Weak solutions of the Cauchy problem for the first order quasi linear differantial equations

In this thesis the general theory for obtaining weak solutions of the Cauchy problem for the first order quasi linear differential equations is investigated. In the first part, the exact solution of the Cauchy problem for one dimensional quasi linear equations with convex state function is developed. The entropy condition has been studied.Using the obtained results the initial value problem of one dimensional differential equation with non convex state function has also been found.Finally, a new method for finding the exact solution of the Cauchy problem for 2- Dimentsional equation is suggested.For this aim, the auxiliary problem having some advantages over the main problem is introduced. Using these advantages of the suggested auxiliary problem the a algorithm for obtaining the numerical solution is constructed.