Vacuum solutions of the gravitational field equations in the brane world model

We consider some classes of solutions of the static, spherically symmetric gravitational field equations in the vacuum in the brane world scenario, in which our Universe is a three-brane embedded in a higher dimensional space-time. The vacuum field equations on the brane are reduced to a system of two ordinary differential equations, which describe all the geometric properties of the vacuum as functions of the dark pressure and dark radiation terms (the projections of the Weyl curvature of the bulk, generating non-local brane stresses). Several classes of exact solutions of the vacuum gravitational field equations on the brane are derived. In the particular case of a vanishing dark pressure the integration of the field equations can be reduced to the integration of an Abel type equation. A perturbative procedure, based on the iterative solution of an integral equation, is also developed for this case. Brane vacuums with particular symmetries are investigated by using Lie group techniques. In the case of a static vacuum brane admitting a one-parameter group of conformal motions the exact solution of the field equations can be found, with the functional form of the dark radiation and pressure terms uniquely fixed by the symmetry. The requirement of the invariance of the field equations with respect to the quasi-homologous group of transformations also imposes a unique, linear proportionality relation between the dark energy and dark pressure. A homology theorem for the static, spherically symmetric gravitational field equations in the vacuum on the brane is also proven.Comment: 13 pages, no figures, to appear in PR

Implementation of variational iteration method for various types of linear and nonlinear partial differential equations

There are various linear and nonlinear one-dimensional partial differential equations that are the focus of this research. There are a large number of these equations that cannot be solved analytically or precisely. The evaluation of nonlinear partial differential equations, even if analytical solutions exist, may be problematic. Therefore, it may be necessary to use approximate analytical methodologies to solve these issues. As a result, a more effective and accurate approach must be investigated and analyzed. It is shown in this study that the Lagrange multiplier may be used to get an ideal value for parameters in a functional form and then used to construct an iterative series solution. Linear and nonlinear partial differential equations may both be solved using the variational iteration method (VIM) method, thanks to its high computing power and high efficiency. Decoding and analyzing possible Korteweg-De-Vries, Benjamin, and Airy equations demonstrates the method’s ability. With just a few iterations, the produced findings are very effective, precise, and convergent to the exact answer. As a result, solving nonlinear equations using VIM is regarded as a viable option

A Novel Third Order Numerical Method for Solving Volterra Integro-Differential Equations

In this paper we introduce a numerical method for solving nonlinear Volterra integro-differential equations. In the first step, we apply implicit trapezium rule to discretize the integral in given equation. Further, the Daftardar-Gejji and Jafari technique (DJM) is used to find the unknown term on the right side. We derive existence-uniqueness theorem for such equations by using Lipschitz condition. We further present the error, convergence, stability and bifurcation analysis of the proposed method. We solve various types of equations using this method and compare the error with other numerical methods. It is observed that our method is more efficient than other numerical methods

Approximations for many-body Green's functions: insights from the fundamental equations

Several widely used methods for the calculation of band structures and photo emission spectra, such as the GW approximation, rely on Many-Body Perturbation Theory. They can be obtained by iterating a set of functional differential equations relating the one-particle Green's function to its functional derivative with respect to an external perturbing potential. In the present work we apply a linear response expansion in order to obtain insights in various approximations for Green's functions calculations. The expansion leads to an effective screening, while keeping the effects of the interaction to all orders. In order to study various aspects of the resulting equations we discretize them, and retain only one point in space, spin, and time for all variables. Within this one-point model we obtain an explicit solution for the Green's function, which allows us to explore the structure of the general family of solutions, and to determine the specific solution that corresponds to the physical one. Moreover we analyze the performances of established approaches like $GW$ over the whole range of interaction strength, and we explore alternative approximations. Finally we link certain approximations for the exact solution to the corresponding manipulations for the differential equation which produce them. This link is crucial in view of a generalization of our findings to the real (multidimensional functional) case where only the differential equation is known.Comment: 17 pages, 7 figure