qFractional calculus and equations
 Responsibility
 Mahmoud H. Annaby, Zeinab S. Mansour.
 Language
 English.
 Imprint
 Heidelberg ; New York : Springer Verlag, c2012.
 Physical description
 xix, 318 p. : ill. ; 23 cm.
 Series
 Lecture notes in mathematics (SpringerVerlag) 2056.
Access
Creators/Contributors
 Author/Creator
 Annaby, Mahmoud H.
 Contributor
 Mansour, Zeinab S.
Contents/Summary
 Bibliography
 Includes bibliographical references (p. 303314) and indexes.
 Contents

 1 Preliminaries. 2 qDifference Equations. 3 qSturm Liouville Problems. 4 RiemannLiouville qFractional Calculi. 5 Other qFractional Calculi. 6 Fractional qLeibniz Rule and Applications. 7 qMittagLeffler Functions. 8 Fractional qDifference Equations. 9 Applications of qIntegral Transforms.
 (source: Nielsen Book Data)
 Publisher's Summary
 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 qSturmLiouville 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 RiemannLiouville; GrunwaldLetnikov; Caputo; ErdelyiKober 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 qMellinBarnes 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 q2Fourier transforms are studied and their applications are investigated.
(source: Nielsen Book Data)
Subjects
 Subject
 Fractional calculus.
Bibliographic information
 Publication date
 2012
 Series
 Lecture notes in mathematics ; 2056
 ISBN
 9783642308970
 364230897X