Analytical Solution to a Third-Order Rational Difference Equation.

Inspired by some open conjectures in a rational dynamical system by G. Ladas and Palladino, in this paper, we consider the problem of solving a third-order difference equation. We comment the conjecture by Ladas. A third-order rational difference equation is solved analytically. The solution is compared with the solution to the linearized equation. We show that the solution to the linearized equation is not good, in general. The methods employed here may be used to solve other rational difference equations. The period of the solution is calculated. We illustrate the accuracy of the obtained solutions in concrete examples.

Analytical Solution to the Generalized Complex Duffing Equation.

Future scientific and technological evolution in many areas of applied mathematics and modern physics will necessarily depend on dealing with complex systems. Such systems are complex in both their composition and behavior, namely, dealing with complex dynamical systems using different types of Duffing equations, such as real Duffing equations and complex Duffing equations. In this paper, we derive an analytical solution to a complex Duffing equation. We extend the Krýlov-Bogoliúbov-Mitropólsky method for solving a coupled system of nonlinear oscillators and apply it to solve a generalized form of a complex Duffing equation.

Perihelion Precessions of Inner Planets in Einstein's Theory and Predicted Values for the Cosmological Constant.

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.

Analytical Approximant to a Quadratically Damped Duffing Oscillator.

The Duffing oscillator of a system with strong quadratic damping is considered. We give an elementary approximate analytical solution to this oscillator in terms of exponential and trigonometric functions. We compare the analytical approximant with the Runge-Kutta numerical solution. We also solve the oscillator by menas of He's homotopy method and the famous Krylov-Bogoliubov-Mitropolsky method. The approximant allows estimating the points at which the solution crosses the horizontal axis.