Tracking of moving front for one dimensional scalar conservation law

In this thesis, the exact solution of the equation describing the motion of a two-phase fluid in a porous medium is obtained, and its properties are investigated. To this end, in the first section, the equations describing the motion of compressible fluids and two-phase mixtures in a porous medium are derived. In the second section, a new method to find the exact solution of the initial value problem of a nonlinear partial differential equation with continuous and piecewise continuous initial function is considered, and the differentiability properties of the solution are investigated. Finally, in the third chapter, the method discussed in the second chapter is applied for the theoretical investigation of the motion of a two-phase mixture. Additionally, a model has been developed for a laboratory experiment, and the waterless and dry periods in the filtration process of oil have been investigated.