q-Fractional Calculus and Equations (Lecture Notes in Mathematics) 〈Vol. 2056〉

Annaby, M. H./ Mansour, Z. S.

Springer（2012/09発売）

• 製本 Paperback:紙装版/ペーパーバック版／ページ数 322 p.／サイズ 6 illus.
• 商品コード 9783642308970

Full Description

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.

Contents

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.