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Mathematics - Analysis | q-Fractional Calculus and Equations

q-Fractional Calculus and Equations

Series: Lecture Notes in Mathematics, Vol. 2056

Annaby, Mahmoud H., Mansour, Zeinab S.

2012, XIX, 318 p. 6 illus.

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  • First detailed rigorous study of q-calculi
  • First detailed rigorous study of q-difference equations
  • First detailed rigorous study of q-fractional calculi and equations
  • Proofs of many classical unproved results are given
  • Illustrative examples and figures helps readers to digest the new approaches

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.

Content Level » Research

Keywords » 33D15, 26A33, 30C15, 39A13, 39A70 - Basic Hypergeometric functions - One variable calculus - Zeros of analytics functions - q$-difference equations

Related subjects » Analysis - Dynamical Systems & Differential Equations - Theoretical, Mathematical & Computational Physics

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