scholarly journals Nonlinear Green’s Functions for Wave Equation with Quadratic and Hyperbolic Potentials

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Asatur Zh. Khurshudyan
Keyword(s):  
Wave Equation ◽  
Differential Equations ◽  
Wave Equations ◽  
Separation Of Variables ◽  
Linear Equations ◽  
Second Order ◽  
Nonlinear Wave Equations ◽  
One Dimensional ◽  
Exponential Nonlinearity ◽  
Second Order Differential Equations

The advantageous Green’s function method that originally has been developed for nonhomogeneous linear equations has been recently extended to nonlinear equations by Frasca. This article is devoted to rigorous and numerical analysis of some second-order differential equations new nonlinearities by means of Frasca’s method. More specifically, we consider one-dimensional wave equation with quadratic and hyperbolic nonlinearities. The case of exponential nonlinearity has been reported earlier. Using the method of generalized separation of variables, it is shown that a hierarchy of nonlinear wave equations can be reduced to second-order nonlinear ordinary differential equations, to which Frasca’s method is applicable. Numerical error analysis in both cases of nonlinearity is carried out for various source functions supporting the advantage of the method.

Geometric characterization of separable second-order differential equations

1993 ◽  
Vol 113 (1) ◽  
pp. 205-224 ◽  
Author(s):  
Eduardo Martínez ◽  
José F. Cariñena ◽  
Willy Sarlet
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Geometric Characterization ◽  
One Dimensional ◽  
Practical Procedure ◽  
Second Order Differential Equations ◽  
Necessary And Sufficient ◽  
Second Order Equations

AbstractWe establish necessary and sufficient conditions for the separability of a system of second-order differential equations into independent one-dimensional second-order equations. The characterization of this property is given in terms of geometrical objects which are directly related to the system and relatively easy to compute. The proof of the main theorem is constructive and thus yields a practical procedure for constructing coordinates in which the system decouples.

scholarly journals Multiple Bounded Positive Solutions to Integral Type BVPs for Singular Second Order ODEs on the Whole Line

2012 ◽  
Vol 2012 ◽  
pp. 1-23 ◽  
Author(s):  
Yuji Liu
Keyword(s):  
Banach Space ◽  
Differential Equations ◽  
Positive Solutions ◽  
Sufficient Conditions ◽  
Second Order ◽  
Integral Type ◽  
One Dimensional ◽  
Second Order Differential Equations ◽  
Type Boundary ◽  
The One

This paper is concerned with the integral type boundary value problems of the second order differential equations with one-dimensionalp-Laplacian on the whole line. By constructing a suitable Banach space and a operator equation, sufficient conditions to guarantee the existence of at least three positive solutions of the BVPs are established. An example is presented to illustrate the main results. The emphasis is put on the one-dimensionalp-Laplacian term[ρ(t)Φ(x’(t))]’involved with the functionρ, which makes the solutions un-concave.

scholarly journals LINEAR AND NONLINEAR TAILS I: GENERAL RESULTS AND PERTURBATION THEORY

2008 ◽  
Vol 05 (04) ◽  
pp. 741-765 ◽  
Author(s):  
NIKODEM SZPAK
Keyword(s):  
Perturbation Theory ◽  
Wave Equation ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Linear Equations ◽  
Perturbation Series ◽  
Nonlinear Wave Equations ◽  
Small Initial Data ◽  
Potential Term ◽  
Full Solution

For nonlinear wave equations with a potential term, we prove pointwise space-time decay estimates and develop a perturbation theory for small initial data. We show that the perturbation series has a positive convergence radius by a method which reduces the wave equation to an algebraic one. We demonstrate that already first and second perturbation orders, satisfying linear equations, can provide precise information about the decay of the full solution to the nonlinear wave equation.

scholarly journals A Simple Example for Linear Partial Differential Equations and Its Solution Using the Method of Separation of Variables

2019 ◽  
Vol 27 (1) ◽  
pp. 25-34
Author(s):  
Sora Otsuki ◽  
Pauline N. Kawamoto ◽  
Hiroshi Yamazaki
Keyword(s):  
Wave Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Partial Derivative ◽  
Separation Of Variables ◽  
Partial Differential ◽  
One Dimensional ◽  
Linear Partial Differential Equations ◽  
Dimensional Wave

Summary In this article, we formalized in Mizar [4], [1] simple partial differential equations. In the first section, we formalized partial differentiability and partial derivative. The next section contains the method of separation of variables for one-dimensional wave equation. In the last section, we formalized the superposition principle.We referred to [6], [3], [5] and [9] in this formalization.

4.—Comparison Theorems for a Class of Second-order Non-linear Ordinary Differential Equations

1975 ◽  
Vol 73 ◽  
pp. 77-84
Author(s):  
Donald C. Benson
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Comparison Theorem ◽  
Linear Equations ◽  
Second Order ◽  
Comparison Theorems ◽  
Linear Ordinary Differential Equations ◽  
Second Order Differential Equations ◽  
Liénard Equations ◽  
Fowler Equation

SynopsisIntegral inequalities are used to obtain comparison theorems for a class of second-order differential equations which includes the Emden-Fowler equation, certain Liénard equations, and linear equations of the form d2y/dx2+f(x)y = 0. For these linear equations the results below imply Sturm's classical comparison theorem.

Banach Lie Algebroids

2011 ◽  
Vol 57 (2) ◽  
pp. 409-416
Author(s):  
Mihai Anastasiei
Keyword(s):  
Differential Equations ◽  
Smooth Manifold ◽  
Vector Bundles ◽  
Second Order ◽  
Lie Algebroids ◽  
Second Order Differential Equations

Banach Lie AlgebroidsFirst, we extend the notion of second order differential equations (SODE) on a smooth manifold to anchored Banach vector bundles. Then we define the Banach Lie algebroids as Lie algebroids structures modeled on anchored Banach vector bundles and prove that they form a category.

scholarly journals New oscillation criteria for second-order neutral differential equations with distributed deviating arguments

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Osama Moaaz ◽  
Choonkil Park ◽  
Elmetwally M. Elabbasy ◽  
Waed Muhsin
Keyword(s):  
Differential Equations ◽  
Second Order ◽  
The Other ◽  
Riccati Transformation ◽  
Transformation Technique ◽  
Deviating Arguments ◽  
Neutral Differential Equations ◽  
Second Order Differential Equations ◽  
Continuous Delay ◽  
New Criteria

AbstractIn this work, we create new oscillation conditions for solutions of second-order differential equations with continuous delay. The new criteria were created based on Riccati transformation technique and comparison principles. Furthermore, we obtain iterative criteria that can be applied even when the other criteria fail. The results obtained in this paper improve and extend the relevant previous results as illustrated by examples.

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