On the Second Boundary-Value Problem for the Airy Equation (original) (raw)

Inhomogeneous Airy’s and Generalized Airy’s Equations with Initial and Bounday Conditions

International Journal of Circuits, Systems and Signal Processing, 2021

Inhomogeneous Airy’s and Generalized Airy’s equations with initial and boundary date are considered in this work. Solutions are obtained for constant and variable forcing functions, and general solutions are expressed in terms of Standard and Generalized Nield-Kuznetsov functions of the first- and second-kinds. Series representations of these functions and their efficient computation methodologies are presented with examples.

The generalized Airy diffusion equation

Electronic Journal of Differential Equations

Solutions of a generalized Airy diffusion equation and an associated nonlinear partial differential equation are obtained. Trigonometric type functions are derived for a third order generalized radial Euler type operator. An associated complex variable theory and generalized Cauchy-Euler equations are obtained. Further, it is shown that the Airy expansions can be mapped onto the Bessel Calculus of Bochner, Cholewinski and Haimo. 2000 Mathematics Subject Classification. 33C10, 35C05, 44A15.

ftp ejde.math.swt.edu (login: ftp) THE GENERALIZED AIRY DIFFUSION EQUATION

2013

Abstract. Solutions of a generalized Airy diffusion equation and an associated nonlinear partial differential equation are obtained. Trigonometric type functions are derived for a third order generalized radial Euler type operator. An associated complex variable theory and generalized Cauchy-Euler equations are obtained. Further, it is shown that the Airy expansions can be mapped onto the Bessel Calculus of Bochner, Cholewinski and Haimo. 1.

A Non-Homogeneous Boundary-Value Problem for the Korteweg-De Vries Equation In a Quarter Plane

Transactions of the American Mathematical …, 2002

The Korteweg-de Vries equation was first derived by Boussinesq and Korteweg and de Vries as a model for long-crested small-amplitude long waves propagating on the surface of water. The same partial differential equation has since arisen as a model for unidirectional propagation of waves in a variety of physical systems. In mathematical studies, consideration has been given principally to pure initial-value problems where the wave profile is imagined to be determined everywhere at a given instant of time and the corresponding solution models the further wave motion. The practical, quantitative use of the Korteweg-de Vries equation and its relatives does not always involve the pure initial-value problem. Instead, initial-boundary-value problems often come to the fore. A natural example arises when modeling the effect in a channel of a wave maker mounted at one end, or in modeling near-shore zone motions generated by waves propagating from deep water. Indeed, the initial-boundary-value problem (0.1)    ηt + ηx + ηηx + ηxxx = 0, for x, t ≥ 0, η(x, 0) = φ(x), η (0, t) = h(t), studied here arises naturally as a model whenever waves determined at an entry point propagate into a patch of a medium for which disturbances are governed approximately by the Korteweg-de Vries equation. The present essay improves upon earlier work on (0.1) by making use of modern methods for the study of nonlinear dispersive wave equations. Speaking technically, local wellposedness is obtained for initial data φ in the class H s (R +) for s > 3 4 and boundary data h in H (1+s)/3 loc (R +), whereas global well-posedness is shown to hold for φ ∈ H s (R +), h ∈ H 7+3s 12 loc (R +) when 1 ≤ s ≤ 3, and for φ ∈ H s (R +), h ∈ H (s+1)/3 loc (R +) when s ≥ 3. In addition, it is shown that the correspondence that associates to initial data φ and boundary data h the unique solution u of (0.1) is analytic. This implies, for example, that solutions may be approximated arbitrarily well by solving a finite number of linear problems.

Solutions to the discrete Airy equation: application to parabolic equation calculations

Journal of Computational and Applied Mathematics, 2004

In the case of the equidistant discretization of the Airy differential equation ("discrete Airy equation") the exact solution can be found explicitly. This fact is used to derive a discrete transparent boundary condition (TBC) for a Schrödinger-type equation with linear varying potential, which can be used in "parabolic equation" simulations in (underwater) acoustics and for radar propagation in the troposphere. We propose different strategies for the discrete TBC and show an efficient implementation. Finally a stability proof for the resulting scheme is given. A numerical example in the application to underwater acoustics shows the superiority of the new discrete TBC.

The Korteweg-De Vries Equation In a Quarter Plane, Continuous Dependence Results

Differential Integral Equations, 1989

Considered herein is an initial-and boundary-value problem that arises in modeling the propagation of small-amplitude, long waves generated by a wavemaker at one end of a homogeneous stretch of nonlinear, dispersive media. The principle accomplishment is to show that the solutions to this problem depend continuously in strong norms on both the initial and the boundary data. 1. Introduction. This paper is a continuation of an earlier one (Bona and Winther 1983) in which an initial-and boundary-value problem for the Korteweg-de Vries equation was analysed. This classical model appears in the study of small amplitude, long wave propagation in an impressive variety of physical situations. It was argued in this previous work that the problem Ut + Ux + UUx + Uxxx = 0, for x, t 2: 0 (l.la) with u(x, 0) = f(x), for x 2: 0, u(O, t) = g(t), for t 2: 0, (l.lb) is especially interesting and apropriate as regards the use of this equation in situations where a wavetrain is created at one end of and travels into an undistured patch of the medium of propagation. A common example to which the Korteweg-de Vries equation might be expected to apply arises in a flume with a wavemaker afixed at one end which, when appropriately oscillated, generates unidirectional, small amplitude, long waves that travel down the channel (cf. Bona, Pritchard and Scott 1981, Hammack and Segur 1974 and Zabusky and Galvin 1971). The problem posed in (1.1) has been investigated by Bona and Heard, as well as in the present authors' earlier paper. The work of Bona and Heard provides existence of relatively weak solutions corresponding to weak assumptions on the initial and boundary data f and

On the boundary‐value problem for the Korteweg–de Vries equation

Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 2003

We consider the initial-boundary value problem on the half-line for the Korteweg-de Vries equation ut + uux + uxxx = 0, t > 0, x > 0, u(x, 0) = u 0 (x), x > 0, u (0, t) = 0, t > 0.

On Energy Waves via Airy Functions in Time-Domain

European Journal of Pure and Applied Mathematics, 2016

The main idea is to solve the system of Maxwell’s equations in accordance with the causality principle to get the energy quantities via Airy functions in a hollow rectangular waveguide. Evolutionary Approach to Electromagnetics which is an analytical time-domain method is used. The boundary-value problem for the system of Maxwell’s equations is reformulated in transverse and longitudinal coordinates. A self-adjoint operator is obtained and the complete set of eigenvectors of the operator initiates an orthonormal basis of the solution space. Hence, the sought electromagnetic field can be presented in terms of this basis. Within the presentation, the scalar coefficients are governed by Klein-Gordon equation. Ultimately, in this study, time-domain waveguide problem is solved analytically in accordance with the causality principle. Moreover, the graphical results are shown for the case when the energy and surplus of the energy for the time-domain waveguide modes are represented via Air...