Variational Iterative Method Applied to Variational Problems with Moving Boundaries

In this paper, He’s variational iterative method has been applied to give exact solution of the Euler Lagrange equation which arises from the variational problems with moving boundaries and isoperimetric problems. In this method, general Lagrange multipliers are introduced to construct correction functional for the variational problems. The initial approximations can be freely chosen with possible unknown constant, which can be determined by imposing the boundary conditions. Illustrative examples have been presented to demonstrate the efficiency and applicability of the variational iterative method.


Introduction
In modeling a large class of problems arising in science, engineering and economics, it is necessary to minimize amounts of a certain functional.Because of the important role of this subject, special attention has been given to these problems.Such problems are called variational problems, see [1,2].
The simplest form of a variational problem can be considered as where is the functional which its extremum must be found.Functional can be considered by two kinds of boundary conditions.In the fixed boundary problems, the admissible function   y x must satisfy following boundary conditions In moving boundary problems, at least one of the boundary points of the admissible function is movable along a boundary curve.Further more many applications of the calculus of variations lead to problems in which not only boundary conditions, but also a quite different type of conditions known as constraints, are imposed on the admissible function.The necessary condition for the admissible solutions of such problems has to satisfy the Euler-Lagrange equation which is generally nonlinear.
In this work we consider He's variational iterative method as a well known method for finding both analytic and approximate solutions of differential equations.Here, the problem is initially approximated with possible unknowns.Then a correction functional is constructed by a general Lagrange multiplier, which can be identified optimally via the variational theory [3].
Variational iterative method is applied on various kinds of problems .
Author of [32] solved variational problems with moving boundaries with Adomian decomposition method.Variational iterative method was applied to solve variational problems with fixed boundaries (see [11,27,30]).In this work we obtain exact solution of variational problems with moving boundaries and isoperimetric problems by variational iterative method.In fact, variational iterative method is applied to solve the Euler-Lagrange equation with prescribed boundary conditions.To present a clear overview of the procedure several illustrative examples are included.

Variational Iterative Method
In variational iterative method which is stated by He [3], solutions of the problems are approximated by a set of functions that may include possible constants to be determined from the boundary conditions.In this method the problem is considered as where is a linear operator, and is a nonlinear operator.
tional is taken into account where  is Lagrange multiplier [5], the subscript n denotes the n-th approximati n y  is as a restricted variation 0 n on, i.e. y   [6][7][8].Taking the variation from both sides of the correct functional with respect to y n and imposin 1 0 n , the stationary conditions are obtained.By using the stationary conditions the optimal valu of the  can be identified.


The successive approximation can be established by determining a general lagrangian multiplier  and initial solution 0 .Since this procedure avoids the discretization of the problem, it is possible to find the closed form solution without any round off error.

y
In the case of m equations, the equations are rewritten in the form of: where i is a linear with respect to i , and i is nonlinear part of the ith equation.In this case the correct functionals are produced as and the optimal values of the , 1, , i i    are obtained by taking the variation from both sides of the correct functionals and finding stationary conditions using

Moving Boundary Problems
The necessary condition for the solution of problem (1) is to satisfy the Euler-Lagrange equation , d 0, d The general form of the variational problem (1) is , , , , , , , , , , , d , Here, the necessary condition for the extermum of the functional (8) is to satisfy the following system of second-order differential equations d 0, 1, 2, , d must be considered by the boundary conditions, but for ith variable boundaries, we have tw As the first case, those problems are consider (9) In fixed boundary problems, Euler-Lagrange equation the problems with variable boundaries, Euler-Lagrange equation must satisfy natural boundary conditions or transversality conditions which will be described in the following theorems.
For the problems w o cases: Type 1: ed for which at least one of the boundary points move freely along a line parallel to the y-axis.Indeed at this point   or Type 2: For the second case, the beginning and end points (or only one of them) can move freely on given curves   and es for some Copyright © 2012 SciRes.
In the case that one of the points is fixed, then the transversality condition has to be held at the oth One can consider transversality conditions for lems with more than one unknown functions.For exampl hich er point.the probe, in to minimize two dimensional case, a vector func- , , , , ', ' d, In which is an admissible vector function.
For further information on transversality con specially for the proofs of Theorems 3.1 and 3.2 and conditions (15), (16), see [2].
1. Consider the following functional: The corresponding Euler-Lagrange equation is: Now natural boundary conditi is as following: itio Therefore, the following boundary cond ns are: By using variational iterative method we consid he fo sides of the correct functional with respect to given: llowing functional is considered: Taking the variation from both For all variations The following stationary conditions are obtained: . Therefore iterative formula can be found as: By imposing ( 18) yields the exact solutions of the problem (see Figure 1).Example 3.2.We want to find the shortest distance fr to the sphe om the point ( This problem is reduced to optimize the following functional:  are: , . 1 1 In above equations "e" and "f" are constant, so they can be rewritten as: The transversality conditions are: The variation from both sides of above equations for finding the optimal value of  is: which yields: So that the following iterative formulas are obtained: By using variational iteration method results: 20) and ( 21) lead to, 0, .
which is the exact solution.

Isoperimetric Problems
Assume that two functions , , G x y y and   , , F x y y ar g all curves e given.Amon x assumes a given value l, determine the one for which the functional ose that F and G have partial derivatives for Gives an extermal value.Supp continuous first and second The neces e solution of this prob- with given boundary conditions in which H F G    for further information (see [2]).ple 3.

Exam
It is aimed to find the minimum of the functional Such that and With exact solution   herefore, the stationary conditions are obtained in the following form: and the desired sequence is   and it is known that in this case imposing (24) on the Euler Lagrange equation yields Hence: . But y must be extremal when , therefore: As it is observed that this so xa solution (see Figure 2).
Example 3.4.The objective is to find an extremum of the functional Such that and With exact solution   2 7 5 2 x x x   ,   z x x  , see [33].By having the following auxiliary functional: The system of Euler-Lagrange equations is in the form: By using Homotopy variational iterative method gives:   which is the exact solution (see Figure 3).

Conclusion
The He's variational iterative method is an efficient method for solving various kinds of problems.In this paper variational iterative method is employed for finding the minimum of a functional with moving boundaries and isoperimetric problems.Using He's variational iterative method the solution of the problem is provided in a closed form.Since this method does not need to the discretize of the variables, there is no computational round off error.Moreover, only a few numbers of iterations are needed to obtain a satisfactory result.
at the point of exit and the point of entrance, llowing transversality conditions to be satisfied: the fo

Figure 1 .
Figure 1.The graphs of approximated and exact solution for Example 3.1.
 is not an extremal of the functional K, there exists a constant  such that the curve   y y x  is an al of the functional e sary condition for th lem is to satisfy the Euler-Lagrange equation xtrem

Figure 2 .
Figure 2. The graphs of approximated and exact solution for Example 3.3.

Figure 3 .
Figure 3.The graphs of approximated and exact solution for Example 3.4.