New solutions for nonlinear perfect fluids.

Master of Science in Mathematics, Statistics and Computer Science. University of KwaZulu-Natal, Durban 2016.We investigate the Einstein system that governs the evolution of uncharged shear-free spherically symmetric fluids. First we present the Einstein equations for the static spherically symmetric gravitational fields in isotropic coordinates. Also the nonstatic spherically symmetric gravitational fields are studied. We have demonstrated that the fundamental differential equation governing the behaviour of the model is of the Emden-Fowler type. Such equations also arise in applications in Newtonian physics. The field equations governing the gravitational behaviour of the model are generated. We integrate the system of partial differential equations and apply a transformation that reduces the system to a second order ordinary differential equation. To solve the resulting ordinary differential equation we employ the method of characteristics to find different expressions for the gravitational potentials. We employ the method of characteristics to obtain first integrals for the Emden-Folwer type equation. To apply the method, we make use of the associated multipliers which are obtained via the Euler operator acting on the arbitrary multiplier and differential equation. These multipliers can be obtained under the various forms of the arbitrary function representing the gravitational potential under which the equation becomes integrable. Thus expanding the differential equation with the associated multiplier, we can find first integrals by solving the system of partial differential equations. The study is comprised of various forms of the multipliers associated to first integrals of the equation in question