32 Results Found

The various forms of Airy’s differential equation are discussed in this work, together with the special functions that arise in the processes of their solutions. Further properties of the arising integral functions are discussed, and their conn...

We construct the Green functions (or Feynman’s propagators) for the Schrödinger equations of the form $i{\psi }_t+{\displaystyle \frac{1}{4}}{\psi }_{xx}\pm t{x}^2\psi =0$ (for the wave function $\psi$ and its time (t) and x-space derivatives) in terms o...

This report bridges fundamental ideas from introductory calculus to advanced concepts in quantum mechanics and nonlinear dynamics. Beginning with the behavior of second derivatives in oscillatory and exponential functions, it introduces the Airy equa...

The propagation dynamics of two-dimensional (2D) ring-Airy beams is studied in the framework of the fractional Schrödinger equation, which includes saturable or cubic self-focusing or defocusing nonlinearity and Lévy index ((LI) alias for...

This paper presents a stress-function-based finite element formulation for two-dimensional elastoplastic plane-stress problems with holes. The Airy stress function is used as the primary global unknown, and the resulting non-homogeneous biharmonic eq...

A generalized model of the variable permeability in the transition layer is introduced in this work. This model can handle flows with a small Darcy number and might be useful in controlling permeability amplifications. A solution method for the resul...

This paper deals with several nonlinear partial differential equations (PDEs) of mathematical physics such as the concatenation model (perturbed concatenation model) from nonlinear fiber optics, the plane hydrodynamic jet theory, the Kadomtsev–...

The notion of k-symbol special functions has recently been introduced. This new concept offers many interesting geometric properties for these special functions including logarithmic convexity. The aim of the present paper is to exploit essentially t...

The aim of this work is to establish a connection between Bohr’s radius and the analytic and normalized solutions of two differential second-order differential equations, namely $y^{″}\left(z\right)+a\left(z\right)y^{\prime }\left(z\right)+b\left(z\right)y\left(z\right)=0$ and $z^2{y}^{″}\left(z\right)+a\left(z\right)y^{\&pri}$...

We present the theoretical models and review the most recent results of a class of experiments in the field of surface gravity waves. These experiments serve as demonstration of an analogy to a broad variety of phenomena in optics and quantum mechani...

This study models the dynamic response of a damped heterogeneous coating layer over a homogeneous elastic half-space via the shear horizontal equation of motion. The so-called partial nonhomogeneous has been considered in the coating, where only the...

The paper concerns the solution of the ordinary differential equation $y^{″}\pm x^{m}y=0$, which may be designated the generalized Airy equation, since the original Airy equation corresponds to the particular case $m=1$ with the + sign. The solutions...

In order to study the transmission characteristics of Airy beams in the plasma sheath, the flow field around a hypersonic vehicle was numerically simulated and analyzed based on the Navier–Stokes (N-S) equation and a turbulence model. Then, acc...

We begin by introducing a new procedure for construction of the exact solutions to Cauchy problem of the real-valued (hyperbolic) Novikov–Veselov equation which is based on the Moutard symmetry. The procedure shown therein utilizes the well-kno...

A question of physical importance is whether finite-energy spatiotemporally localized (i.e., pulsed) generalizations of monochromatic accelerating Airy beams are feasible. For luminal solutions, this question has been answered within the framework of...

The propagation characteristics of Airy beams in an inhomogeneous medium with periodic potential are studied theoretically and numerically. The Gross–Pitaevskii equation was solved with periodic potential using the separating variables method,...

Lagrangian formalism is established for differential equations with special functions of mathematical physics as solutions. Formalism is based on either standard or non-standard Lagrangians. This work shows that the procedure of deriving the standard...

Thin plates subjected to transverse load and undergoing large deflections have been widely studied and published in the literature. However, there is still a lack of information and understanding about the membrane stresses created under large deflec...

In their treatment of the Airy-Laguerre-Gaussian equation, Zhong, Belic and Zhang constructed an alternative solution for the azimuthal case [...]

Researchers frequently utilize the method of optical initial structure (MOIS) of Czerny–Turner (C–T) spectrometers for aberration-correction studies based on the coma-free equation. While effective, this method has limitations: small nume...

In this study, we use a multi-parameter perturbation method to solve the problem of a functionally graded piezoelectric cantilever beam under combined loads, in which three piezoelectric coefficients are selected as the perturbation parameters. First...

We study the quantum radiation of particle production by vacuum from an ultra-relativistic moving mirror (dynamical Casimir effect) solution that allows (possibly for the first time) analytically calculable time evolution of particle creation and an...

The two-parameter Wright special function is an interesting mathematical object that arises in the theory of the space and time-fractional diffusion equations. Moreover, many other special functions are particular instantiations of the Wright functio...

A class of problems of wave propagation in waveguides consisting of one or several layers that are characterized by linear variation of the squared refractive index along the normal to the interfaces between them is considered in this paper. In vario...

Polynomials that are orthogonal with respect to a perturbation of the Freud weight function by some parameter, known to be modified Freudian orthogonal polynomials, are considered. In this contribution, we investigate certain properties of semi-class...

In information transfer, the dissipation of a signal is of crucial importance. The feasibility of reconstructing the distorted signal depends on the related permanent loss. Therefore, understanding the quantized dissipative transversal mechanical wav...

In this work, based on Newton’s second law, taking into account heredity, an equation is derived for a linear hereditary oscillator (LHO). Then, by choosing a power-law memory function, the transition to a model equation with Gerasimov–Ca...

Currently, divers face significant safety risks when cleaning marine organisms from the steel structures of offshore underwater platform jackets. Consequently, utilizing robots instead of divers to carry out underwater biofouling removal operations w...

This paper proposes a finite element approach to the numerical solution of boundary value problems of linear elasticity theory formulated directly in terms of the stress tensor components. In contrast to the classical finite element formulation, in w...

We study an effective quantum cosmological model with a scale-dependent gravitational coupling $G\left(a\right)$. Starting from a Newtonian cosmological construction inspired by a corrected two-body gravitational potential, we derive an effective Friedmann-type e...

Numerical models are useful tools for studying complex wave–wave and wave–current interactions in coastal areas. They are also very useful for assessing the potential risks of flooding, hydrodynamic actions on coastal protection structures, bathymetr...

Earlier quantum calculations of the optical response of Nakamura’s blue laser diode, assuming Kronig–Penney-like band-edge profiles, omitted the effects of charge polarization, cladding-layer asymmetry, and recombination delay times, whil...