الفهرس 
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Abstract
Most important physical processes in nature are governed by partial differential equation (PDEs) The analytic solutions for some physical problems are not known. So the numerical methods are used to get an approximate solutions.
In this thesis, we examine in detail various numerical schemes that can be used, to solve the different partial differential equations (elliptic, parabolic, and hyperbolic partial differential equations). Also we introduce some applications for each scheme.
Construction of the grid points is one of the most important steps that are ’specified before starting the numerical solution of partial differential ’equations. In most cases, the domain of solution is irregular and hence the mesh is no uniform. In order to simplify the domain of solution and have uniform grids, the physical domain is transformed into a rectangular domain. The Original partial differential equations are also’ transformed to the computational. Domain and then solved on uniform grids.
Adaptive grid generation is one of the most efficient techniques to find the numerical solution of partial differential equations. The adaptive techniques are used in a construction of finer grids in the. Regions of large gradient.
In this thesis; we study one of the adaptive techniques, which is known as Vinokur distribution and apply this technique iii solving some boundary value, problems. Also we apply, that adaptive technique in solving the unsteady incompressible Navier Stokes equations in the vorticity stream. function formulation for the flow inside a cavity. Also we study’ the unsteady incompressible Navier Stokes equations in the vorticity stream function formulation for the flom, inside contraction geometry.