q-Fractional Calculus and Equations [Elektronisk resurs] / by Mahmoud H. Annaby, Zeinab S. Mansour.

• 1 Preliminaries -- 2 q-Difference Equations -- 3 q-Sturm Liouville Problems -- 4 Riemann–Liouville q-Fractional Calculi -- 5 Other q-Fractional Calculi -- 6 Fractional q-Leibniz Rule and Applications -- 7 q-Mittag–Leffler Functions -- 8 Fractional q-Difference Equations -- 9 Applications of q-Integral Transforms.

• <p>This nine-chapter monograph introduces a rigorous investigation of <i>q-</i>difference operators in standard and fractional settings. It starts with elementary calculus of <i>q-</i>differences and integration of Jackson’s type before turning to <i>q-</i>difference equations. The existence and uniqueness theorems are derived using successive approximations, leading to systems of equations with retarded arguments. Regular  <i>q-</i>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 <i>q</i>-calculi. Hence fractional <i>q-</i>calculi of the types Riemann–Liouville; Grünwald–Letnikov;  Caputo;  Erdélyi–Kober and Weyl are defined analytically. Fractional <i>q-</i>Leibniz rules with applications  in <i>q-</i>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 <i>q-</i>fractional difference equations; families of <i>q-</i>Mittag-Leffler functions are defined and their properties are investigated, especially the <i>q-</i>Mellin–Barnes integral  and Hankel contour integral representation of  the <i>q-</i>Mittag-Leffler functions under consideration,  the distribution, asymptotic and reality of their zeros, establishing <i>q-</i>counterparts of Wiman’s results. Fractional <i>q-</i>difference equations are studied; existence and uniqueness theorems are given and classes of Cauchy-type problems are completely solved in terms of families of <i>q-</i>Mittag-Leffler functions. Among many <i>q-</i>analogs of classical results and concepts, <i>q-</i>Laplace, <i>q-</i>Mellin and <i>q</i>²<i>-</i>Fourier transforms are studied and their applications are investigated.</p> 

Ämnesord

Mathematics.  (LCSH)

Global analysis (Mathematics).  (LCSH)

Functional equations.  (LCSH)

Functions of complex variables.  (LCSH)

Integral equations.  (LCSH)

Integral Transforms.  (LCSH)

Mathematical physics.  (LCSH)

Mathematics. 

Analysis. 

Difference and Functional Equations. 

Functions of a Complex Variable. 

Integral Transforms, Operational Calculus. 

Integral Equations. 

Mathematical Methods in Physics. 

Klassifikation

QA299.6-433 (LCC)

MAT034000 (ämneskategori)

515 (DDC)

Td (kssb/8 (machine generated))