الفهرس 
IntroductionOnly 14 pages are availabe for public view
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Abstract
The present study dealt with the hyperbolic system of conservation laws . We have two famous example of these systems, the Burger’s equation and the ultra-relativistic Euler equations for an ideal gas, which is a system of nonlinear hyperbolic conservation laws. These equations are described in terms of the pressure p, the spatial part u R3of the dimensional four-velocity . We analyze the single shocks and rarefaction waves and give the solution of the Riemann problem in a constructive way. The contents and results of this thesis are organized as follows: 1- Give an introduction about the hyperbolic systems of conservation laws and their types and methods of their derivation, and some examples of the conservation laws. 2- Consider the ultra-relativistic Euler equations. These equations are written in differential form as well as in a weak integral form. 3-An entropy inequality is given in weak integral form with an entropy function which satisfies Gibbs equation. 4- The Rankine-Hugoniot jump conditions and the entropy inequality were used in order to derive a simple parameter representation for the admissible shocks. 5- which gives a very simple characterization for the Lax entropy conditions of single shock waves. 6- We also present other known parameterizations for single shocks and rarefaction waves. 7- We use these shock and rarefaction parameterizations in order to derive an exact Riemann solution for the one-dimensional ultra-relativistic Euler equations. 8- We present another system of hyperbolic systems of conservation laws, which is equivalent to the ultra- relativistic Euler equations. This equivalent system describes a phonon-Bose gas in terms of the energy density e and the heat flux Q. 9-We also compute the Riemann invariants for the ultra-relativistic Euler equations. 10-Finally, we give an interesting figure to show the equivalent between three deferent systems the ultra-relativistic Euler equations (p, u), the Riemann invariants (z,w) and a phonon-Bose gas (e,Q).