Crossing the phantom divide line from a generalized time-dependent Hubble parameter and its dynamical evolution a la Riccati

2013 ◽  
Vol 91 (8) ◽  
pp. 623-631 ◽  
Author(s):  
Rami Ahmad El-Nabulsi
Keyword(s):  
Differential Equation ◽  
Hubble Parameter ◽  
Dynamical Evolution ◽  
Time Dependent ◽  
Effective Equation ◽  
Riccati Differential Equation ◽  
Phantom Divide ◽  
Phantom Divide Line ◽  
Generalized Time ◽  
Friedmann Robertson Walker

We introduce the notion of a “generalized time-dependent Hubble parameter” for the case of Friedmann–Robertson–Walker cosmology. We obtain a Riccati differential equation for the Hubble parameter, H, and it was observed that the effective equation of state in our framework can cross the phantom divide line as supported by recent astrophysical observations. In addition, the model is able to evolve without initial singularity.

scholarly journals Bulk viscosity in F(T, TG) gravity

2017 ◽  
Vol 95 (3) ◽  
pp. 262-266
Author(s):  
M. Sharif ◽  
Kanwal Nazir
Keyword(s):  
Equation Of State ◽  
Bulk Viscosity ◽  
State Parameter ◽  
Time Dependent ◽  
Effective Equation ◽  
Phantom Divide ◽  
Equation Of State Parameter ◽  
Phantom Divide Line ◽  
Dependent Viscosity ◽  
The Universe

The present paper is devoted to exploring the effect of bulk viscosity in the context of F(T, TG) gravity. We consider a time-dependent viscosity model with a particular expression of Hubble parameter. We evaluate viscous effective equation of state parameter for three well-known F(T, TG) models. The behavior of the accelerated expanding universe is explored graphically through the viscous equation of state parameter. This parameter indicates the phantom-dominated era as well as crosses the phantom divide line for all three models. We conclude that the universe shows a transition from quintessence to phantom region in the presence of bulk viscosity.

scholarly journals Quadratic First Integrals of Time-dependent Dynamical Systems of the Form q¨a=-Γbcaq˙bq˙c-ω(t)Qa(q)

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1503
Author(s):  
Antonios Mitsopoulos ◽  
Michael Tsamparlis
Keyword(s):  
Dynamical System ◽  
Differential Equation ◽  
First Integrals ◽  
Time Dependent ◽  
Killing Tensor ◽  
Emden Equation ◽  
General Polynomial ◽  
Linear Differential ◽  
Generalized Time ◽  
New System

We consider the time-dependent dynamical system q¨a=−Γbcaq˙bq˙c−ω(t)Qa(q) where ω(t) is a non-zero arbitrary function and the connection coefficients Γbca are computed from the kinetic metric (kinetic energy) of the system. In order to determine the quadratic first integrals (QFIs) I we assume that I=Kabq˙aq˙b+Kaq˙a+K where the unknown coefficients Kab,Ka,K are tensors depending on t,qa and impose the condition dIdt=0. This condition leads to a system of partial differential equations (PDEs) involving the quantities Kab,Ka,K,ω(t) and Qa(q). From these PDEs, it follows that Kab is a Killing tensor (KT) of the kinetic metric. We use the KT Kab in two ways: a. We assume a general polynomial form in t both for Kab and Ka; b. We express Kab in a basis of the KTs of order 2 of the kinetic metric assuming the coefficients to be functions of t. In both cases, this leads to a new system of PDEs whose solution requires that we specify either ω(t) or Qa(q). We consider first that ω(t) is a general polynomial in t and find that in this case the dynamical system admits two independent QFIs which we collect in a Theorem. Next, we specify the quantities Qa(q) to be the generalized time-dependent Kepler potential V=−ω(t)rν and determine the functions ω(t) for which QFIs are admitted. We extend the discussion to the non-linear differential equation x¨=−ω(t)xμ+ϕ(t)x˙(μ≠−1) and compute the relation between the coefficients ω(t),ϕ(t) so that QFIs are admitted. We apply the results to determine the QFIs of the generalized Lane–Emden equation.

Interacting phantom cosmology via Noether symmetry approach

2020 ◽  
Vol 17 (12) ◽  
pp. 2050179
Author(s):  
Yusuf Kucukakca
Keyword(s):  
Scalar Field ◽  
Cosmological Solution ◽  
State Parameter ◽  
Noether Symmetry ◽  
Accelerated Expansion ◽  
Effective Equation ◽  
Phantom Divide ◽  
Symmetry Approach ◽  
Phantom Divide Line ◽  
Phantom Scalar

In this paper, we have presented a cosmological model where a phantom scalar field is minimally coupled to dark matter component. Noether symmetry method was applied both to investigate the cosmological solution and to find out what is the form of the potential of scalar field and the unknown function in the considered model. By using this method, these forms are resulted as trigonometric functions. Also, the obtained cosmological solutions are compatible with observations describing the accelerated expansion of the Universe. Furthermore, it turns out that the effective equation of state parameter in the model can cross the phantom divide line.

scholarly journals GHOST DARK ENERGY IN f(R) MODEL OF GRAVITY

2012 ◽  
Vol 21 (06) ◽  
pp. 1250057 ◽  
Author(s):  
KH. SAAIDI ◽  
A. AGHAMOHAMMADI ◽  
B. SABET ◽  
O. FAROOQ
Keyword(s):  
Dark Energy ◽  
Equation Of State ◽  
Dynamical Evolution ◽  
State Parameter ◽  
Jordan Frame ◽  
Density Parameter ◽  
Eos Parameter ◽  
Phantom Divide ◽  
Equation Of State Parameter ◽  
Phantom Divide Line

We study a correspondence between f(R) model of gravity in the Jordan frame and a phenomenological kind of dark energy (DE), which is known as QCD ghost DE. Since this kind of DE is not stable in the context of Einsteinian theory of gravity and Brans–Dicke model of gravity, we consider two kinds of correspondence between modified gravity and DE. By studding the dynamical evolution of model and finding relevant quantities such as, equation of state parameter, deceleration parameter, dimensionless density parameter, we show that the model can describe the present Universe and also the EoS parameter can cross the phantom divide line without needs to any kinetic energy with negative sign. Furthermore, by obtaining the adiabatic squared sound speed of the model for different cases of interaction, we show that this model is stable. Finally, we fit this model with supernova observational data in a noninteraction case and we find the best values of parameter at 1σ confidence interval as; [Formula: see text], [Formula: see text] and [Formula: see text]. These best-fit values show that DE equation of state parameter, ωd0, can cross the phantom divide line at the present time.

scholarly journals New Approaches to the General Linearization Problem of Jacobi Polynomials Based on Moments and Connection Formulas

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1573
Author(s):  
Waleed Mohamed Abd-Elhameed ◽  
Badah Mohamed Badah
Keyword(s):  
Differential Equation ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
Tau Method ◽  
Riccati Differential Equation ◽  
Shifted Jacobi Polynomials ◽  
Linearization Problem ◽  
Linearization Coefficients ◽  
New Formula ◽  
Connection Formulas

This article deals with the general linearization problem of Jacobi polynomials. We provide two approaches for finding closed analytical forms of the linearization coefficients of these polynomials. The first approach is built on establishing a new formula in which the moments of the shifted Jacobi polynomials are expressed in terms of other shifted Jacobi polynomials. The derived moments formula involves a hypergeometric function of the type 4F3(1), which cannot be summed in general, but for special choices of the involved parameters, it can be summed. The reduced moments formulas lead to establishing new linearization formulas of certain parameters of Jacobi polynomials. Another approach for obtaining other linearization formulas of some Jacobi polynomials depends on making use of the connection formulas between two different Jacobi polynomials. In the two suggested approaches, we utilize some standard reduction formulas for certain hypergeometric functions of the unit argument such as Watson’s and Chu-Vandermonde identities. Furthermore, some symbolic algebraic computations such as the algorithms of Zeilberger, Petkovsek and van Hoeij may be utilized for the same purpose. As an application of some of the derived linearization formulas, we propose a numerical algorithm to solve the non-linear Riccati differential equation based on the application of the spectral tau method.

scholarly journals Precise Integration Method for Solving Noncooperative LQ Differential Game

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Hai-Jun Peng ◽  
Sheng Zhang ◽  
Zhi-Gang Wu ◽  
Biao-Song Chen
Keyword(s):  
Differential Equation ◽  
Differential Game ◽  
Integration Method ◽  
Transformation Method ◽  
Linear Quadratic ◽  
Riccati Differential Equation ◽  
Precise Integration ◽  
Precise Integration Method ◽  
Direct Integration Method ◽  
Two Parameters

The key of solving the noncooperative linear quadratic (LQ) differential game is to solve the coupled matrix Riccati differential equation. The precise integration method based on the adaptive choosing of the two parameters is expanded from the traditional symmetric Riccati differential equation to the coupled asymmetric Riccati differential equation in this paper. The proposed expanded precise integration method can overcome the difficulty of the singularity point and the ill-conditioned matrix in the solving of coupled asymmetric Riccati differential equation. The numerical examples show that the expanded precise integration method gives more stable and accurate numerical results than the “direct integration method” and the “linear transformation method”.

scholarly journals Sequential eigenfunction expansion for a problem in combustion theory

2003 ◽  
Vol 45 (1) ◽  
pp. 35-48 ◽  
Author(s):  
M. Al-Refai ◽  
K. K. Tam
Keyword(s):  
Differential Equation ◽  
Eigenfunction Expansion ◽  
Time Dependent ◽  
Linear Parabolic Equation ◽  
System Of Equations ◽  
Multiple Steady State ◽  
Definitive Answer ◽  
Combustion Theory ◽  
The Galerkin Method ◽  
Steady State Solutions

AbstractA method of sequential eigenfunction expansion is developed for a semi-linear parabolic equation. It allows the time-dependent coefficients of the eigenfunctions to be determined sequentially and iterated to reach convergence. At any stage, only a single ordinary differential equation needs to be considered, in contrast to the Galerkin method which requires the consideration of a system of equations. The method is applied to a central problemin combustion theory to provide a definitive answer to the question of the influence of the initial data in determining whether the solution is sub- or super-critical, in the case of multiple steady-state solutions. It is expected this method will prove useful in dealing with similar problems.

On-line T-S fuzzy control using Riccati differential equation

2017 ◽  
Vol 33 (6) ◽  
pp. 3871-3881 ◽  
Author(s):  
Jorge Cervantes ◽  
Wen Yu ◽  
Sergio Salazar
Keyword(s):  
Differential Equation ◽  
Fuzzy Control ◽  
Riccati Differential Equation ◽  
On Line

Dynamic Stability of an Axially Moving Yarn with Time-Dependent Tension

2014 ◽  
Vol 900 ◽  
pp. 753-756 ◽  
Author(s):  
You Guo Li
Keyword(s):  
Differential Equation ◽  
Homotopy Perturbation Method ◽  
Time Dependent ◽  
Second Law ◽  
Order Ordinary Differential Equation ◽  
Vibration Equation ◽  
Transversal Vibration ◽  
First Order ◽  
Homotopy Perturbation ◽  
Axially Moving

In this paper the nonlinear transversal vibration of axially moving yarn with time-dependent tension is investigated. Yarn material is modeled as Kelvin element. A partial differential equation governing the transversal vibration is derived from Newtons second law. Galerkin method is used to truncate the governing nonlinear differential equation, and thus first-order ordinary differential equation is obtained. The periodic vibration equation and the natural frequency of moving yarn are received by applying homotopy perturbation method. As a result, the condition which should be avoided in the weaving process for resonance is obtained.

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