Annaby / Mansour q-Fractional Calculus and Equations

Erscheinungsjahr 2012
ISBN: 978-3-642-30898-7
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 318 Seiten

Reihe: Mathematics and Statistics (R0)

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 q- Fourier transforms are studied and their applications are investigated.

Zielgruppe

Research

Autoren/Hrsg.

Annaby, Mahmoud H.

Mansour, Zeinab S.

Inhaltsverzeichnis

1 Preliminaries.- 2 q-Difference Equations.- 3 q-Sturm Liouville Problems.- 4 Riemann–Liouville q-Fractional Calculi.- 5 Other q-Fractional Calculi.- 6 Fractional q-Leibniz Rule and Applications.- 7 q-Mittag–Leffler Functions.- 8 Fractional q-Difference Equations.- 9 Applications of q-Integral Transforms.