Airy Equation Research Articles

Revivals, or the Talbot effect, for the Airy equation

AbstractWe study Dirichlet‐type problems for the simplest third‐order linear dispersive partial differential equations (PDE), often referred to as the Airy equation. Such problems have not been extensively studied, perhaps due to the complexity of the spectral structure of the spatial operator. Our specific interest is to determine whether the peculiar phenomenon of revivals, also known as Talbot effect, is supported by these boundary conditions, which for third‐order problems are not reducible to periodic ones. We prove that this is the case only for a very special choice of the boundary conditions, for which a new type of weak cusp revival phenomenon has been recently discovered. We also give some new results on the functional class of the solution for other cases.

Integral representations for products of two solutions of the Airy equation with shifted arguments and their applications in physics

Integral representations for products of two solutions of the Airy equation with shifted arguments and their applications in physics

The Airy equation with nonlocal conditions

AbstractWe study a third‐order dispersive linear evolution equation on the finite interval subject to an initial condition and inhomogeneous boundary conditions but, in place of one of the three boundary conditions that would typically be imposed, we use a nonlocal condition, which specifies a weighted integral of the solution over the spatial interval. Via adaptations of the Fokas transform method (or unified transform method), we obtain a solution representation for this problem. We also study the time periodic analog of this problem, and thereby obtain long time asymptotics for the original problem with time periodic boundary and nonlocal data.

Some Approaches for Solving Multiplicative Second-Order Linear Differential Equations with Variable Exponentials and Multiplicative Airy’s Equation

This paper offers several approaches for solving multiplicative second-order linear differential equations with variable exponentials, such as normalization and reduction to Riccati equations. In addition, in this paper, the multiplicative version of the Airy equation, which emerges in fluid mechanics, geophysics, and atomic physics, is solved using the multiplicative power series solution method.

Analytical Solution of the Schrödinger Equation with an Exponential Type Mass Depending on the Spatial Variable

In the present work, we proceed to study the Schrödinger equation with dependent mass position. From the resulting partial differential equations, we obtain exact analytical solutions governed by Bessel functions. The exact solution of Schrödinger's equation for a particle with dependent position of the mass (PDM) is a "half-harmonic potential" defined in a Hilbert space. The harmonic oscillator is carried by the wave function ψ(x) through the Bessel function. The magnitude of ψ(x) increases dramatically as the values of the spatial coordinate become larger and larger. This growth is all the more marked as the mass parameter increases. It appears that the wave function has the same asymptotic behavior as the solution of Airy's equation of the second kind. Their oscillations have the same amplitudes near infinity. The mass parameter has a great importance and influence on the mass and also on the wave potential. It is a control parameter in the Schrödinger equations (PDM).

Quantum Riemannian geometry of the discrete interval and q-deformation

We solve for quantum Riemannian geometries on the finite lattice interval • – • –⋯– • with n nodes (the Dynkin graph of type An) and find that they are necessarily q-deformed with q=eıπn+1. This comes out of the intrinsic geometry and not by assuming any quantum group in the picture. Specifically, we discover a novel “boundary effect” whereby, in order to admit a quantum Levi-Cività connection, the “metric weight” at any edge is forced to be greater pointing toward the bulk compared to toward the boundary, with the ratio given by (i + 1)q/(i)q at node i, where (i)q is a q-integer. The Christoffel symbols are also q-deformed. The limit q → 1 likewise forces the quantum Riemannian geometry of the natural numbers N to have rational metric multiples (i + 1)/i in the direction of increasing i. In both cases, there is a unique Ricci-scalar flat metric up to normalization. Elements of quantum field theory and quantum gravity are exhibited for n = 3 and for the continuum limit of the geometry of N. The Laplacian for the scalar-flat metric becomes the Airy equation operator 1xd2dx2 in so far as a limit exists. Scaling this metric by a conformal factor eψ(i) gives a limiting Ricci scalar curvature proportional to e−ψxd2ψdx2.

Implementation of variational iteration method for various types of linear and nonlinear partial differential equations

There are various linear and nonlinear one-dimensional partial differential equations that are the focus of this research. There are a large number of these equations that cannot be solved analytically or precisely. The evaluation of nonlinear partial differential equations, even if analytical solutions exist, may be problematic. Therefore, it may be necessary to use approximate analytical methodologies to solve these issues. As a result, a more effective and accurate approach must be investigated and analyzed. It is shown in this study that the Lagrange multiplier may be used to get an ideal value for parameters in a functional form and then used to construct an iterative series solution. Linear and nonlinear partial differential equations may both be solved using the variational iteration method (VIM) method, thanks to its high computing power and high efficiency. Decoding and analyzing possible Korteweg-De-Vries, Benjamin, and Airy equations demonstrates the method’s ability. With just a few iterations, the produced findings are very effective, precise, and convergent to the exact answer. As a result, solving nonlinear equations using VIM is regarded as a viable option.

Investigation of Airy equations with random initial conditions

The paper investigates properties of mean-square solutions to the Airy equation with random initial data given by stationary processes. The result on the modulus of continuity of the solution is stated and properties of the covariance function are described. Bounds for the distributions of the suprema of solutions under φ-sub-Gaussian initial conditions are presented. Several examples are provided to illustrate the results. Possible extensions of the results are discussed.

Трехмерная ловушка с возбуждением колебаний ионов на границе устойчивости диаграммы Матье

Vibrations of charged particles in compositions of three-dimensional high-frequency quadrupole and static homogeneous electric fields in the stable region and in the vicinity of the stability boundary of the Mathieu diagram are investigated. Using a pseudopotential model of a rapidly oscillating field, it is shown that the motion of charged particles during linear scanning of a secular frequency is described by the Airy differential equation. Based on the properties of solutions of the Airy equation, a method of ion mass separation with resonant excitation of oscillations at the stability boundary of the Mathieu diagram has been developed. To implement the method, the ion-optical system of the three-dimensional trap is supplemented with corrective electrodes. Computer modeling has determined the optimal potentials of the correcting electrodes, at which the errors of the distributions of quadrupole and homogeneous fields do not exceed 10E-4 and 2·10E-3. Keywords: superposition of quadrupole and homogeneous fields, Airy differential equation, mode of resonant excitation of oscillations, three-dimensional ion trap with correcting electrodes.

ЕЛЕКТРОМАГНІТНЕ ПОЛЕ БОЧКОПОДІБНОГО ПОЛОГО РЕЗОНАТОРА

In the article is investigates the electromagnetic field of a barrel-shaped hollow resonator. It is significant that in the articles of K.M. Tsibizov developed an approach for determining the fields of waveguide resonator devices with a variable cross section. In this article, this approach is extended to a barrel-shaped hollow resonator. The solution of the Airy equation, which was used by K.M. Tsibizov are approximate solutions obtained through solving an equation that reduces to one of the forms of the hypergeometric equation or the Weber equation. The radial and azimuthal components of the field of a barrel-shaped hollow resonator are the corresponding components of the field of a hollow cylindrical resonator, and the longitudinal component is described by a degenerate hypergeometric function. In this article presents the design of a barrel-shaped waveguide resonator with metal ends and writes expressions for the resonant wavelengths of a barrel-shaped resonator with metal ends for H- and E-oscillations. The resonant wavelengths and resonant frequencies for two barrel-shaped resonators with metal ends are given in accordance with a cylindrical resonator with an equal end radius of the barrel-shaped resonator and equal longitudinal dimensions. The article also presents a waveguide resonator with an open left end and writes down the first five roots of the Airy functions. The consideration of such a resonator aims at demonstrating the generality of the considered approach.The design of a barrel-shaped resonator with open ends, the sections of which are critical, is presented.It is indicated that finding solutions to the equations of functions that describe the longitudinal distribution of the field is a demonstration of the generalization of the approach developed by K.M. Tsibizov, and as examples of the study of resonators of variable cross section: a barrel-shaped waveguide resonator with metal ends, a waveguide resonator with an open left end, and a barrel-shaped resonator with open ends, which are shown in the figures.

Stokes matrices for Airy equations

We compute Stokes matrices for generalised Airy equations and prove that they are regular unipotent (up to multiplication with the formal monodromy). This class of differential equations was defined by Katz and includes the classical Airy equation. In addition, it includes differential equations which are not rigid. Our approach is based on the topological computation of Stokes matrices of the enhanced Fourier-Sato transform of a perverse sheaf due to D'Agnolo, Hien, Morando and Sabbah.

Strichartz and Local Smoothing Estimates for Stochastic Dispersive Equations with Linear Multiplicative Noise

We study a quite general class of stochastic dispersive equations with linear multiplicative noise, including especially the Schrodinger and Airy equations. The pathwise Strichartz and local smoothing estimates are derived here in both the conservative and nonconservative case. In particular, we obtain the P-integrability of constants in these estimates, where P is the underlying probability measure. Several applications are given to nonlinear problems, including local well-posedness of stochastic nonlinear Schrodinger equations with variable coefficients and lower order perturbations, integrability of global solutions to stochastic nonlinear Schrodinger equations with constant coefficients. As another consequence, we prove as well the large deviation principle for the small noise asymptotics.

The Deligne–Simpson problem for connections on [formula omitted] with a maximally ramified singularity

The classical additive Deligne–Simpson problem is the existence problem for Fuchsian connections with residues at the singular points in specified adjoint orbits. Crawley-Boevey found the solution in 2003 by reinterpreting the problem in terms of quiver varieties. A more general version of this problem, solved by Hiroe, allows additional unramified irregular singularities. We apply the theory of fundamental and regular strata due to Bremer and Sage to formulate a version of the Deligne–Simpson problem in which certain ramified singularities are allowed. These allowed singular points are called toral singularities; they are singularities whose leading term with respect to a lattice chain filtration is regular semisimple. We solve this problem in the special case of connections on Gm with a maximally ramified singularity at 0 and possibly an additional regular singular point at infinity. Examples of such connections arise from Airy, Bessel, and Kloosterman differential equations. They play an important role in recent work in the geometric Langlands program. We also give a complete characterization of all such connections which are rigid, under the additional hypothesis of unipotent monodromy at infinity.

The modified KdV equation for a nonlinear evolution problem with perturbation technique

This paper examines nonlinear partial differential equation (PDE) solutions. Scientists and engineers have struggled to solve nonlinear differential equations. Nonlinear equations arrive in nearly all problems in nature. There are no well-established techniques for solving all nonlinear equations, and efforts have been made to enhance approaches for a specific class of problems. Keeping this in mind, we shall investigate the perturbation method’s efficiency in solving nonlinear PDEs. Several techniques work well for diverse issues. We recognize that there may be several solutions to a given nonlinear issue. Methods include homotropy analysis, tangent hyperbolic function, factorization and trial function. However, some of these strategies do not cover all nonlinear issue solutions. In this paper, we use the perturbation technique to solve the zeroth-order Airy equation and also find the Bessel function in the first-order nonhomogeneous differential equation by using self-similar solutions that appears in modified Korteweg–de Vries (KdV) equation. This approach will be used for nonlinear equations in physics and applied mathematics.

The Nonlinear Schrödinger–Airy equation in weighted Sobolev spaces

We study the persistence property of the solution for the nonlinear Schrödinger–Airy equation with initial data in the weighted Sobolev space H1/4(R)∩L2(|x|2mdx), 0<m≤1/8, via the contraction principle.

Airy function solution of two classes of nonlinear evolution equations

In this paper , we study the problem of constructing the new solutions of the mKdV equation and STO equation. By using the Airy equation and its solution, the new solutions of mKdV equation and STO equation are constructed. First of all, through function transformation, mKdV equation and STO equation can be turned into Airy function to solve. Then, construct the solution of mKdV equation and STO equation by using the solution of Airy function. Finally, analyze the properties of these solutions with the help of symbolic computation system: Mathematica.

Generating Resonant and Repeated Root Solutions to Ordinary Differential Equations Using Perturbation Methods

In the study of ordinary differential equations (ODEs) of the form $\hat{L}[y(x)]=f(x)$, where $\hat{L}$ is a linear differential operator, two related phenomena can arise: resonance, where $f(x)\propto u(x)$ and $\hat{L}[u(x)]=0$, and repeated roots, where $f(x)=0$ and $\hat{L}=\hat{D}^n$ for $n\geq 2$. We illustrate a method to generate exact solutions to these problems by taking a known homogeneous solution $u(x)$, introducing a parameter $\epsilon$ such that $u(x)\rightarrow u(x;\epsilon)$, and Taylor expanding $u(x;\epsilon)$ about $\epsilon = 0$. The coefficients of this expansion $\frac{\partial^k u}{\partial\epsilon^k}\big{|}_{\epsilon=0}$ yield the desired resonant or repeated root solutions to the ODE. This approach, whenever it can be applied, is more insightful and less tedious than standard methods such as reduction of order or variation of parameters. We provide examples of many common ODEs, including constant coefficient, equidimensional, Airy, Bessel, Legendre, and Hermite equations. While the ideas can be introduced at the undergraduate level, we could not find any existing elementary or advanced text that illustrates these ideas with appropriate generality.

The Airy equations with impulsive effect: multi-valued nonlinear perturbations

The Airy equations with impulsive effect: multi-valued nonlinear perturbations

Властивостi розв’язкiв лiнiйного рiвняння KdV iзφ-субгауссовими початковими умовами

In this paper, there are studied sample paths properties of stochastic processes representing solutions (in L_2(Ω) sense) to the linear Korteweg–de Vries equation (called also the Airy equation) with random initial conditions given by φ-sub-Gaussian stationary processes. The main results are the bounds for the distributions of the suprema for such stochastic processes considered over bounded domains. Also, there are presented some examples to illustrate the results of the study.

Airy eigenstates and their relation to coordinate eigenstates

We study the eigenvalue problem for a linear potential Hamiltonian and, by writing the Airy equation in terms of momentum and position operators define Airy states. By using the unit operator in terms of Airy states we find a relation between them and position eigenstates.