Couverture : q-Fractional Calculus and Equations

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q-Fractional Calculus and Equations

Mahmoud H. Annaby, Zeinab S. Mansour

Résumé

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.

Détails de l'édition

ISBN-13
9783642308970
ISBN-10
364230897X
EAN
9783642308970
Éditeur
Springer (Heidelberg)
Date de publication
26 août 2012
Langue
en
Pages
318
Dimensions
24 cm
Provenance des données
  • dimensions BnF Catalogue général · 23 juillet 2026
  • language Google Books · 4 août 2026
  • page_count Google Books · 4 août 2026
  • publication_date Google Books · 4 août 2026
  • summary Google Books · 4 août 2026
  • title Google Books · 4 août 2026