q-Fractional calculus and equations / / Mahmoud H. Annaby, Zeinab S. Mansour

• Autore: Annaby Mahmoud H
• Pubblicazione: Berlin ; ; Heidelberg, : Springer, c2012
• Edizione: 1st ed. 2012.
• Descrizione fisica: 1 online resource (xix, 318 pages) : illustrations
• Disciplina: 515.83
• Soggetto topico: Fractional calculus; Difference equations
• Altri autori: MansourZeinab S
• Nota di bibliografia: Includes bibliographical references (p. 303-314) and indexes.
• Nota di contenuto: 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.
• Sommario/riassunto: 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.
• Titolo autorizzato: Q-fractional calculus and equations
• ISBN: 3-642-30898-8
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910484622003321
• Lo trovi qui: Univ. Federico II
• Serie: Lecture notes in mathematics (Springer-Verlag) ; ; 2056.