q-Fractional calculus and equations / [Book] Mahmoud H. Annaby, Zeinab S. Mansour, editors.
Material type:
- 9783642308970 (paperback)
- 364230897X
- 515.83
- 515.83
Item type | Current library | Call number | Status | Date due | Barcode | Item holds |
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Junaid Zaidi Library, COMSATS University Islamabad Ground Floor | 515.83 ANN-F (Browse shelf(Opens below)) | Available | 56419 |
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515.8 VAT-I Introduction of real analysis | 515.8 VAT-I Introduction of real analysis | 515.8 VAT-I Introduction of real analysis | 515.83 ANN-F q-Fractional calculus and equations / | 515.83 FLA-E Elementary functions and analytic geometry | 515.83 HIR-C Calculus of one variable | 515.83 MEE-S Stochastic models for fractional calculus |
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; Erdľyi–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.
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