{"product_id":"9783642308970","title":"Lecture Notes in Mathematics","description":"\u003ch1\u003eLecture Notes in Mathematics\u003c\/h1\u003e \u003ch2\u003eAnnaby, Mahmoud H.; Mansour, Zeinab S.\u003c\/h2\u003e \u003cp\u003e\u003c\/p\u003e\u003cp\u003eThis nine-chapter monograph introduces a rigorous investigation of \u003ci\u003eq-\u003c\/i\u003edifference operators in standard and fractional settings. It starts with elementary calculus of \u003ci\u003eq-\u003c\/i\u003edifferences and integration of Jackson’s type before turning to \u003ci\u003eq-\u003c\/i\u003edifference equations. The existence and uniqueness theorems are derived using successive approximations, leading to systems of equations with retarded arguments. Regular  \u003ci\u003eq-\u003c\/i\u003eSturm–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 \u003ci\u003eq\u003c\/i\u003e-calculi. Hence fractional \u003ci\u003eq-\u003c\/i\u003ecalculi of the types Riemann–Liouville; Grünwald–Letnikov;  Caputo;  Erdélyi–Kober and Weyl are defined analytically. Fractional \u003ci\u003eq-\u003c\/i\u003eLeibniz rules with applications  in \u003ci\u003eq-\u003c\/i\u003eseries 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 \u003ci\u003eq-\u003c\/i\u003efractional difference equations; families of \u003ci\u003eq-\u003c\/i\u003eMittag-Leffler functions are defined and their properties are investigated, especially the \u003ci\u003eq-\u003c\/i\u003eMellin–Barnes integral  and Hankel contour integral representation of  the \u003ci\u003eq-\u003c\/i\u003eMittag-Leffler functions under consideration,  the distribution, asymptotic and reality of their zeros, establishing \u003ci\u003eq-\u003c\/i\u003ecounterparts of Wiman’s results. Fractional \u003ci\u003eq-\u003c\/i\u003edifference equations are studied; existence and uniqueness theorems are given and classes of Cauchy-type problems are completely solved in terms of families of \u003ci\u003eq-\u003c\/i\u003eMittag-Leffler functions. Among many \u003ci\u003eq-\u003c\/i\u003eanalogs of classical results and concepts, \u003ci\u003eq-\u003c\/i\u003eLaplace, \u003ci\u003eq-\u003c\/i\u003eMellin and \u003ci\u003eq\u003c\/i\u003e\u003csup\u003e2\u003c\/sup\u003e\u003ci\u003e-\u003c\/i\u003eFourier transforms are studied and their applications are investigated.\u003c\/p\u003e \u003ch3\u003eDetails\u003c\/h3\u003e \u003cp\u003ePublished by: Springer\u003c\/p\u003e \u003cp\u003ePublication Date: 2012-08-26\u003c\/p\u003e \u003cp\u003eFormat: Paperback\u003c\/p\u003e \u003cp\u003eISBN-13: 9783642308970\u003c\/p\u003e \u003cp\u003eDOI: 10.1007\/978-3-642-30898-7\u003c\/p\u003e \u003cp\u003eDimensions: 235cm x155cm\u003c\/p\u003e \u003cp\u003ePages: 318\u003c\/p\u003e ","brand":"Springer Berlin Heidelberg","offers":[{"title":"Default Title","offer_id":45369821364364,"sku":"9783642308970","price":49.49,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0710\/9545\/1788\/files\/9783642308970.jpg?v=1775675403","url":"https:\/\/lateknightbooks.com\/products\/9783642308970","provider":"Late Knight Books and Services, LLC","version":"1.0","type":"link"}