Homotopy Perturbation Method for Solving Moving Boundary and Isoperimetric Problems

In this paper, homotopy perturbation method is applied to solve moving boundary and isoperimetric problems. This method does not depend upon a small parameter in the equation, homotopy is constructed with an imbedding parameter p, which is considered as a “small parameter”. Finally, we use combined homotopy perturbation method and Green’s function method for solving second order problems. Some examples are given to illustrate the effectiveness of methods. The results show that these methods provides a powerful mathematical tools for solving problems.


Introduction
In the modeling of a large class of problems which comes up in science, engineering and economics, it is necessary to minimize amounts of a certain functional.Because of the key role of this subject, it has been considerable attention has been devoted to these kinds of problems.Such problems are called variational problems (see [1,2]).
Consider the simplest form of a variational problem as: where v is the functional that its extremum must be achieved.There are two kinds of boundary conditions that functional v can be considered by.In the case of fixed boundary problems, the admissible function   y x must satisfy the boundary conditions In moving boundary problems at least one of the boundary points of the admissible function is movable along a boundary curve.As a matter of fact, many applications of the calculus of variations lead to problems in which not only boundary conditions, but also conditions of quite a different type, known as constraints are imposed on the admissible function.The necessary condition for the admissible solutions at this problems is to satisfy the Euler-Lagrange equation which is mainly consider as nonlinear.
Author of [13] solved variational problems with moving boundaries with Adomian decomposition method.
In [14] Homotopy perturbation method applied to solve variational problems with fixed boundaries.In this paper solution of variational problems with moving boundaries problems can be obtained by Homotopy perturbation method first.Then we obtain solution of them by using combined homotopy perturbation method and Green's function method.This algorithm is offered for the solution of second-order boundary value problems with two-point boundary conditions.To transform the ordinary differential equation into an equivalent integral one, which has already satisfied the boundary conditions, we apply the Green's function method first.Then, the homotopy perturbation method is used to the resulting equation to construct the numerical solution for such problems.To illustrate a clear overview of the procedure several illustrative examples are involved.

Moving Boundary Problems
The essential condition for the solution of problem (1) has been fulfilled the Euler-Lagrange equation d 0, d The general form of the variational problem (1) is Here the necessary condition for the extremum of the functional (4) is satisfying the system of second-order differential equations below d 0, 1, 2, , d In the fixed boundary problems, Euler-Lagrange equation must be considered by the boundary conditions, but for the problems with variable boundaries, Euler-Lagrange equation has to satisfy natural boundary conditions or transversality conditions that has been discussed in the following theorems.
Type 1: Firstly, we consider problems for which at least one of the boundary points move freely along a line parallel to the y-axis, actually at this point   y x is not specified.In this case all admissible functions have the same domain of definition   Type 2: Secondly, we ought to turn to the beginning and end points (or only one of them) that move freely on given curves In this case, we look for a function y x , which emanates at some and minimizes the functional (1).In this problem the points 0 1 , x x are unknown, they must satisfy the necessary conditions called transversality conditions, prescribed in the following theorem.
Theorem 2.2.If the function 1 which emanates at some 0 x x  from the curve and terminates for some on the curve yields a relative minimum for functional (1), where x y y space that contains all lineal elements of , x x and that at the point of departure and the point of arrival, the transversality conditions: are satisfied.In such a state that one of the points is fixed, then the transversality condition has to hold at the other point.One can consider transversality conditions for the problems with more than one unknown functions.For example, in the two dimensional case we seek a vector , , ,  v y x y y y y x in which and the endpoint lies on a two-dimensional surface that is given by Here the transversality conditions at x x  are:   In which , y x 0 1 y x is an admissible vector function.

Isoperimetric Problems
Assume that two functions and  , , x assumes a given value l, determine the one for which the functional x assumes an extremal value.We assume that F and G have continuous first and second partial derivatives for 0 1 x x x   y and for arbitrary values of the variables y and  .

Euler's theorem:
is not an extremal of the functional K, then there exists a constant λ, such that the curve y y x x  is an extremal of the functional The vital condition for the solution of this problem is to satisfy the Euler-Lagrange equation see [15] for further information.

Homotopy Perturbation Method
We consider the following nonlinear differential equation with natural boundary conditions or transversality conditions where A is a general differential operator, B is a boundary operator,   f r is a known analytic function and  is the boundary of the domain  .
The operator A can, generally, be divided into two parts L and N, where L is Linear, while N is nonlinear, so that we can write By Homotopy perturbation technique [3] and [4], we create a homotopy which satisfies Or where is an embeding parameter, and 0 is an initial approximation of Equation (13).Obviously from Equation ( 16): In this method, using the homotopy parameter p, we have the following power series in p Setting 1 p  results in the approximate solution of Equation ( 13) We know the natural boundary condition at Therefore, we have the following boundary conditions for (3.1.2): We can readily construct a homotopy which satisfies With initial approximation , Suppose that the solution of Equation (25) has the (20).

  1 t y t e  
Now we solve this problem with Adomian decomposition method.Using the operator form of (23), we have: Applying the Adomian decomposition: Now we use ( ) as the approximation of y(x), for example, for we have: n   120 720 5040 40320 We solve the equation , for the determination of A. Table 1 shows the error of 3 g .Solutions of two methods show that the homotopy perturbation method is better than adomian decomposition method.
Example 3.1.2.We want to find the minimum of the integral From the auxiliary function [15]: A homotopy can be constructed as follows Our initial approximation is:   0 y t at  b.Supposing the approximate solution of Equation ( 29) has the form of (20), by the same manipulation like above example, we have Table 1.Shows the error of g 3 .

Combined Homotopy Perturbation Method and Green's Function Method for Solving Second-Order Moving Boundary Problems
Assume that [16]  with boundary conditions that obtained from transversality conditions.Where  is a real constant number,

 
, f y y is a nonlinear function, and We can write Equation (32) to an equivalent integral equation; By adding a nonhomogeneous term, in moving boundary problems, the above equation is modified because boundary conditions are nonhomogeneous.Here, the function In Equation (33), if 0   , then Green's function amended as below: We construct the homotopy form of Equation (33), which satisfies: The solution of the Equation ( 32) is assumed to be the d power series of p as expressed in Equation (20) and substituting Equation (20) into (36) and equating the term with identical powers of p, The following equations can be obtained: We obtain the approximate solution from Equation (20) by solving these integral equation.If we construct the homotopy as: exa exa  and by imposing boundary condition on that, we have: x is the exact solution of (32), then we have:  and im- posing boundary condition on that, we have:   38) will give the exact solution. 6.

Numerical Examples
Also But from (37) we have: Also we have: That is approach to exact solution.And also we obtain: , , , v x r x r x r x has been shown by imposing boundary condition on that, we have:  

Conclusion
In this paper, we solve the moving boundary and isoperimetric problems by using Homotopy perturbation method.Embedding parameter   0,1 p  can be taken into account as a perturbation parameter.By the application of Green's function, the problem concerned is transformed into an equivalent integral equation, which is solved using the homotopy perturbation method.Numerical examples show that these proposed methods were valid and effective for solving problems.

0 1 , 2 . 1 .
x x and satisfy the Euler-Lagrange equation in this interval.Furthermore such functions must satisfy conditions called natural boundary conditions prescribed in the following theorem.Theorem Suppose the function , respectively, the following natural boundary conditions: changing process of p from zero to unity.In topology, this is named deformation, and      0 , L v L y A v f r  are called homotopic.
into (25), and equating the terms with the identical powers of p: