13 research outputs found
Forced oscillation of second order nonlinear dynamic equations on time scales
By means of the Kartsatos technique and generalized Riccati transformation techniques, we establish some new oscillation criteria for a second order nonlinear dynamic equations with forced term on time scales in terms of the coefficients
Discrete analogues of some classical special functions
Analogues of special functions on time scales are studied with special focus on the time scale = hℤ. Functions investigated in particular include complex monomials, new trigonometric functions, Gaussian bell, Hermite and Laguerre polynomials, Bessel functions, and hypergeometric series --Abstract, page iii
The Jacobi operator on and its various -functions
We offer a detailed treatment of spectral and Weyl-Titchmarsh-Kodaira theory
for all self-adjoint Jacobi operator realizations of the differential
expression \begin{align*} \tau_{\alpha,\beta} = - (1-x)^{-\alpha}
(1+x)^{-\beta}(d/dx) \big((1-x)^{\alpha+1}(1+x)^{\beta+1}\big) (d/dx),& \\
\alpha, \beta \in \mathbb{R}, \; x \in (-1,1),& \end{align*} in
, . In addition to discussing the separated boundary conditions that
lead to Jacobi orthogonal polynomials as eigenfunctions in detail, we
exhaustively treat the case of coupled boundary conditions and illustrate the
latter with the help of the general -periodic and Krein--von Neumann
extensions. In particular, we treat all underlying Weyl-Titchmarsh-Kodaira and
Green's function induced -functions and revisit their Nevanlinna-Herglotz
property. We also consider connections to other differential operators
associated with orthogonal polynomials such as Laguerre, Gegenbauer, and
Chebyshev.Comment: 59 pages. arXiv admin note: text overlap with arXiv:2102.00685,
arXiv:2110.15913, arXiv:1910.1311
Differential/Difference Equations
The study of oscillatory phenomena is an important part of the theory of differential equations. Oscillations naturally occur in virtually every area of applied science including, e.g., mechanics, electrical, radio engineering, and vibrotechnics. This Special Issue includes 19 high-quality papers with original research results in theoretical research, and recent progress in the study of applied problems in science and technology. This Special Issue brought together mathematicians with physicists, engineers, as well as other scientists. Topics covered in this issue: Oscillation theory; Differential/difference equations; Partial differential equations; Dynamical systems; Fractional calculus; Delays; Mathematical modeling and oscillations
Maximal Hardy Fields
We show that all maximal Hardy fields are elementarily equivalent as
differential fields, and give various applications of this result and its
proof. We also answer some questions on Hardy fields posed by Boshernitzan.Comment: 470 pp. This document is not intended for publication in its current
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