scholarly journals 1 + 3 covariant perturbations in power-law f(R) gravity

2021 ◽  
Vol 81 (4) ◽  
Author(s):  
Beatrice Murorunkwere ◽  
Joseph Ntahompagaze ◽  
Edward Jurua
Keyword(s):  
Differential Equations ◽  
Evolution Equations ◽  
Numerical Solutions ◽  
Background Solution ◽  
Scalar Tensor Theory ◽  
Covariant Approach ◽  
Tensor Theory ◽  
Harmonic Decomposition ◽  
Density Perturbations ◽  
Perturbation Equations

AbstractWe applied the 1+3 covariant approach around the Friedmann–Lemaître–Robertson–Walker (FLRW) background, together with the equivalence between f(R) gravity and scalar-tensor theory to study cosmological perturbations. We defined the gradient variables in the 1 + 3 covariant approach which we used to derive a set of evolution equations. Harmonic decomposition was applied to partial differential equations to obtain ordinary differential equations used to analyse the behavior of the perturbation quantities. We focused on dust dominated area and the perturbation equations were applied to background solution of $$\alpha R+\beta R^{n}$$ α R + β R n model, n being a positive constant. The transformation of the perturbation equations into redshift dependence was done. After numerical solutions, it was found that the evolution of energy-density perturbations in a dust-dominated universe for different values of n decays with increasing redshift.

scholarly journals Controllability for Fuzzy Fractional Evolution Equations in Credibility Space

2021 ◽  
Vol 5 (3) ◽  
pp. 112
Author(s):  
Azmat Ullah Khan Niazi ◽  
Naveed Iqbal ◽  
Rasool Shah ◽  
Fongchan Wannalookkhee ◽  
Kamsing Nonlaopon
Keyword(s):  
Differential Equations ◽  
Evolution Equations ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Exact Controllability ◽  
Fractional Evolution Equations ◽  
Liu Process ◽  
Bounded Space ◽  
Fractional Differential ◽  
Derivatives Of

This article addresses exact controllability for Caputo fuzzy fractional evolution equations in the credibility space from the perspective of the Liu process. The class or problems considered here are Caputo fuzzy differential equations with Caputo derivatives of order β∈(1,2), 0CDtβu(t,ζ)=Au(t,ζ)+f(t,u(t,ζ))dCt+Bx(t)Cx(t)dt with initial conditions u(0)=u0,u′(0)=u1, where u(t,ζ) takes values from U(⊂EN),V(⊂EN) is the other bounded space, and EN represents the set of all upper semi-continuously convex fuzzy numbers on R. In addition, several numerical solutions have been provided to verify the correctness and effectiveness of the main result. Finally, an example is given, which expresses the fuzzy fractional differential equations.

A Multi-Domain Bivariate Pseudospectral Method for Evolution Equations

2017 ◽  
Vol 14 (04) ◽  
pp. 1750041 ◽  
Author(s):  
V. M. Magagula ◽  
S. S. Motsa ◽  
P. Sibanda
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Evolution Equations ◽  
Numerical Solutions ◽  
Nonlinear Evolution ◽  
Kdv Equation ◽  
Pseudospectral Method ◽  
Analytic Solutions ◽  
Partial Differential ◽  
Various Boundary Conditions

In this paper, we present a new general approach for solving nonlinear evolution partial differential equations. The novelty of the approach is in the combination of spectral collocation and Lagrange interpolation polynomials with Legendre–Gauss–Lobatto grid points to descritize and solve equations in piece-wise defined intervals. The method is used to solve several nonlinear evolution partial differential equations, namely, the modified KdV–Burgers equation, modified KdV equation, Fisher’s equation, Burgers–Fisher equation, Burgers–Huxley equation and the Fitzhugh–Nagumo equation. The results are compared with known analytic solutions to confirm accuracy, convergence and to get a general understanding of the performance of the method. In all the numerical experiments, we report a high degree of accuracy of the numerical solutions. Strategies for implementing various boundary conditions are discussed.

scholarly journals On multifluid perturbations in scalar–tensor cosmology

2020 ◽  
pp. 2050120
Author(s):  
Joseph Ntahompagaze ◽  
Shambel Sahlu ◽  
Amare Abebe ◽  
Manasse R. Mbonye
Keyword(s):  
Scalar Field ◽  
Energy Density ◽  
Power Law ◽  
Scalar Tensor Theory ◽  
Frw Cosmology ◽  
Static Approximation ◽  
Covariant Approach ◽  
Tensor Theory ◽  
Text Model ◽  
Friedmann Robertson Walker

In this paper, the scalar–tensor theory is applied to the study of perturbations in a multifluid universe, using the [Formula: see text] covariant approach. Both scalar and harmonic decompositions are instituted on the perturbation equations. In particular, as an application, we study perturbations on a background Friedmann-Robertson-Walker (FRW) cosmology consisting of both radiation and dust in the presence of a scalar field. We consider both radiation-dominated and dust-dominated epochs, respectively, and study the results. During the analysis, quasi-static approximation is instituted. It is observed that the fluctuations of the energy density decrease with increasing redshift, for different values of [Formula: see text] of a power-law [Formula: see text] model.

scholarly journals Formation and Clustering of Primordial Black Holes in Brans-Dicke Theory

Universe ◽  
2020 ◽  
Vol 6 (10) ◽  
pp. 158
Author(s):  
Victor Berezin ◽  
Vyacheslav Dokuchaev ◽  
Yury Eroshenko ◽  
Alexey Smirnov
Keyword(s):  
Black Holes ◽  
Scalar Field ◽  
Gravitational Wave ◽  
Significant Contribution ◽  
Threshold Value ◽  
Primordial Black Holes ◽  
Cosmological Horizon ◽  
Scalar Tensor Theory ◽  
Tensor Theory ◽  
Density Perturbations

The formation of primordial black holes in the early universe in the Brans-Dicke scalar-tensor theory of gravity is investigated. Corrections to the threshold value of density perturbations are found. Above the threshold, the gravitational collapse occurs after the cosmological horizon crossing. The corrections depend in a certain way on the evolving scalar field. They affect the probability of primordial black holes formation, and can lead to their clustering at large scales if the scalar field is inhomogeneous. The formation of the clusters, in turn, increases the probability of black holes merge and the corresponding rate of gravitational wave bursts. The clusters can provide a significant contribution to the LIGO/Virgo gravitational wave events, if part of the observed events are associated with primordial black holes.

scholarly journals Gravitational memory effects and Bondi-Metzner-Sachs symmetries in scalar-tensor theories

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Shaoqi Hou ◽  
Zong-Hong Zhu
Keyword(s):  
General Relativity ◽  
Angular Momentum ◽  
Evolution Equations ◽  
Equations Of Motion ◽  
Memory Effects ◽  
Scalar Tensor Theory ◽  
Null Infinity ◽  
Infinite Dimensional ◽  
Tensor Theory ◽  
Bondi Mass

Abstract The relation between gravitational memory effects and Bondi-Metzner-Sachs symmetries of the asymptotically flat spacetimes is studied in the scalar-tensor theory. For this purpose, the solutions to the equations of motion near the future null infinity are obtained in the generalized Bondi-Sachs coordinates with a suitable determinant condition. It turns out that the Bondi-Metzner-Sachs group is also a semi-direct product of an infinite dimensional supertranslation group and the Lorentz group as in general relativity. There are also degenerate vacua in both the tensor and the scalar sectors in the scalar-tensor theory. The supertranslation relates the vacua in the tensor sector, while in the scalar sector, it is the Lorentz transformation that transforms the vacua to each other. So there are the tensor memory effects similar to the ones in general relativity, and the scalar memory effect, which is new. The evolution equations for the Bondi mass and angular momentum aspects suggest that the null energy fluxes and the angular momentum fluxes across the null infinity induce the transition among the vacua in the tensor and the scalar sectors, respectively.

scholarly journals A study of perturbations in scalar–tensor theory using 1 + 3 covariant approach

2018 ◽  
Vol 27 (03) ◽  
pp. 1850033 ◽  
Author(s):  
Joseph Ntahompagaze ◽  
Amare Abebe ◽  
Manasse Mbonye
Keyword(s):  
Energy Density ◽  
Radiation Energy ◽  
Linear Perturbation ◽  
Scalar Tensor Theory ◽  
Radiation Energy Density ◽  
Tensor Theory ◽  
Density Perturbations ◽  
Theories Of Gravity ◽  
Theory Of Gravitation ◽  
Second Order Equations

This work discusses scalar–tensor theories of gravity, with a focus on the Brans–Dicke sub-class, and one that also takes note of the latter’s equivalence with [Formula: see text] gravitation theories. A [Formula: see text] covariant formalism is used in this case to discuss covariant perturbations on a background Friedmann–Laimaître–Robertson–Walker (FLRW) spacetime. Linear perturbation equations are developed based on gauge-invariant gradient variables. Both scalar and harmonic decompositions are applied to obtain second-order equations. These equations can then be used for further analysis of the behavior of the perturbation quantities in such a scalar–tensor theory of gravitation. Energy density perturbations are studied for two systems, namely for a scalar fluid-radiation system and for a scalar fluid-dust system, for [Formula: see text] models. For the matter-dominated era, it is shown that the dust energy density perturbations grow exponentially, a result which agrees with those already existing in the literatures. In the radiation-dominated era, it is found that the behavior of the radiation energy–density perturbations is oscillatory, with growing amplitudes for [Formula: see text], and with decaying amplitudes for [Formula: see text]. This is a new result.

Normal form method in search for periodic solutions of ordinary differential equations. Case of the fourth order

2018 ◽  
pp. 24-34
Author(s):  
V. F. Edneral ◽  
O. D. Timofeevskaya
Keyword(s):  
Normal Form ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Solutions ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Numerical Calculations ◽  
Computational Time ◽  
Resonant Normal Form ◽  
Normal Form Method

Introduction:The method of resonant normal form is based on reducing a system of nonlinear ordinary differential equations to a simpler form, easier to explore. Moreover, for a number of autonomous nonlinear problems, it is possible to obtain explicit formulas which approximate numerical calculations of families of their periodic solutions. Replacing numerical calculations with their precalculated formulas leads to significant savings in computational time. Similar calculations were made earlier, but their accuracy was insufficient, and their complexity was very high.Purpose:Application of the resonant normal form method and a software package developed for these purposes to fourth-order systems in order to increase the calculation speed.Results:It has been shown that with the help of a single algorithm it is possible to study equations of high orders (4th and higher). Comparing the tabulation of the obtained formulas with the numerical solutions of the corresponding equations shows good quantitative agreement. Moreover, the speed of calculation by prepared approximating formulas is orders of magnitude greater than the numerical calculation speed. The obtained approximations can also be successfully applied to unstable solutions. For example, in the Henon — Heyles system, periodic solutions are surrounded by chaotic solutions and, when numerically integrated, the algorithms are often unstable on them.Practical relevance:The developed approach can be used in the simulation of physical and biological systems.

Existence results of mild solutions for impulsive fractional integro-differential evolution equations with infinite delay

2014 ◽  
Vol 17 (4) ◽  
Author(s):  
Shengli Xie
Keyword(s):  
Differential Equations ◽  
Differential Evolution ◽  
Existence Theorem ◽  
Banach Spaces ◽  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Infinite Delay ◽  
Mild Solutions ◽  
Existence Results ◽  
Order Case

AbstractIn this paper we prove the existence and uniqueness of mild solutions for impulsive fractional integro-differential evolution equations with infinite delay in Banach spaces. We generalize the existence theorem for integer order differential equations to the fractional order case. The results obtained here improve and generalize many known results.

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