The purpose of this thesis is to study the finite analytic method for the major classes of partial differential equations .parabolic and elliptic and show the accuracy of this method by comparison with the exact solution .In the parabolic' case, the 'linear Diffusion and Heat equations with constant coefficients are considered. For these problems, high accuracy and more efficient solutions are obtained . Also in the Heat equation it is found that this solution can be used to examine or to explain the instability behavior and the accuracy of various finite difference approximations. It is shown also that the solution for linear equations with constant coefficients can be used to solve linear equations with non constant coefficients and attractive accuracy is found here. In the elliptic case the results are the same as in the parabolic for both constant and non constant coefficients. Then the finite analytic method is used to solve nonlinear problems in connection with Predictor-Corrector technique for parabolic type and with Picard and Newton techniques for elliptic type equations. For the typical study we consider important examples such as Burger's equation, nonlinear Heat equation and nonlinear poisson equation. Favorable results appear from the comparison between the finite analytic and the exact solutions. After seeing the effectiveness of the finite analytic method , il is applied to the problem of flow of compressible gas in pipe lines network specifically tree like network. A new form of the governing model partial differential equations for fluid flow is presented {see [3]}. The solution process is divided into two alternating procedures, the first is a flow calculation and the other is a pressure calculation. The nonlinear equation for the flow calculation is solved by the finite analytic technique, and the pressure calculation is by explicit integration. These two procedures are undertaken iteratively until convergence is obtained. To shew the general procedure step by step, one pipe example is considered at first, then three pipes connected at a junction, then five pipes with two junctions and finally a network consisting of ten pipes are solved utilizing the above techniques.