000 04620nam a22005775i 4500
001 978-3-642-30898-7
003 DE-He213
005 20190213151307.0
007 cr nn 008mamaa
008 120825s2012 gw | s |||| 0|eng d
020 _a9783642308987
_9978-3-642-30898-7
024 7 _a10.1007/978-3-642-30898-7
_2doi
050 4 _aQA299.6-433
072 7 _aPBK
_2bicssc
072 7 _aMAT034000
_2bisacsh
072 7 _aPBK
_2thema
082 0 4 _a515
_223
100 1 _aAnnaby, Mahmoud H.
_eauthor.
_4aut
_4http://id.loc.gov/vocabulary/relators/aut
245 1 0 _aq -Fractional Calculus and Equations
_h[electronic resource] /
_cby Mahmoud H. Annaby, Zeinab S. Mansour.
264 1 _aBerlin, Heidelberg :
_bSpringer Berlin Heidelberg :
_bImprint: Springer,
_c2012.
300 _aXIX, 318 p. 6 illus.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
490 1 _aLecture Notes in Mathematics,
_x0075-8434 ;
_v2056
505 0 _a1 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.
520 _aThis 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.
650 0 _aGlobal analysis (Mathematics).
650 0 _aFunctional equations.
650 0 _aFunctions of complex variables.
650 0 _aIntegral Transforms.
650 0 _aIntegral equations.
650 0 _aMathematical physics.
650 1 4 _aAnalysis.
_0http://scigraph.springernature.com/things/product-market-codes/M12007
650 2 4 _aDifference and Functional Equations.
_0http://scigraph.springernature.com/things/product-market-codes/M12031
650 2 4 _aFunctions of a Complex Variable.
_0http://scigraph.springernature.com/things/product-market-codes/M12074
650 2 4 _aIntegral Transforms, Operational Calculus.
_0http://scigraph.springernature.com/things/product-market-codes/M12112
650 2 4 _aIntegral Equations.
_0http://scigraph.springernature.com/things/product-market-codes/M12090
650 2 4 _aMathematical Methods in Physics.
_0http://scigraph.springernature.com/things/product-market-codes/P19013
700 1 _aMansour, Zeinab S.
_eauthor.
_4aut
_4http://id.loc.gov/vocabulary/relators/aut
710 2 _aSpringerLink (Online service)
773 0 _tSpringer eBooks
776 0 8 _iPrinted edition:
_z9783642308970
776 0 8 _iPrinted edition:
_z9783642308994
830 0 _aLecture Notes in Mathematics,
_x0075-8434 ;
_v2056
856 4 0 _uhttps://doi.org/10.1007/978-3-642-30898-7
912 _aZDB-2-SMA
912 _aZDB-2-LNM
999 _c10178
_d10178