In this paper, a new analytical solution to the undamped Helmholtz oscillator equation in terms of the Weierstrass elliptic function is reported. The solution is given for any arbitrary initial conditions. A comparison between our new solution and the numerical approximate solution using the Range Kutta approach is performed. We think that the methodology employed here may be useful in the study of several nonlinear problems described by a differential equation of the form
z
″
=
F
z
in the sense that
z
=
z
t
. In this context, our solutions are applied to some physical applications such as the signal that can propagate in the LC series circuits. Also, these solutions were used to describe and investigate some oscillations in plasma physics such as oscillations in electronegative plasma with Maxwellian electrons and negative ions.
In this paper, we obtain the approximate value of 42.9815 arcsec/century for Mercury’s perihelion precession by solving both numerically and analytically the nonlinear ordinary differential equation derived from the geodesic equation in Einstein’s Theory of Relativity. We also compare our result with known results, and we illustrate graphically the way Mercury’s perihelion moves. The results we obtained are applicable to any body that moves around the Sun. We give predictions about the value of the Cosmological Constant. Simple algebraic formulas allow to estimate perihelion shifts with high accuracy.
We use Gamma Ray Bursts (GRBs) data from Y. Wang (2008) to put additional constraints on a set of cosmological dark energy models based on the holographic principle. GRBs are among the most complex and energetic astrophysical events known in the universe offering us the opportunity to obtain information from the history of cosmic expansion up to about redshift of z ∼ 6. These astrophysical objects provide us a complementary observational test to determine the nature of dark energy by complementing the information of data from Supernovas (e.g. Union 2.1 compilation). We found that the ΛCDM model gives the best fit to the observational data, although our statistical analysis (∆AIC and ∆BIC) shows that the models studied in this work ("Hubble Radius Scale" and "Ricci Scale Q") have a reasonable agreement with respect to the most successful, except for the "Ricci Scale CPL" and "Future Event Horizon" models, which can be ruled out by the present study. However, these results reflect the importance of GRBs measurements to provide additional observational constraints to alternative cosmological models, which are mandatory to clarify the way in which the paradigm of dark energy or any alternative model is correct.
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