Browsing by Author "Nyonyi, Yusuf."

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Exact solutions for spherical relativistic models.
(2011) Nyonyi, Yusuf.; Govinder, Keshlan Sathasiva.; Maharaj, Sunil Dutt.
In this thesis we study relativistic models of gravitating uids with heat ow and electric charge. Firstly, we derive the model of a charged shear-free spherically symmetric cosmological model with heat ow. The solution of the Einstein-Maxwell equations of the system is governed by the pressure isotropy condition. This condition is a highly nonlinear partial di erential equation. We analyse this master equation using Lie's group theoretic approach. The Lie symmetry generators that leave the equation invariant are found. We provide exact solutions to the gravitational potentials using the rst symmetry admitted by the equation. Our new exact solutions contain the earlier results of Msomi et al (2011) without charge. Using the second symmetry we are able to reduce the order of the master equation to a rst order highly nonlinear di erential equation. Secondly, we study a shear-free spherically symmetric cosmological model with heat ow in higher dimensions. We establish the Einstein eld equations and nd the governing pressure isotropy condition. We use an algorithm due to Deng (1989) to provide several new classes of solutions to the model. The four-dimensional case is contained in our general result. Solutions due to Bergmann (1981), Maiti (1982), Modak (1984) and Sanyal and Ray (1984) for the four-dimensional case are regained. We also establish a new class of solutions that contains the results of Deng (1989) from four dimensions.
Group theoretic approach to heat conducting gravitating systems.
(2013) Nyonyi, Yusuf.; Maharaj, Sunil Dutt.; Govinder, Keshlan Sathasiva.
We study shear-free heat conducting spherically symmetric gravitating fluids defined in four and higher dimensional spacetimes. We analyse models that are both uncharged and charged via the pressure isotropy condition emanating from the Einstein field equations and the Einstein-Maxwell system respectively. Firstly, we consider the uncharged model defined in higher dimensions, and we use the algorithm due to Deng to generate new exact solutions. Three new metrics are identified which contain the results of four dimensions as special cases. We show graphically that the matter variables are well behaved and the speed of sound is causal. Secondly, we use Lie's group theoretic approach to study the condition of pressure isotropy of a charged relativistic model in four dimensions. The Lie symmetry generators that leave the equation invariant are found. We provide exact solutions to the gravitational potentials using the symmetries admitted by the equation. The new exact solutions contain earlier results without charge. We show that new charged solutions related to the Lie symmetries, that are generalizations of conformally at metrics, may be generated using the algorithm of Deng. Finally, we extend our study to find models of charged gravitating fluids defined in higher dimensional manifolds. The Lie symmetry generators related to the generalized pressure isotropy condition are found, and exact solutions to the gravitational potentials are generated. The new exact solutions contain earlier results obtained in four dimensions. Using particular Lie generators, we are able to provide forms for the gravitational potentials or reduce the order of the master equation to a first order nonlinear differential equation. Exact expressions for the temperature pro les, from the transport equation for both the causal and noncausal cases, in higher dimensions are obtained, generalizing previous results. In summary, the Deng algorithm and Lie analysis prove to be useful approaches in generating new models for gravitating fluids.