Q-fractional calculus and equations /
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| Author / Creator: | Annaby, Mahmoud H. |
|---|---|
| Imprint: | Berlin ; New York : Springer, ©2012. |
| Description: | 1 online resource. |
| Language: | English |
| Series: | Lecture notes in mathematics, 0075-8434 ; 2056 Lecture notes in mathematics (Springer-Verlag) ; 2056. |
| Subject: | |
| Format: | E-Resource Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/11077262 |
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| 050 | 4 | |a QA314 |b .A56 2012 | |
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| 100 | 1 | |a Annaby, Mahmoud H. |0 http://id.loc.gov/authorities/names/nb2012027985 |1 http://viaf.org/viaf/264642733 | |
| 245 | 1 | 0 | |a Q-fractional calculus and equations / |c Mahmoud H. Annaby, Zeinab S. Mansour. |
| 260 | |a Berlin ; |a New York : |b Springer, |c ©2012. | ||
| 300 | |a 1 online resource. | ||
| 336 | |a text |b txt |2 rdacontent |0 http://id.loc.gov/vocabulary/contentTypes/txt | ||
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| 490 | 1 | |a Lecture notes in mathematics, |x 0075-8434 ; |v 2056 | |
| 505 | 0 | 0 | |t Preliminaries -- |t q-Difference Equations -- |t q-Sturm-Liouville Problems -- |t Riemann-Liouville q-Fractional Calculi -- |t Other q-Fractional Calculi -- |t Fractional q-Leibniz Rule and Applications -- |t q-Mittag-Leffler Functions -- |t Fractional q-Difference Equations -- |t q-Integral Transforms for Solving Fractional q-Difference Equations. |
| 504 | |a Includes bibliographical references and index. | ||
| 520 | |a 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. | ||
| 650 | 0 | |a Fractional calculus. |0 http://id.loc.gov/authorities/subjects/sh93004015 | |
| 650 | 7 | |a Fractional calculus. |2 fast |0 (OCoLC)fst00933515 | |
| 653 | 4 | |a Mathematics. | |
| 653 | 4 | |a Global analysis (Mathematics) | |
| 653 | 4 | |a Functional equations. | |
| 653 | 4 | |a Functions of complex variables. | |
| 653 | 4 | |a Integral equations. | |
| 653 | 4 | |a Integral Transforms. | |
| 653 | 4 | |a Analysis. | |
| 653 | 4 | |a Difference and Functional Equations. | |
| 655 | 4 | |a Electronic books. | |
| 700 | 1 | |a Mansour, Zeinab S. |0 http://id.loc.gov/authorities/names/nb2012027987 |1 http://viaf.org/viaf/262532111 | |
| 830 | 0 | |a Lecture notes in mathematics (Springer-Verlag) ; |v 2056. | |
| 856 | 4 | 0 | |u http://link.springer.com/10.1007/978-3-642-30898-7 |y SpringerLink |
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