q-Fractional Calculus and Equations

Nonfiction, Science & Nature, Mathematics, Mathematical Analysis, Science, Physics, Mathematical Physics
Cover of the book q-Fractional Calculus and Equations by Mahmoud H. Annaby, Zeinab S. Mansour, Springer Berlin Heidelberg
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Author: Mahmoud H. Annaby, Zeinab S. Mansour ISBN: 9783642308987
Publisher: Springer Berlin Heidelberg Publication: August 27, 2012
Imprint: Springer Language: English
Author: Mahmoud H. Annaby, Zeinab S. Mansour
ISBN: 9783642308987
Publisher: Springer Berlin Heidelberg
Publication: August 27, 2012
Imprint: Springer
Language: English

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.

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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.

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