New models in general relativity and Einstein-Gauss-Bonnet gravity.

Masters Degree. University of KwaZulu-Natal, Durban.We generate the Einstein-Gauss-Bonnet field equations in five dimensions for a spherically symmetric static spacetime. The matter distributions considered are both neutral and charged. The introduction of a coordinate transformation brings the condition of isotropic pressure to a single master ordinary differential equation that is an Abel equation of the second kind. We demonstrate that the master equation can be reduced to a first order nonlinear canonical differential equation. Firstly, we consider uncharged gravitating matter. Several new classes of exact solutions are found in explicit and implicit forms. One of the potentials is determined completely. The second potential satisfies a constraint equation. Secondly, we study charged gravitating matter with Maxwell’s equations. We find new classes of exact charged solutions in explicit and implicit forms using two approaches. In the first approach, we can find new exact models without integration. In the second approach the Abelian pressure isotropy equation has to be integrated, which we demonstrate is possible in a number of cases. The inclusion of the electromagnetic field provides an extra degree of freedom that leads to viable exact solutions. An interesting feature characterising the new models is that a general relativity limit does not exist. Our new solutions exist only in Einstein-Gauss-Bonnet gravity. In addition, we have considered the dynamics of a shear-free fluid in Einstein gravity in higher dimensions with nonvanishing heat flux in a spherically symmetric manifold. This endeavour generates new exact models, being a generalisation of models developed in earlier treatments

A Chiellini type integrability condition for the generalized first kind Abel differential equation

The Chiellini integrability condition of the first order first kind Abel equation $dy/dx=f(x)y^2+g(x)y^3$ is extended to the case of the general Abel equation of the form $dy/dx=a(x)+b(x)y+f(x)y^{\alpha -1}+g(x)y^{\alpha}$, where $\alpha \in \Re$, and $\alpha > 1$. In the case $\alpha =2$ the generalized Abel equations reduces to a Riccati type equation, for which a Chiellini type integrability condition is obtained.Comment: 4 pages, no figure

Integrable Abel equations and Vein's Abel equation

We first reformulate and expand with several novel findings some of the basic results in the integrability of Abel equations. Next, these results are applied to Vein's Abel equation whose solutions are expressed in terms of the third order hyperbolic functions and a phase space analysis of the corresponding nonlinear oscillator is also providedComment: 12 pages, 4 figures, 17 references, online at Math. Meth. Appl. Sci. since 7/28/2015, published 4/201

Exact Solutions of a (2+1)-Dimensional Nonlinear Klein-Gordon Equation

The purpose of this paper is to present a class of particular solutions of a C(2,1) conformally invariant nonlinear Klein-Gordon equation by symmetry reduction. Using the subgroups of similitude group reduced ordinary differential equations of second order and their solutions by a singularity analysis are classified. In particular, it has been shown that whenever they have the Painlev\'e property, they can be transformed to standard forms by Moebius transformations of dependent variable and arbitrary smooth transformations of independent variable whose solutions, depending on the values of parameters, are expressible in terms of either elementary functions or Jacobi elliptic functions.Comment: 16 pages, no figures, revised versio