Equations (19)-(24) are highly ordered and nonlinear, so we use different techniques to calculate the 3-curvature

perturbation equation. First, we will use [theta] + [psi] = [THETA] and find the solution of differential equation.

Here, [v.sub.t] = dF([bar.u])v is referred to as the linearization of (4) about [bar.u]; so, from this linearization, we obtain the linear

perturbation equation.

The

perturbation equation adopted in this study is the same as that used to perturb the global best particle solution in [17], where the global best particle is perturbed unconditionally.

which is a singular

perturbation equation since the highest-order derivative has been multiplied by a small parameter, where 0 < [member of] = 1/[[mu].sup.2] [much less than] 1.

Now, we use their results to write the following

perturbation equation corresponding to flat FRW universe which expands with acceleration.

Let us construct the homotopy

perturbation equationsThe zeroth-order

perturbation equations describe a steady, zero-eccentricity flow condition and are listed below.

In the next section, we present the governing equations of motion and derive the associated

perturbation equations. We introduce the linear and nonlinear analysis of our system in Section 3 and 4, respectively.

Abstract: I will describe in detail a package we developed recently that uses a fully geometrical method to derive

perturbation equations about a spatially homogeneous background.

Based on the expansion (10), the

perturbation equations could be obtained from (1)-(6) in the form:

The linearized

perturbation equations are conveniently solved by a two-dimensional Fourier transform in the xy-plane and a Laplace transform in time, which result in ordinary differential equations in z for the transformed velocity and pressure perturbations, which can be solved analytically.