Dissertation (MSc Mathematics)In this thesis we shall be dealing with some basic known approximated solution of model problems involving Navier-stoke equations for incompressible fluid as well as linear elasticity theory.these include well known poiseuilleflow,couetteflow,flow between concentric cylinders,boundary layer flow over impulsively stated plate,simply supported beam with uniform distributed load,cantilevered beam with uniformly distributed load,linear membrane problems, where also an integro-differential equation for the MAC solution will be introduced. To obtain the MAC solution for the 2D Laplace equation the conformal mapping will be used [9]. The invariant integral was used in [8] to introduce the MAC solution for the Dirichlet problem. The method of cones is used in this paper to obtain the MAC model for the linear elasticity equations. The integro-differential equations for the MAC model of elasticity are introduced using the principle of superposition. A rectangular membrane with fixed boundary conditions under applied a transversal force has the solution with singularity. That is the Green’s function of this problem. A number of journals and problems concerning the membrane theories are presented in references [1] - [12]. We can conclude that the membrane problem is important and it is under consideration of many research groups.The MAC model of the membrane will be obtained, which solution is called the MAC solution. If the classical equation of the membrane under small deformation is a wave equation then the MAC equation is an integro-differential equation. The conformal mapping is used to create the MAC Green’s function and the method of superposition is applied to create the MAC model.