q-Fractional Calculus and Equations

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-SturmLiouville 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 RiemannLiouville; GrünwaldLetnikov; Caputo; ErdélyiKober 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-MellinBarnes 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 q 2 -Fourier transforms are studied and their applications are investigated.

First detailed rigorous study of q-calculi First detailed rigorous study of q-difference equations First detailed rigorous study of q-fractional calculi and equations Proofs of many classical unproved results are given Illustrative examples and figures helps readers to digest the new approaches Includes supplementary material: sn.pub/extras Includes supplementary material: sn.pub/extras

Inhalt
1 Preliminaries.- 2 q-Difference Equations.- 3 q-Sturm Liouville Problems.- 4 RiemannLiouville q-Fractional Calculi.- 5 Other q-Fractional Calculi.- 6 Fractional q-Leibniz Rule and Applications.- 7 q-MittagLeffler Functions.- 8 Fractional q-Difference Equations.- 9 Applications of q-Integral Transforms.