The numerical solutions of the Euler?s system in a class of discontinuous functions

Euler?s system of differential equations is one of the basic systems of hydrodynamics which models many important practical problems. As it is known that the solution of the Euler?s system even in one dimensional case has the points of discontinuities whose locations are unknown beforehand. The function of this type can be expressed by the concept of a weak solution.The necessary theoretical background for further investigations is given in the first section.In the second and third sections, the auxiliary problems having some advantages over the main problem for both two and three dimensional Euler?s system are introduced.It is also proved that the suggested auxiliary problems are the first Bernoulli and Cauchy integrals of the two and three dimensional Euler?s system of equations, respectively.Finally, the boundary layer equations which are the special cases of the Euler?s system of equations are investigated. Using the Von Mises transform, the boundary layer equations are reduced to the quasi linear parabolic equation in the coordinates. Moreover, the exact solution of this problem is obtained, and the differentiable properties of this solution are studied.