arXiv subjects

Asatur Khurshudyan

Publications and source records attributed to Asatur Khurshudyan.

5 recordsLinked to original sources

arXiv2018

A General Representation for the Green's Function of Second Order Nonlinear Differential Equations

In this paper we study some classes of second order non-homogeneous nonlinear differential equations allowing a specific representation for nonlinear Green's function. In particular, we show that if the nonlinear term possesses a special multiplicativity property, then its Green's function is represented as the product of the Heaviside function and the general solution of the corresponding homogeneous equations subject to non-homogeneous Cauchy conditions. Hierarchies of specific non-linearities admitting this representation are derived. The nonlinear Green's function solution is numerically justified for the sinh-Gordon and Liouville equations. We also list two open problems leading to a more thorough characterizations of non-linearities admitting the obtained representation for the nonlinear Green's function.

math-ph↗

arXiv2018

General Representation of Nonlinear Green's Function for Second Order Differential Equations Nonlinear in the First Derivative

The Green's function method which has been originally proposed for linear systems has several extensions to the case of nonlinear equations. A recent extension has been proposed to deal with certain applications in quantum field theory. The general solution of second order nonlinear differential equations is represented in terms of a so-called short time expansion. The first term of the expansion has been shown to be an efficient approximation of the solution for small values of the state variable. The proceeding terms contribute to the error correction. This paper is devoted to extension of the short time expansion solution to non-linearities depending on the first derivative of the unknown function. Under a proper assumption on the nonlinear term, a general representation for Green's function is derived. It is also shown how the knowledge of nonlinear Green's function can be used to study the spectrum of the nonlinear operator. Particular cases and their numerical analysis support the advantage of the method. The technique we discuss grants to obtain a closed form analytic solution for non-homogeneous non-linear PDEs so far amenable just to numerical solutions. This opens up the possibility of several applications in physics and engineering.

math-ph↗

arXiv2018

New Green's functions for some nonlinear oscillating systems and related PDEs

During the past three decades, the advantageous concept of the Green's function has been extended from linear systems to nonlinear ones. At that, there exist a rigorous and an approximate extensions. The rigorous extension introduces the so-called backward and forward propagators, which play the same role for nonlinear systems as the Green's function plays for linear systems. The approximate extension involves the Green's formula for linear systems with a Green's function satisfying the corresponding nonlinear equation. For the numerical evaluation of nonlinear ordinary differential equations the second approach seems to be more convenient. In this article we study a hierarchy of nonlinear partial differential equations that can be approximated by the second approach. Green's functions for particular non-linearities are derived explicitly. Numerical error analysis in the case of exponential non-linearity for different source functions supports the advantage of the approach.

math-ph↗

arXiv2017

Some interacting dark energy models

In this paper we study new cosmological models involving new forms of non-gravitational interaction between cold dark matter and dark energy. The main purpose is to demonstrate the applicability of the forms of interaction to the problem in modern cosmology known as accelerated expansion of the large scale universe. It is well known, that a non-gravitational interaction can improve theoretical prediction, but a fundamental theory allowing to derive the form of the interaction is missing. Therefore, an active sought to find a correct form of the interaction from various phenomenological assumptions is still one of the active branches of research in modern cosmology. Besides cosmographic analysis, we perform $Om$ analysis.

gr-qc↗

arXiv2017

On cosmology of interacting varying polytropic dark fluids

In this paper a possibility of the accelerated expansion of the large scale universe with a varying polytropic fluid of a certain type is presented. About a special role of non-gravitational interactions between dark energy and dark matter, in particular, about a possibility of improvement and solution of problems arising in modern cosmology, has been discussed for a long time. This motivates us to consider new models, where non-gravitational interactions between varying polytropic fluid and cold dark matter are allowed. Mainly non-linear interactions of a specific type is considered, found in recent literature. In order to understand the behavior of suggested cosmological models, besides cosmographic analysis, $Om$ analysis is applied. Moreover, with different datasets, including a strong gravitational lensing dataset, the observational constraints on the model parameters are obtained using $χ^{2}$ analysis.

gr-qc↗