03819nam 2200553Ia 450 991048462200332120200520144314.03-642-30898-810.1007/978-3-642-30898-7(CKB)3400000000085876(SSID)ssj0000746101(PQKBManifestationID)11430138(PQKBTitleCode)TC0000746101(PQKBWorkID)10863032(PQKB)10159859(DE-He213)978-3-642-30898-7(MiAaPQ)EBC3070565(PPN)165117087(EXLCZ)99340000000008587620120730d2012 uy 0engurnn#008mamaatxtrdacontentcrdamediacrrdacarrierq-Fractional calculus and equations /Mahmoud H. Annaby, Zeinab S. Mansour1st ed. 2012.Berlin ;Heidelberg Springerc20121 online resource (xix, 318 pages) illustrationsLecture notes in mathematics,0075-8434 ;20563-642-30897-X Includes bibliographical references (p. 303-314) and indexes.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.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.Lecture notes in mathematics (Springer-Verlag) ;2056.Fractional calculusDifference equationsFractional calculus.Difference equations.515.83Annaby Mahmoud H477683Mansour Zeinab S518042MiAaPQMiAaPQMiAaPQBOOK9910484622003321Q-fractional calculus and equations848040UNINA