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In this paper, the homoto pyanalysis method (HAM) is presented to solve some of engineering problems. The homotopy analysis method is applied in obtaining exact solutions for equations of the type Δ<sup>2</sup>=b(x,y) and Δ<sup>2</sup>u=b(x,y,u) on an elliptical domain. Exact solutions are presented for several examples involving to demon strate the applic ability and efficiency of HAM.

The homotopy analysis method is developed in 1992 by Liao [

We refer to the problem given by Partridge and Brebbia [

where

In all applications, the domain bounded by the ellipse given in

The equation of the ellipse is

We apply the HAM to equations of the type

where

where

To both sides of Equation (5), we get the following mth-order deformation equation

where

And

Note that

We apply Homotopy Analysis Method to equations of the type

Equation (1) suggests that we define an equation of linear operator as

And Equation (2) suggests that we define an equation of linear operator as

Using the above definitions, the zeroth-orderde formation equation is constructed as

Applying the homotopy-derivative to the zeroth-orderde formation equation, we obtain the following mth- orderde formation equations

Since

where

Example 1. Consider the equation of the type

With initial guess

using HAM, were cursively obtain

When

Example 2. Consider the equation of the type

with initial guess

using HAM, were cursively obtain

When

Example 3. Consider the equation of the type

with initial guess

using HAM, were cursively obtain

For

which is the exact solution.

Example 4. Consider the equation of the type

with initial guess

using HAM, were cursively obtain

For

which is the exact solution.

Example 5. Consider the equation of the type

with initial guess

using HAM, were cursively obtain

For

which is the exact solution.

In this paper, the homotopy analysis method has been applied to solve some of engineering problems defined on an elliptical domain. Exact solutions for equations of the type

Selcuk Yildirim, (2015) Homotopy Analysis Method for Equations of the Type ∇^{2}=b(x,y) and ∇^{2}u=b(x,y,u). Journal of Applied Mathematics and Physics,03,391-398. doi: 10.4236/jamp.2015.34049