The matter collineations of plane symmetric spacetimes are studied according to the degenerate energy-momentum tensor. We have found many cases where the energy-momentum tensor is degenerate but the group of matter collineations is finite. Further we obtain different constraint equations on the energy-momentum tensor. Solving these constraints may provide some new exact solutions of Einstein field equations.

In this paper, we establish two new iterative methods of order four and five by using modified homotopy perturbation technique. We also present the convergence analysis of these iterative methods. To assess the validity and performance of these iterative methods, we have applied to solve some nonlinear problems.

This paper studies the reducibility of almost-periodic Hamiltonian systems with small perturbation near the equilibrium which is described by the following Hamiltonian system: It is proved that, under some non-resonant conditions, non-degeneracy conditions, the suitable hypothesis of analyticity and for the sufficiently small ε, the system can be reduced to a constant coefficients system with an equilibrium by means of an almost-periodic symplectic transformation.

<abstract><p>In the present paper, we focus on the reducibility of an almost-periodic linear Hamiltonian system</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \frac{dX}{dt} = J[A+\varepsilon Q(t)]X, X\in \mathbb{R}^{2d} , $\end{document} </tex-math></disp-formula></p> <p>where $ J $ is an anti-symmetric symplectic matrix, $ A $ is a symmetric matrix, $ Q(t) $ is an analytic almost-periodic matrix with respect to $ t $, and $ \varepsilon $ is a parameter which is sufficiently small. Using some non-resonant and non-degeneracy conditions, rapidly convergent methods prove that, for most sufficiently small $ \varepsilon $, the Hamiltonian system is reducible to a constant coefficients Hamiltonian system through an almost-periodic symplectic transformation with similar frequencies as $ Q(t) $. At the end, an application to Schrödinger equation is given.</p></abstract>

In this paper, we have modified fixed point method and have established two new iterative methods of order two and three. We have discussed their convergence analysis and comparison with some other existing iterative methods for solving nonlinear equations.