### Abstract:

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.