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-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 q 2 -Fourier transforms are studied and their applications are investigated.
ISBN/EAN | 9783642308970 |
Auteur | Mansour, Zeinab S. |
Uitgever | Van Ditmar Boekenimport B.V. |
Taal | Engels |
Uitvoering | Paperback / gebrocheerd |
Pagina's | 340 |
Lengte | 250.0 mm |
Breedte | 159.0 mm |