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 , 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 equations
The 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.
Following Ujevic and Letelier (2004), and for the sake of self-consistency of the current paper, in what follows we present the derivation (of the first-order perturbation equations
for relativistic thin disks.
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.