics: Analysis, Difference and Functional Equations, Functions of a Complex Variable, Integral Transforms and Operational Calculus, Integral Equations, Mathematical Methods in Physics