Integrable systems on regular time scales

Ankara : The Department of Mathematics and the Institute of Engineering and Science of Bilkent University, 2009.Thesis (Ph.D.) -- Bilkent University, 2009.Includes bibliographical references leaves 98-102.We present two approaches to unify the integrable systems. Both approaches are based on the classical R-matrix formalism. The first approach proceeds from the construction of (1 + 1)-dimensional integrable ∆-differential systems on regular time scales together with bi-Hamiltonian structures and conserved quantities. The second approach is established upon the general framework of integrable discrete systems on R and integrable dispersionless systems. We discuss the deformation quantization scheme for the dispersionless systems. We also apply the theories presented in this dissertation, to several well-known examples.Yantır, Burcu SilindirPh.D

Generalized quantum exponential function and its applications

This article aims to present (q,h) analogue of exponential function which unifies, extends h-and q-exponential functions in a convenient and efficient form. For this purpose, we introduce generalized quantum binomial which serves as an analogue of an ordinary polynomial. We state (q, h) analogue of Taylor series and introduce generalized quantum exponential function which is determined by Taylor series in generalized quantum binomial. Furthermore, we prove existence and uniqueness theorem for a first order, linear, homogeneous IVP whose solution produces an infinite product form for generalized quantum exponential function. We conclude that both representations of generalized quantum exponential function are equivalent. We illustrate our results by ordinary and partial difference equations. Finally, we present a generic dynamic wave equation which admits generalized trigonometric, hyperbolic type of solutions and produces various kinds of partial differential/difference equations

Alpha extension of (q,h)-time scales

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