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Book synopsis

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.

Book details

Book category Books in English Mathematics & science Physics Mathematical physics

• Full title: q-Fractional Calculus and Equations
• Author: Mahmoud H. Annaby, Zeinab S. Mansour
• Language: English
• Binding: Paperback
• Number of pages: 332
• EAN: 9783642308970
• ISBN: 364230897X
• ID: 01661381
• Publisher: Springer, Berlin
• Weight: 522 g
• Dimensions: 235 × 155 × 18 mm
• Date of publishing: 11. May 2012

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