Applying He's variational iteration method to FRW cosmology

This work is devoted to the investigation of Friedmann-Robertson-Walker (FRW) cosmological models with the help of the so-called Variational Iteration Method (VIM). For this end, we briefly recall the main equations of the cosmological models and the basic idea of VIM. In order to approbate the VIM in FRW cosmology and demonstrate the main steps in solving by this method, we consider the test example of the universe with dust for which the exact solution of the model is known. Then, a solution for the spatially flat FRW model of the universe filled with the dust and quintessence is obtained when the exact analytic solution cannot be found. A comparison of our solution with the corresponding numerical solution shows that it is of a high degree of accuracy. Moreover, the Dynamical System Analysis to the dynamics of the homogeneous and isotropic FRW universes is used as a special case of generalized Lotka–Volterra system where the competitive species are the barotropic fluids filling the Universe. With the help of VIM, we have found the iterative formulae for the density parameters of the cosmological analog of the generalized Lotka–Volterra set of equations. All solutions illustrated graphically by means of Maple software.

He's Variational Iteration Method

In this chapter, a variational iteration method (VIM) has been applied to nonlinear heat transfer equation. The concept of the variational iteration method is introduced briefly for applying this method for problem solving. The proposed iterative scheme finds the solution without any discretization, linearization, or restrictive assumptions. The results reveal that the VIM is very effective and convenient in predicting the solution of such problems.

Variational Iteration Method for Solving Riccati Matrix Differential Equations

<p>Riccati matrix differential equation has long been known to be so difficult to solve analytically and/or numerically. In this connection, most of the recent studies are concerned with the derivation of the necessary conditions that ensure the existence of the solution. Therefore, in this paper, He’s Variational iteration method is used to derive the general form of the iterative approximate sequence of solutions and then proved the convergence of the obtained sequence of approximate solutions to the exact solution. This proof is based on using the mathematical induction to derive a general formula for the upper bound proved to be converge to zero under certain conditions. </p>