scholarly journals On the Green function of higher order differential equation with normal operator coefficients on the semi-axis

2018 ◽  
Keyword(s):  
Differential Equation ◽  
Green Function ◽  
Order Differential Equation ◽  
Normal Operator ◽  
Higher Order ◽  
The Green Function

Electrostatic oscillations in cold inhomogeneous plasma I. Differential equation approach

1971 ◽  
Vol 5 (2) ◽  
pp. 239-263 ◽  
Author(s):  
Z. Sedláček
Keyword(s):  
Differential Equation ◽  
Green Function ◽  
Normal Mode Analysis ◽  
Inhomogeneous Plasma ◽  
Mode Analysis ◽  
Complex Frequency ◽  
Plasma Oscillations ◽  
The Laplace Transform ◽  
Collective Oscillations ◽  
The Green Function

Small amplitude electrostatic oscillations in a cold plasma with continuously varying density have been investigated. The problem is the same as that treated by Barston (1964) but instead of his normal-mode analysis we employ the Laplace transform approach to solve the corresponding initial-value problem. We construct the Green function of the differential equation of the problem to show that there are branch-point singularities on the real axis of the complex frequency-plane, which correspond to the singularities of the Barston eigenmodes and which, asymptotically, give rise to non-collective oscillations with position-dependent frequency and damping proportional to negative powers of time. In addition we find an infinity of new singularities (simple poles) of the analytic continuation of the Green function into the lower half of the complex frequency-plane whose position is independent of the spatial co-ordinate so that they represent collective, exponentially damped modes of plasma oscillations. Thus, although there may be no discrete spectrum, in a more general sense a dispersion relation does exist but must be interpreted in the same way as in the case of Landau damping of hot plasma oscillations.

On the spectral properties and positivity of solutionsof a periodic boundary value problemfor a second-order functional differential equation

2020 ◽  
pp. 123-130
Author(s):  
Manuel J. Alves ◽  
Sergey M. Labovskiy
Keyword(s):  
Differential Equation ◽  
Green Function ◽  
Differential Operator ◽  
Boundary Value Problem ◽  
Functional Differential Equation ◽  
Spectral Properties ◽  
Periodic Boundary ◽  
Boundary Value ◽  
Functional Differential ◽  
The Green Function

For a functional-differential operator Lu = (1/ρ)(-(pu')' + ∫_0^l▒〖u(s)d_s r(x,s)〗) with symmetry, the completeness and orthogonality of the eigenfunctions is shown. Thepositivity conditions of the Green function of the periodic boundary value problem areobtained.

scholarly journals Oscillation of even-order neutral differential equations via comparison principles

2014 ◽  
Vol 30 (3) ◽  
pp. 293-300
Author(s):  
J. DZURINA ◽  
◽  
B. BACULIKOVA ◽  
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Order Differential Equation ◽  
Higher Order ◽  
Comparison Principles ◽  
Neutral Differential Equations ◽  
First Order ◽  
Even Order ◽  
Delay Differential

In the paper we offer oscillation criteria for even-order neutral differential equations, where z(t) = x(t) + p(t)x(τ(t)). Establishing a generalization of Philos and Staikos lemma, we introduce new comparison principles for reducing the examination of the properties of the higher order differential equation onto oscillation of the first order delay differential equations. The results obtained are easily verifiable.

scholarly journals Asymptotic and Oscillatory Behavior of Solutions of a Class of Higher Order Differential Equation

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1434 ◽  
Author(s):  
Elmetwally M. Elabbasy ◽  
Clemente Cesarano ◽  
Omar Bazighifan ◽  
Osama Moaaz
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Delay Differential Equations ◽  
Order Differential Equation ◽  
Higher Order ◽  
Oscillatory Behavior ◽  
Future Developments ◽  
Correct Direction ◽  
Delay Differential ◽  
The Right

The objective of this paper is to study asymptotic behavior of a class of higher-order delay differential equations with a p-Laplacian like operator. Symmetry ideas are often invisible in these studies, but they help us decide the right way to study them, and show us the correct direction for future developments. New oscillation criteria are obtained by employing a refinement of the generalized Riccati transformations and comparison principles. This new theorem complements and improves a number of results reported in the literature. Some examples are provided to illustrate the main results.

A global random walk on grid algorithm for second order elliptic equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Karl K. Sabelfeld ◽  
Dmitrii Smirnov
Keyword(s):  
Differential Equation ◽  
Random Walk ◽  
Green Function ◽  
Elliptic Equations ◽  
Symmetry Property ◽  
Variable Coefficients ◽  
Second Order ◽  
Grid Algorithm ◽  
The Green Function ◽  
Second Order Elliptic Equations

Abstract We suggest in this paper a global random walk on grid (GRWG) method for solving second order elliptic equations. The equation may have constant or variable coefficients. The GRWS method calculates the solution in any desired family of m prescribed points of the gird in contrast to the classical stochastic differential equation based Feynman–Kac formula, and the conventional random walk on spheres (RWS) algorithm as well. The method uses only N trajectories instead of mN trajectories in the RWS algorithm and the Feynman–Kac formula. The idea is based on the symmetry property of the Green function and a double randomization approach.

scholarly journals A second-order differential equation for a point charged particle

2017 ◽  
Vol 14 (04) ◽  
pp. 1750049 ◽  
Author(s):  
Ricardo Gallego Torromé
Keyword(s):  
Differential Equation ◽  
Charged Particle ◽  
Order Differential Equation ◽  
Higher Order ◽  
Second Order ◽  
Second Order Differential Equation ◽  
New Equation ◽  
Power Radiation ◽  
Classical Point

A model for the dynamics of a classical point charged particle interacting with higher order jet fields is introduced. In this model, the dynamics of the charged particle is described by an implicit ordinary second-order differential equation. Such equation is free of run-away and pre-accelerated solutions of Dirac’s type. The theory is Lorentz invariant, compatible with the first law of Newton and Larmor’s power radiation formula. Few implications of the new equation in the phenomenology of non-neutral plasmas is considered.

scholarly journals On the Complex Oscillation of Meromorphic Solutions of Nonhomogeneous Linear Differential Equations with Meromorphic Coefficients

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Chuang-Xin Chen ◽  
Ning Cui ◽  
Zong-Xuan Chen
Keyword(s):  
Differential Equation ◽  
Meromorphic Function ◽  
Differential Equations ◽  
Rational Function ◽  
Finite Order ◽  
Order Differential Equation ◽  
Higher Order ◽  
Meromorphic Solutions ◽  
Complex Oscillation ◽  
Linear Differential

In this paper, we study the higher order differential equation f k + B f = H , where B is a rational function, having a pole at ∞ of order n > 0 , and H ≡ 0 is a meromorphic function with finite order, and obtain some properties related to the order and zeros of its meromorphic solutions.

Sign in / Sign up

Export Citation Format

Share Document